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• 101.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Some applications of representation theory in homogeneous dynamics and automorphic functions2018Doctoral thesis, comprehensive summary (Other academic)

This thesis consists of an introduction and five papers in the general area of dynamics and functions on homogeneous spaces. A common feature is that representation theory plays a key role in all articles.

Papers I-IV are concerned with the effective equidistribution of translates of pieces of subgroup orbits in quotient spaces of semisimple Lie groups by discrete subgroups. In Paper I we focus on finite-volume quotients of SL(2,C) and study the speed of equdistribution for expanding translates orbits of horospherical subgroups. Paper II also studies the effective equidistribution of translates of horospherical orbits, though now in the setting of a quotient of a general semisimple Lie group by a lattice subgroup. Like Paper II, Paper III considers effective equidistribution in quotients of general semisimple Lie groups, but now studies translates of orbits of symmetric subgroups. In all these papers we show that the translates equidistribute with the same exponential rate as for the decay of the corresponding matrix coefficients of the translating subgroup. In Paper IV we consider the effective equidistribution of translates of pieces of horospheres in infinite-volume quotients of groups SO(n,1) by geometrically finite subgroups, and improve the dependency on the spectral gap for certain known effective equidistribution results.

In Paper V we study the Fourier coefficients of Eisenstein series for generic non-cocompact cofinite Fuchsian groups. We use Zagier's renormalization of certain divergent integrals to enable use of the so-called triple product method, and then combine this with the analytic continuation of irreducible representations of SL(2,R) due to Bernstein and Reznikov.

1. On the rate of equidistribution of expanding translates of horospheres in finite-volume quotients of SL(2,C)
Open this publication in new window or tab >>On the rate of equidistribution of expanding translates of horospheres in finite-volume quotients of SL(2,C)
2017 (English)In: Journal of Modern Dynamics, ISSN 1930-5311, E-ISSN 1930-532X, Vol. 11, p. 155-188Article in journal (Refereed) Published
##### Abstract [en]

Let Gamma be a lattice in G = SL(2, C). We give an effective equidistribution result with precise error terms for expanding translates of pieces of horospherical orbits in Gamma\G. Our method of proof relies on the theory of unitary representations.

##### Keywords
Effective equidistribution, translates, horospheres
##### National Category
Mathematical Analysis
##### Identifiers
urn:nbn:se:uu:diva-315688 (URN)10.3934/jmd.2017008 (DOI)000396538600008 ()
##### Funder
Swedish Research Council, 621-2011-3629Göran Gustafsson Foundation for promotion of scientific research at Uppala University and Royal Institute of Technology Available from: 2017-02-18 Created: 2017-02-18 Last updated: 2018-04-11Bibliographically approved
2. On the rate of equidistribution of expanding translates of horospheres in Γ\G
Open this publication in new window or tab >>On the rate of equidistribution of expanding translates of horospheres in Γ\G
(English)Manuscript (preprint) (Other academic)
##### Abstract [en]

Let G be a semisimple Lie group and Γ a lattice in G. We generalize a method of Burger to prove precise effective equidistribution results for translates of pieces of horospheres in the homogeneous space Γ\G.

##### National Category
Mathematical Analysis
##### Identifiers
urn:nbn:se:uu:diva-347856 (URN)
Available from: 2018-04-09 Created: 2018-04-09 Last updated: 2018-04-11
3. On the equidistribution of translates of orbits of symmetric subgroups in Γ\G
Open this publication in new window or tab >>On the equidistribution of translates of orbits of symmetric subgroups in Γ\G
(English)Manuscript (preprint) (Other academic)
##### Abstract [en]

We use the method of Burger to study the rate of equidistribution for translates of orbits of symmetric subgroups in homogeneous spaces Γ\G for semisimple Lie groups G and lattices Γ.

##### National Category
Mathematical Analysis
##### Identifiers
urn:nbn:se:uu:diva-347857 (URN)
Available from: 2018-04-09 Created: 2018-04-09 Last updated: 2018-04-11
4. Effective equidistribution of horospheres in infinite-volume quotients of SO(n, 1) by geometrically finite groups
Open this publication in new window or tab >>Effective equidistribution of horospheres in infinite-volume quotients of SO(n, 1) by geometrically finite groups
(English)Manuscript (preprint) (Other academic)
##### Abstract [en]

We use the method of Burger to study the rate of equidistribution for translates of pieces of horospheres in Γ\ SO0(n, 1) for geometrically finite discrete subgroups Γ < SO0(n, 1) with infinite covolume.

##### National Category
Mathematical Analysis
##### Identifiers
urn:nbn:se:uu:diva-347858 (URN)
Available from: 2018-04-09 Created: 2018-04-09 Last updated: 2018-04-11
5. Renormalization of integrals of products of Eisenstein series and analytic continuation of representations
Open this publication in new window or tab >>Renormalization of integrals of products of Eisenstein series and analytic continuation of representations
(English)Manuscript (preprint) (Other academic)
##### Abstract [en]

We combine Zagier’s theory of renormalizable automorphic functions on the hyperbolic plane with the analytic continuation of representations of SL(2, R) due to Bernstein and Reznikov to study triple products of Eisenstein series of generic (in particular, non-arithmetic) non-compact finite-volume hyperbolic surfaces.

