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  • 1.
    Gut, Allan
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Gnedenko-Raikov's theorem, central limit theory, and the weak law of large numbers2006In: Statistics and Probability Letters, ISSN 0167-7152, E-ISSN 1879-2103, Vol. 76, no 17, p. 1935-1939Article in journal (Refereed)
    Abstract [en]

    This note is devoted to the connection between a theorem due to Gnedenko, classical central limit theory, and the weak law of large numbers.

  • 2.
    Gut, Allan
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    On convergence of randomly indexed sequences; a counterexample based on the St. Petersburg game2014In: Statistics and Probability Letters, ISSN 0167-7152, E-ISSN 1879-2103, Vol. 87, p. 105-107Article in journal (Refereed)
    Abstract [en]

    If Y-1, Y-2,... is a sequence of random variables such that Y-n -> Y as n -> infinity, and {tau(t), t >= 0} is a family of "indices" such that tau(t) -> infinity t -> infinity, then it is pretty obvious that Y-tau(t) -> Y as t -> infinity. However, if one relaxes one of -> to -> and lets the other one remain as is, then one of the resulting conclusions holds, whereas the other one does not. In this note we provide a "more natural" counterexample than the original one due to Richter (1965), followed by a minor extension.

  • 3.
    Gut, Allan
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Mathematical Statistics.
    Renewal theory with a trend2011In: Statistics and Probability Letters, ISSN 0167-7152, E-ISSN 1879-2103, Vol. 81, no 8, p. 1292-1299Article in journal (Refereed)
    Abstract [en]

    We prove some analogs of results from renewal theory for random walks in the case when there is a drift, more precisely when the mean of the kth summand equals k(