##### National Category
Mathematical Analysis
##### Identifiers
urn:nbn:se:uu:diva-347859 (URN)
Available from: 2018-04-09 Created: 2018-04-09 Last updated: 2018-04-24
• 102.
Aarhus Univ, Fac Sci, Ctr Quantum Geometry Moduli Spaces, DK-8000 Aarhus C, Denmark..
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
The homological content of the Jones representations at q =-12016In: Journal of knot theory and its ramifications, ISSN 0218-2165, Vol. 25, no 11, article id 1650062Article in journal (Refereed)

We generalize a discovery of Kasahara and show that the Jones representations of braid groups, when evaluated at q = -1, are related to the action on homology of a branched double cover of the underlying punctured disk. As an application, we prove for a large family of pseudo-Anosov mapping classes a conjecture put forward by Andersen, Masbaum, and Ueno [Topological quantum field theory and the Nielsen-Thurston classification of M(0, 4), Math. Proc. Cambridge Philos. Soc. 141(3) (2006) 477-488] by extending their original argument for the sphere with four marked points to our more general case.

• 103.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Univ Technol Sydney, POB 123, Broadway, NSW 2007, Australia.. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Dynkin Games With Heterogeneous Beliefs2017In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 54, no 1, p. 236-251Article in journal (Refereed)

We study zero-sum optimal stopping games (Dynkin games) between two players who disagree about the underlying model. In a Markovian setting, a verification result is established showing that if a pair of functions can be found that satisfies some natural conditions, then a Nash equilibrium of stopping times is obtained, with the given functions as the corresponding value functions. In general, however, there is no uniqueness of Nash equilibria, and different equilibria give rise to different value functions. As an example, we provide a thorough study of the game version of the American call option under heterogeneous beliefs. Finally, we also study equilibria in randomized stopping times.

• 104.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
The inverse first-passage problem and optimal stopping2016In: The Annals of Applied Probability, ISSN 1050-5164, E-ISSN 2168-8737, Vol. 26, no 5, p. 3154-3177Article in journal (Refereed)

Given a survival distribution on the positive half-axis and a Brownian motion, a solution of the inverse first-passage problem consists of a boundary so that the first passage time over the boundary has the given distribution. We show that the solution of the inverse first-passage problem coincides with the solution of a related optimal stopping problem. Consequently, methods from optimal stopping theory may be applied in the study of the inverse first passage problem. We illustrate this with a study of the associated integral equation for the boundary.

• 105.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Feynman-Kac Theorems for Generalized Diffusions2015In: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 367, no 11, p. 8051-8070, article id PII S0002-9947(2015)06278-3Article in journal (Refereed)

We find Feynman-Kac type representation theorems for generalized diffusions. To do this we need to establish existence, uniqueness and regularity results for equations with measure-valued coefficients.

• 106.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Systems and Control. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Automatic control.
Monotonicity and robustness in Wiener disorder detection2019In: Sequential Analysis, ISSN 0747-4946, E-ISSN 1532-4176, Vol. 38, no 1, p. 57-68Article in journal (Refereed)
• 107.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Momentum liquidation under partial information2016In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 53, no 2, p. 341-359Article in journal (Refereed)

Momentum is the notion that an asset that has performed well in the past will continue to do so for some period. We study the optimal liquidation strategy for a momentum trade in a setting where the drift of the asset drops from a high value to a smaller one at some random change-point. This change-point is not directly observable for the trader, but it is partially observable in the sense that it coincides with one of the jump times of some exogenous Poisson process representing external shocks, and these jump times are assumed to be observable. Comparisons with existing results for momentum trading under incomplete information show that the assumption that the disappearance of the momentum effect is triggered by observable external shocks significantly improves the optimal strategy.

• 108.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Skattning av avstånd mellan arter i fylogenetiska träd2018Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
• 109.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Hodge Decomposition for Manifolds with Boundary and Vector Calculus2017Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
• 110.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Asymptotic Expansions of Integrals and the Method of Steepest Descent2016Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
• 111.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Recent development in conditioned Galton-Watson trees2019Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
• 112.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
The p-Laplace equation – general properties and boundary behaviour2018Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
• 113.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Farey Fractions2017Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
• 114.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Information Diffusion-Based Modeling of Oil Futures Prices2014Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
• 115.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
On the global stability of a peer-to-peer network model2012In: Operations Research Letters, ISSN 0167-6377, E-ISSN 1872-7468, Vol. 40, no 3, p. 190-194Article in journal (Refereed)

In this paper, we analyze the stability properties of a system of ordinary differential equations describing the thermodynamic limit of a microscopic and stochastic model for file sharing in a peer-to-peer network introduced by Kesidis et al. We show, under certain assumptions, that this BitTorrent-like system has a unique locally attracting equilibrium point which is also computed explicitly. Local and global stability are also shown.

• 116.
Heriot Watt Univ, Sch Math & Comp Sci, Edinburgh EH14 4AS, Midlothian, Scotland;Sobolev Inst Math, Novosibirsk, Russia;Novosibirsk State Univ, Novosibirsk, Russia.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Ecole Polytech Fed Lausanne, Inst Math, Stn 8, CH-1015 Lausanne, Switzerland.
Power Law Condition for Stability of Poisson Hail2018In: Journal of theoretical probability, ISSN 0894-9840, E-ISSN 1572-9230, Vol. 31, no 2, p. 684-704Article in journal (Refereed)

The Poisson hail model is a space-time stochastic system introduced by Baccelli and Foss (J Appl Prob 48A:343-366, 2011) whose stability condition is nonobvious owing to the fact that it is spatially infinite. Hailstones arrive at random points of time and are placed in random positions of space. Upon arrival, if not prevented by previously accumulated stones, a stone starts melting at unit rate. When the stone sizes have exponential tails, then stability conditions exist. In this paper, we look at heavy tailed stone sizes and prove that the system can be stabilized when the rate of arrivals is sufficiently small. We also show that the stability condition is, in a weak sense, optimal. We use techniques and ideas from greedy lattice animals.

• 117.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
The Happy Ending Problem and its connection to Ramsey theory2019Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
• 118.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Convergence of directed random graphs to the Poisson-weighted infinite tree2016In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 53, no 2, p. 463-474Article in journal (Refereed)

We consider a directed graph on the integers with a directed edge from vertex i to j present with probability n-1, whenever i<j, independently of all other edges. Moreover, to each edge (i,j) we assign weight n-1(j - i). We show that the closure of vertex 0 in such a weighted random graph converges in distribution to the Poisson-weighted infinite tree as n→∞. In addition, we derive limit theorems for the length of the longest path in the subgraph of the Poisson-weighted infinite tree which has all vertices at weighted distance of at most ρ from the root.

• 119.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Distribution of the smallest visited point in a greedy walk on the line2016In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 53, no 3, p. 880-887Article in journal (Refereed)

We consider a greedy walk on a Poisson process on the real line. It is known that the walk does not visit all points of the process. In this paper we first obtain some useful independence properties associated with this process which enable us to compute the distribution of the sequence of indices of visited points. Given that the walk tends to +∞, we find the distribution of the number of visited points in the negative half-line, as well as the distribution of the time at which the walk achieves its minimum.

• 120.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Greedy walks on two lines2016Article in journal (Other academic)
• 121.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
On Directed Random Graphs and Greedy Walks on Point Processes2016Doctoral thesis, comprehensive summary (Other academic)

This thesis consists of an introduction and five papers, of which two contribute to the theory of directed random graphs and three to the theory of greedy walks on point processes.

We consider a directed random graph on a partially ordered vertex set, with an edge between any two comparable vertices present with probability p, independently of all other edges, and each edge is directed from the vertex with smaller label to the vertex with larger label. In Paper I we consider a directed random graph on ℤ2 with the vertices ordered according to the product order and we show that the limiting distribution of the centered and rescaled length of the longest path from (0,0) to (n, [na] ), a<3/14, is the Tracy-Widom distribution. In Paper II we show that, under a suitable rescaling, the closure of vertex 0 of a directed random graph on ℤ with edge probability n−1 converges in distribution to the Poisson-weighted infinite tree. Moreover, we derive limit theorems for the length of the longest path of the Poisson-weighted infinite tree.

The greedy walk is a deterministic walk on a point process that always moves from its current position to the nearest not yet visited point. Since the greedy walk on a homogeneous Poisson process on the real line, starting from 0, almost surely does not visit all points, in Paper III we find the distribution of the number of visited points on the negative half-line and the distribution of the index at which the walk achieves its minimum. In Paper IV we place homogeneous Poisson processes first on two intersecting lines and then on two parallel lines and we study whether the greedy walk visits all points of the processes. In Paper V we consider the greedy walk on an inhomogeneous Poisson process on the real line and we determine sufficient and necessary conditions on the mean measure of the process for the walk to visit all points.

1. Convergence to the Tracy-Widom distribution for longest paths in a directed random graph
Open this publication in new window or tab >>Convergence to the Tracy-Widom distribution for longest paths in a directed random graph
2013 (English)In: Latin American Journal of Probability and Mathematical Statistics, ISSN 1980-0436, E-ISSN 1980-0436, Vol. 10, no 2, p. 711-730Article in journal (Refereed) Published
##### Abstract [en]

We consider a directed graph on the 2-dimensional integer lattice, placing a directed edge from vertex (i1,i2) to (j1,j2), whenever i1 ≤ j1, i2 ≤ j2, with probability p, independently for each such pair of vertices. Let Ln,m denote the maximum length of all paths contained in an n×m rectangle. We show that there is a positive exponent a, such that, if m/na→1, as n→∞, then a properly centered/rescaled version of Ln,m converges weakly to the Tracy-Widom distribution. A generalization to graphs with non-constant probabilities is also discussed.

##### Keywords
Random graph, last passage percolation, strong approximation, Tracy- Widom distribution
##### National Category
Probability Theory and Statistics
##### Identifiers
urn:nbn:se:uu:diva-304258 (URN)
Available from: 2016-10-03 Created: 2016-10-03 Last updated: 2017-11-30
2. Convergence of directed random graphs to the Poisson-weighted infinite tree
Open this publication in new window or tab >>Convergence of directed random graphs to the Poisson-weighted infinite tree
2016 (English)In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 53, no 2, p. 463-474Article in journal (Refereed) Published
##### Abstract [en]

We consider a directed graph on the integers with a directed edge from vertex i to j present with probability n-1, whenever i<j, independently of all other edges. Moreover, to each edge (i,j) we assign weight n-1(j - i). We show that the closure of vertex 0 in such a weighted random graph converges in distribution to the Poisson-weighted infinite tree as n→∞. In addition, we derive limit theorems for the length of the longest path in the subgraph of the Poisson-weighted infinite tree which has all vertices at weighted distance of at most ρ from the root.

##### Keywords
Directed random graph, Poisson-weighted infinite tree, rooted geometric graph
##### National Category
Probability Theory and Statistics
##### Identifiers
urn:nbn:se:uu:diva-299908 (URN)10.1017/jpr.2016.13 (DOI)000378598700012 ()
Available from: 2016-07-29 Created: 2016-07-29 Last updated: 2017-11-28
3. Distribution of the smallest visited point in a greedy walk on the line
Open this publication in new window or tab >>Distribution of the smallest visited point in a greedy walk on the line
2016 (English)In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 53, no 3, p. 880-887Article in journal (Refereed) Published
##### Abstract [en]

We consider a greedy walk on a Poisson process on the real line. It is known that the walk does not visit all points of the process. In this paper we first obtain some useful independence properties associated with this process which enable us to compute the distribution of the sequence of indices of visited points. Given that the walk tends to +∞, we find the distribution of the number of visited points in the negative half-line, as well as the distribution of the time at which the walk achieves its minimum.

##### National Category
Probability Theory and Statistics
##### Identifiers
urn:nbn:se:uu:diva-305791 (URN)10.1017/jpr.2016.46 (DOI)000386349900016 ()
Available from: 2016-10-21 Created: 2016-10-21 Last updated: 2017-11-29Bibliographically approved
4. Greedy walks on two lines
Open this publication in new window or tab >>Greedy walks on two lines
2016 (English)Article in journal (Other academic) Submitted
##### National Category
Probability Theory and Statistics
##### Identifiers
urn:nbn:se:uu:diva-305792 (URN)
Available from: 2016-10-21 Created: 2016-10-21 Last updated: 2016-10-23
5. The greedy walk on an inhomogeneous Poisson process
Open this publication in new window or tab >>The greedy walk on an inhomogeneous Poisson process
2016 (English)Article in journal (Other academic) Submitted
##### National Category
Probability Theory and Statistics
##### Identifiers
urn:nbn:se:uu:diva-305795 (URN)
Available from: 2016-10-21 Created: 2016-10-21 Last updated: 2016-10-28
• 122.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
The greedy walk on an inhomogeneous Poisson process2016Article in journal (Other academic)
• 123.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
The greedy walk on an inhomogeneous Poisson process2018In: Electronic Communications in Probability, ISSN 1083-589X, E-ISSN 1083-589X, Vol. 23, article id 14Article in journal (Refereed)

The greedy walk is a deterministic walk that always moves from its current position to the nearest not yet visited point. In this paper we consider the greedy walk on an inhomogeneous Poisson point process on the real line. We prove that the property of visiting all points of the point process satisfies a 0-1 law and determine explicit sufficient and necessary conditions on the mean measure of the point process for this to happen. Moreover, we provide precise results on threshold functions for the property of visiting all points.

• 124.
Federal University of Rio de Janeiro, Institute of Mathematics.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Random iterations of homeomorphisms on the circle2017In: MODERN STOCHASTICS-THEORY AND APPLICATIONS, ISSN 2351-6054, Vol. 4, no 3, p. 253-271Article in journal (Refereed)

We study random independent and identically distributed iterations of functions from an iterated function system of homeomorphisms on the circle which is minimal. We show how such systems can be analyzed in terms of iterated function systems with probabilities which are non-expansive on average.

• 125.
Univ Cambridge, Stat Lab, Ctr Math Sci, Wilberforce Rd, Cambridge CB3 0WB, England..
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Univ Cambridge, Stat Lab, Ctr Math Sci, Wilberforce Rd, Cambridge CB3 0WB, England..
Influence in product spaces2016In: Advances in Applied Probability, ISSN 0001-8678, E-ISSN 1475-6064, Vol. 48, no A, p. 145-152Article in journal (Refereed)

The theory of influence and sharp threshold is a key tool in probability and probabilistic combinatorics, with numerous applications. One significant aspect of the theory is directed at identifying the level of generality of the product probability space that accommodates the event under study. We derive the influence inequality for a completely general product space, by establishing a relationship to the Lebesgue cube studied by Bourgain, Kahn, Kalai, Katznelson and Linial (BKKKL) in 1992. This resolves one of the assertions of BKKKL. Our conclusion is valid also in the setting of the generalized influences of Keller.

• 126.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Evaluating the Longstaff-Schwartz method for pricing of American options2015Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
• 127.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Conditioning of Gaussian processes and a zero area Brownian bridgeManuscript (preprint) (Other academic)

We generalize the notion of Gaussian bridges by conditioning Gaussian processes given that certain linear functionals of the sample paths vanish. We show the equivalence of the laws of the unconditioned and the conditioned process and by an application of Girsanov's theorem, we show that the conditioned process follows a stochastic differential equation (SDE) whenever the unconditioned process does. In the Markovian case, we are able to determine the coefficients in the SDE of the conditioned process explicitly. Our main example is Brownian motion on [0,1] pinned down in 0 at time 1 and conditioned to have vanishing area spanned by the sample paths. Finally, the generalization to arbitrary separable Banach spaces is studied.

• 128.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Gaussian Bridges: Modeling and Inference2014Doctoral thesis, comprehensive summary (Other academic)

This thesis consists of a summary and five papers, dealing with the modeling of Gaussian bridges and membranes and inference for the α-Brownian bridge.

In Paper I we study continuous Gaussian processes conditioned that certain functionals of their sample paths vanish. We deduce anticipative and non-anticipative representations for them. Generalizations to Gaussian random variables with values in separable Banach spaces are discussed. In Paper II we present a unified approach to the construction of generalized Gaussian random fields. Then we show how to extract different Gaussian processes, such as fractional Brownian motion, Gaussian bridges and their generalizations, and Gaussian membranes from them.

In Paper III we study a simple decision problem on the scaling parameter in α-Brownian bridges. We generalize the Karhunen-Loève theorem and obtain the distribution of the involved likelihood ratio based on Karhunen-Loève expansions and Smirnov's formula. The presented approach is applied to a simple decision problem for Ornstein-Uhlenbeck processes as well. In Paper IV we calculate the bias of the maximum likelihood estimator for the scaling parameter and propose a bias-corrected estimator. We compare it with the maximum likelihood estimator and two alternative Bayesian estimators in a simulation study. In Paper V we solve an optimal stopping problem for the α-Brownian bridge. In particular, the limiting behavior as α tends to zero is discussed.

1. Conditioning of Gaussian processes and a zero area Brownian bridge
Open this publication in new window or tab >>Conditioning of Gaussian processes and a zero area Brownian bridge
(English)Manuscript (preprint) (Other academic)
##### Abstract [en]

We generalize the notion of Gaussian bridges by conditioning Gaussian processes given that certain linear functionals of the sample paths vanish. We show the equivalence of the laws of the unconditioned and the conditioned process and by an application of Girsanov's theorem, we show that the conditioned process follows a stochastic differential equation (SDE) whenever the unconditioned process does. In the Markovian case, we are able to determine the coefficients in the SDE of the conditioned process explicitly. Our main example is Brownian motion on [0,1] pinned down in 0 at time 1 and conditioned to have vanishing area spanned by the sample paths. Finally, the generalization to arbitrary separable Banach spaces is studied.

##### National Category
Probability Theory and Statistics
##### Research subject
Mathematical Statistics
##### Identifiers
urn:nbn:se:uu:diva-232543 (URN)
Available from: 2014-09-20 Created: 2014-09-19 Last updated: 2014-10-30
2. Gaussian processes, bridges and membranes extracted from selfsimilar random fields
Open this publication in new window or tab >>Gaussian processes, bridges and membranes extracted from selfsimilar random fields
(English)Manuscript (preprint) (Other academic)
##### Abstract [en]

We consider the class of selfsimilar Gaussian generalized random fields introduced by Dobrushin in 1979. These fields are indexed by Schwartz functions on Rd and parametrized by a self-similarity index and the degree of stationarity of their increments. We show that such Gaussian fields arise in explicit form by letting Gaussian white noise, or Gaussian random balls white noise, drive a shift and scale shot-noise mechanism on Rd, covering both isotropic and anisotropic situations. In some cases these fields allow indexing with a wider class of signed measures, and by using families of signed measures parametrized by the points in euclidean space we are able to extract pointwise defined Gaussian processes, such as fractional Brownian motion on Rd. Developing this method further, we construct Gaussian bridges and Gaussian membranes on a finite domain, which vanish on the boundary of the domain.

##### National Category
Probability Theory and Statistics
##### Research subject
Mathematical Statistics
##### Identifiers
urn:nbn:se:uu:diva-232545 (URN)
Available from: 2014-09-23 Created: 2014-09-19 Last updated: 2015-04-23Bibliographically approved
3. Inference for α-Brownian bridge based on Karhunen-Loève expansions
Open this publication in new window or tab >>Inference for α-Brownian bridge based on Karhunen-Loève expansions
(English)Article in journal (Refereed) Submitted
##### Abstract [en]

We study a simple decision problem for the scaling parameter in the α-Brownian bridge X(α) on the interval [0,1]: given two values α0, α1 ≥ 0 with α0 + α1 ≥ 1 and some time 0 ≤ T ≤ 1 we want to test H0: α = α0 vs. H1: α = α1 based on an observation of X(α) until time T. The likelihood ratio can be written as a functional of a quadratic form ψ(X(α)) of X(α). In order to calculate the distribution of ψ(X(α)) under the null hypothesis, we generalize the Karhunen-Loève Theorem to positive finite measures on [0,1] and compute the Karhunen-Loève expansion of X(α) under such a measure. Based on this expansion, the distribution of ψ(X(α)) follows by Smirnov's formula.

##### National Category
Probability Theory and Statistics
##### Research subject
Mathematical Statistics
##### Identifiers
urn:nbn:se:uu:diva-232541 (URN)
Available from: 2014-09-19 Created: 2014-09-19 Last updated: 2014-10-30
4. Bias-correction of the maximum likelihood estimator for the α-Brownian bridge
Open this publication in new window or tab >>Bias-correction of the maximum likelihood estimator for the α-Brownian bridge
2014 (English)In: Statistics and Probability Letters, ISSN 0167-7152, E-ISSN 1879-2103, Vol. 93, p. 78-86Article in journal (Refereed) Published
##### Abstract [en]

The bias of the maximum likelihood estimator of the parameter α in the α-Brownian bridge is derived. A bias-correction which improves the estimator substantially is proposed. The corrected estimator and Bayesian estimators are compared in a simulation study.

##### Keywords
α-Brownian bridge, Bias-correction, Estimation, Scaled Brownian bridge
##### National Category
Probability Theory and Statistics
##### Identifiers
urn:nbn:se:uu:diva-229177 (URN)10.1016/j.spl.2014.06.020 (DOI)000341479500012 ()
##### Note

Title also written as: Bias-correction of the maximum likelihood estimator for the alpha-Brownian bridge

Available from: 2014-08-04 Created: 2014-08-04 Last updated: 2017-12-05Bibliographically approved
5. Optimal stopping of an α-Brownian bridge
Open this publication in new window or tab >>Optimal stopping of an α-Brownian bridge
(English)Article in journal (Refereed) Submitted
##### Abstract [en]

We study the problem of stopping an α-Brownian bridge as close as possible to its global maximum. This extends earlier results found for the Brownian bridge (the case α = 1). The exact behavior for α close to 0 is investigated.

##### National Category
Probability Theory and Statistics
##### Research subject
Mathematical Statistics
##### Identifiers
urn:nbn:se:uu:diva-232542 (URN)
Available from: 2014-09-19 Created: 2014-09-19 Last updated: 2014-10-30
• 129.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Inference for α-Brownian bridge based on Karhunen-Loève expansionsArticle in journal (Refereed)

We study a simple decision problem for the scaling parameter in the α-Brownian bridge X(α) on the interval [0,1]: given two values α0, α1 ≥ 0 with α0 + α1 ≥ 1 and some time 0 ≤ T ≤ 1 we want to test H0: α = α0 vs. H1: α = α1 based on an observation of X(α) until time T. The likelihood ratio can be written as a functional of a quadratic form ψ(X(α)) of X(α). In order to calculate the distribution of ψ(X(α)) under the null hypothesis, we generalize the Karhunen-Loève Theorem to positive finite measures on [0,1] and compute the Karhunen-Loève expansion of X(α) under such a measure. Based on this expansion, the distribution of ψ(X(α)) follows by Smirnov's formula.

• 130.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Optimal stopping of an α-Brownian bridgeArticle in journal (Refereed)

We study the problem of stopping an α-Brownian bridge as close as possible to its global maximum. This extends earlier results found for the Brownian bridge (the case α = 1). The exact behavior for α close to 0 is investigated.

• 131.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Gaussian processes, bridges and membranes extracted from selfsimilar random fieldsManuscript (preprint) (Other academic)

We consider the class of selfsimilar Gaussian generalized random fields introduced by Dobrushin in 1979. These fields are indexed by Schwartz functions on Rd and parametrized by a self-similarity index and the degree of stationarity of their increments. We show that such Gaussian fields arise in explicit form by letting Gaussian white noise, or Gaussian random balls white noise, drive a shift and scale shot-noise mechanism on Rd, covering both isotropic and anisotropic situations. In some cases these fields allow indexing with a wider class of signed measures, and by using families of signed measures parametrized by the points in euclidean space we are able to extract pointwise defined Gaussian processes, such as fractional Brownian motion on Rd. Developing this method further, we construct Gaussian bridges and Gaussian membranes on a finite domain, which vanish on the boundary of the domain.

• 132.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Social Sciences, Department of Statistics. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.
Bias-correction of the maximum likelihood estimator for the α-Brownian bridge2014In: Statistics and Probability Letters, ISSN 0167-7152, E-ISSN 1879-2103, Vol. 93, p. 78-86Article in journal (Refereed)

The bias of the maximum likelihood estimator of the parameter α in the α-Brownian bridge is derived. A bias-correction which improves the estimator substantially is proposed. The corrected estimator and Bayesian estimators are compared in a simulation study.

• 133.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Pricing exotic power options2014Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
• 134. Hart, M. W.
Uppsala University, Disciplinary Domain of Science and Technology, Biology, Department of Ecology and Genetics, Evolutionary Biology. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Selection on coevolving human gamete recognition genes2016In: Integrative and Comparative Biology, ISSN 1540-7063, E-ISSN 1557-7023, Vol. 56, p. E84-E84Article in journal (Other academic)
• 135.
Simon Fraser University, Department of Biological Science, Burnaby.
Arizona State University Colleges at Lake Havasu City, School of Mathematical and Natural Sciences. Simon Fraser University, Department of Biological Science, Burnaby. University of Chicago, Department of Human Genetics. University of Chicago, Department of Human Genetics. Uppsala University, Disciplinary Domain of Science and Technology, Biology, Department of Ecology and Genetics, Evolutionary Biology. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Positive selection on human gamete-recognition genes2018In: PeerJ, ISSN 2167-8359, E-ISSN 2167-8359, Vol. 6, article id e4259Article in journal (Refereed)

Coevolution of genes that encode interacting proteins expressed on the surfaces of sperm and eggs can lead to variation in reproductive compatibility between mates and reproductive isolation between members of different species. Previous studies in mice and other mammals have focused in particular on evidence for positive or diversifying selection that shapes the evolution of genes that encode sperm-binding proteins expressed in the egg coat or zona pellucida (ZP). By fitting phylogenetic models of codon evolution to data from the 1000 Genomes Project, we identified candidate sites evolving under diversifying selection in the human genes ZP3 and ZP2. We also identified one candidate site under positive selection in C4BPA, which encodes a repetitive protein similar to the mouse protein ZP3R that is expressed in the sperm head and binds to the ZP at fertilization. Results from several additional analyses that applied population genetic models to the same data were consistent with the hypothesis of selection on those candidate sites leading to coevolution of sperm- and egg-expressed genes. By contrast, we found no candidate sites under selection in a fourth gene (ZP1) that encodes an egg coat structural protein not directly involved in sperm binding. Finally, we found that two of the candidate sites (in C4BPA and ZP2) were correlated with variation in family size and birth rate among Hutterite couples, and those two candidate sites were also in linkage disequilibrium in the same Hutterite study population. All of these lines of evidence are consistent with predictions from a previously proposed hypothesis of balancing selection on epistatic interactions between C4BPA and ZP3 at fertilization that lead to the evolution of co-adapted allele pairs. Such patterns also suggest specific molecular traits that may be associated with both natural reproductive variation and clinical infertility.

• 136.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Elliptic Curves: A journey through theory and its applications2019Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
• 137.
McGill Univ, Sch Comp Sci, Montreal, PQ, Canada..
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Univ Toronto, Dept Math, St George St 40, Toronto, ON M5R 2E4, Canada..
Graph properties, graph limits, and entropy2018In: Journal of Graph Theory, ISSN 0364-9024, E-ISSN 1097-0118, Vol. 87, no 2, p. 208-229Article in journal (Refereed)

We study the relation between the growth rate of a graph property and the entropy of the graph limits that arise from graphs with that property. In particular, for hereditary classes we obtain a new description of the coloring number, which by well-known results describes the rate of growth. We study also random graphs and their entropies. We show, for example, that if a hereditary property has a unique limiting graphon with maximal entropy, then a random graph with this property, selected uniformly at random from all such graphs with a given order, converges to this maximizing graphon as the order tends to infinity.

• 138.
Univ Missouri, Dept Math, Columbia, MO 65211 USA..
Syracuse Univ, Math Dept, 215 Carnegie Bldg, Syracuse, NY 13244 USA.. CSIC, CSIC, Inst Ciencias Matemat, UAM,UC3M,UCM, C Nicolas Cabrera 13-15, E-28049 Madrid, Spain.. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
The weak-A∞ PROPERTY OF HARMONIC AND p-HARMONIC MEASURES IMPLIES UNIFORM RECTIFIABILITY2017In: Analysis & PDE, ISSN 2157-5045, E-ISSN 1948-206X, Vol. 10, no 3, p. 513-558Article in journal (Refereed)

Let E subset of Rn+1, n >= 2, be an Ahlfors-David regular set of dimension n. We show that the weak- A 1 property of harmonic measure, for the open set Omega: =Rn+1 \ E, implies uniform rectifiability of E. More generally, we establish a similar result for the Riesz measure, p-harmonic measure, associated to the p-Laplace operator, 1 < p < infinity.

• 139.
University of Missouri, Columbia, USA.
University of Missouri, Columbia, USA. Instituto de Ciencias Matematicas, Madrid, Spain. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
The weak-A property of harmonic and p-harmonic measures implies uniform rectifiability2015In: Analysis & PDE, ISSN 2157-5045, E-ISSN 1948-206XArticle in journal (Refereed)

Let $E\subset \ree$, $n\ge 2$, be an Ahlfors-David regular set of dimension $n$. We show that the weak-$A_\infty$ property of harmonic measure, for the open set$\Omega:= \ree\setminus E$, implies uniform rectifiability of $E$. More generally, we establish a similar result for the Riesz measure, $p$-harmonic measure,associated to the $p$-Laplace operator, $1<p<\infty$.

• 140.
Stockholms universitet.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Asymptotic distribution of two-protected nodes in ternary search trees2015In: Electronic Journal of Probability, ISSN 1083-6489, E-ISSN 1083-6489, Vol. 20, article id 9Article in journal (Refereed)

We study protected nodes in m-ary search trees, by putting them in context of generalised Polya urns. We show that the number of two-protected nodes (the nodes that are neither leaves nor parents of leaves) in a random ternary search tree is asymptotically normal. The methods apply in principle to m-ary search trees with larger m as well, although the size of the matrices used in the calculations grow rapidly with m; we conjecture that the method yields an asymptotically normal distribution for all m <= 26. The one-protected nodes, and their complement, i.e., the leaves, are easier to analyze. By using a simpler Polya urn (that is similar to the one that has earlier been used to study the total number of nodes in m-ary search trees), we prove normal limit laws for the number of one-protected nodes and the number of leaves for all m <= 2 6

• 141.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Fringe trees, Crump-Mode-Jagers branching processes and m-ary search trees2017In: Probability Surveys, ISSN 1549-5787, E-ISSN 1549-5787, Vol. 14, p. 53-154Article in journal (Refereed)

This survey studies asymptotics of random fringe trees and extended fringe trees in random trees that can be constructed as family trees of a Crump-Mode-Jagers branching process, stopped at a suitable time. This includes random recursive trees, preferential attachment trees, fragmentation trees, binary search trees and (more generally) m-ary search trees, as well as some other classes of random trees.

We begin with general results, mainly due to Aldous (1991) and Jagers and Nerman (1984). The general results are applied to fringe trees and extended fringe trees for several particular types of random trees, where the theory is developed in detail. In particular, we consider fringe trees of m-ary search trees in detail; this seems to be new.

Various applications are given, including degree distribution, protected nodes and maximal clades for various types of random trees. Again, we emphasise results for m-ary search trees, and give for example new results on protected nodes in m-ary search trees.

A separate section surveys results on the height of the random trees due to Devroye (1986), Biggins (1995, 1997) and others.

This survey contains well-known basic results together with some additional general results as well as many new examples and applications for various classes of random trees.

• 142.
Stockholms universitet.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Limit laws for functions of fringe trees for binary search trees and random recursive trees2015In: Electronic Journal of Probability, ISSN 1083-6489, E-ISSN 1083-6489, Vol. 20, article id 4Article in journal (Refereed)

We prove general limit theorems for sums of functions of subtrees of (random) binary search trees and random recursive trees. The proofs use a new version of a representation by Devroye, and Stein's method for both normal and Poisson approximation together with certain couplings. As a consequence, we give simple new proofs of the fact that the number of fringe trees of size k = k(n) in the binary search tree or in the random recursive tree (of total size n) has an asymptotical Poisson distribution if k -> infinity, and that the distribution is asymptotically normal for k = o(root n). Furthermore, we prove similar results for the number of subtrees of size k with some required property P, e.g., the number of copies of a certain fixed subtree T. Using the Cramer-Wold device, we show also that these random numbers for different fixed subtrees converge jointly to a multivariate normal distribution. We complete the paper by giving examples of applications of the general results, e.g., we obtain a normal limit law for the number of l-protected nodes in a binary search tree or in a random recursive tree.

• 143.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Univ Cambridge, Dept Pure Math & Math Stat, Cambridge CB2 1TN, England..
Univ Memphis, Dept Math Sci, Memphis, TN 38152 USA.. Univ Cambridge, Dept Pure Math & Math Stat, Cambridge CB2 1TN, England..
Majority Bootstrap Percolation on G(n, p)2017In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 24, no 1, article id P1.1Article in journal (Refereed)

Majority bootstrap percolation on a graph G is an epidemic process defined in the following manner. Firstly, an initially infected set of vertices is selected. Then step by step the vertices that have at least half of its neighbours infected become infected. We say that percolation occurs if eventually all vertices in G become infected. In this paper we provide sharp bounds for the critical size of the initially infected set in majority bootstrap percolation on the Erdos-Renyi random graph G(n,p). This answers an open question by Janson, Luczak, Turova and Vallier (2012). Our results obtained for p = clog(n)/n are close to the results obtained by Balogh, Bollobas and Morris (2009) for majority bootstrap percolation on the hypercube. We conjecture that similar results will be true for all regular-like graphs with the same density and sufficiently strong expansion properties.

• 144.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Basic quantisation rules of semiclassical analysis2016Independent thesis Advanced level (professional degree), 20 credits / 30 HE creditsStudent thesis

Semiclassical analysis is the study of how to connect classical mechanics with quantummechanics in a mathematically rigorous way. What is crucial for quantum mechanics isthat the operators that occur are self-adjoint and map a subset of L2 into L2. This hasbeen proven in this thesis. Also, a technique of asymptotic expansion of functions interms of Planck's constant h is developed, in order to study the so called semiclassicallimit (i.e. the limit as h approaches 0) We also develop different approaches toquantisation, and a calculus of operators (i.e. quantum observables) based on acalculus for their corresponding symbols (or classical observables).

• 145. Izhakian, Zur
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Superboolean rank and the size of the largest triangular submatrix of a random matrix2015In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 143, no 1, p. 407-418Article in journal (Refereed)

We explore the size of the largest (permuted) triangular submatrix of a random matrix, and more precisely its asymptotical behavior as the size of the ambient matrix tends to infinity. The importance of such permuted triangular submatrices arises when dealing with certain combinatorial algebraic settings in which these submatrices determine the rank of the ambient matrix and thus attract special attention.

• 146.
Univ Calif Santa Barbara, Dept Stat & Appl Probabil, Santa Barbara, CA 93106 USA..
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Asymptotic Distribution Of The Maximum Interpoint Distance In A Sample Of Random Vectors With A Spherically Symmetric Distribution2015In: The Annals of Applied Probability, ISSN 1050-5164, E-ISSN 2168-8737, Vol. 25, no 6, p. 3571-3591Article in journal (Refereed)

Extreme value theory is part and parcel of any study of order statistics in one dimension. Our aim here is to consider such large sample theory for the maximum distance to the origin, and the related maximum "interpoint distance,'' in multidimensions. We show that for a family of spherically symmetric distributions, these statistics have a Gumbel-type limit, generalizing several existing results. We also discuss the other two types of limit laws and suggest some open problems. This work complements our earlier study on the minimum interpoint distance.

• 147.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Asymptotic Normality of Fringe Subtrees and Additive Functionals in Conditioned Galton-Watson Trees2016In: Random structures & algorithms (Print), ISSN 1042-9832, E-ISSN 1098-2418, Vol. 48, no 1, p. 57-101Article in journal (Refereed)

We consider conditioned Galton-Watson trees and show asymptotic normality of additive functionals that are defined by toll functions that are not too large. This includes, as a special case, asymptotic normality of the number of fringe subtrees isomorphic to any given tree, and joint asymptotic normality for several such subtree counts. Another example is the number of protected nodes. The offspring distribution defining the random tree is assumed to have expectation 1 and finite variance; no further moment condition is assumed.

• 148.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Asymptotics Of Fluctuations In Crump-Mode-Jagers Processes: The Lattice Case2018In: Advances in Applied Probability, ISSN 0001-8678, E-ISSN 1475-6064, Vol. 50, no A, p. 141-171Article in journal (Refereed)

Consider a supercritical Crump-Mode-Jagers process in which all births are at integer times (the lattice case). Let (mu) over cap (z) be the generating function of the intensity of the offspring process, and consider the complex roots of (mu) over cap (z) = 1. The root of smallest absolute value is e(-alpha) = 1/m, where alpha > 0 is the Malthusian parameter; let gamma* be the root of second smallest absolute value. Subject to some technical conditions, the second-order fluctuations of the age distribution exhibit one of three types of behaviour: (i) when gamma* > e(-alpha/2) = m(-1/2), they are asymptotically normal; (ii) when gamma* = e(-alpha/2), they are still asymptotically normal, but with a larger variance; and (iii) when gamma* < e(-alpha/2), the fluctuations are in general oscillatory and (degenerate cases excluded) do not converge in distribution. This trichotomy is similar to what has been observed in related situations, such as some other branching processes and for Polya urns. The results lead to a symbolic calculus describing the limits. The asymptotic results also apply to the total of other (random) characteristics of the population.

• 149.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Graph limits and hereditary properties2016In: European journal of combinatorics (Print), ISSN 0195-6698, E-ISSN 1095-9971, Vol. 52, p. 321-337Article in journal (Refereed)

We give a survey of some general results on graph limits associated to hereditary classes of graphs. As examples, we consider some classes defined by forbidden subgraphs and some classes of intersection graphs, including triangle-free graphs, chordal graphs, cographs, interval graphs, unit interval graphs, threshold graphs, and line graphs.

• 150.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Maximal clades in random binary search trees2015In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 22, no 1, article id P1.31Article in journal (Refereed)

We study maximal clades in random phylogenetic trees with the Yule Harding model or, equivalently, in binary search trees. We use probabilistic methods to reprove and extend earlier results on moment asymptotics and asymptotic normality. In particular, we give an explanation of the curious phenomenon observed by Drmota, Fuchs and Lee (2014) that asymptotic normality holds, but one should normalize using half the variance.

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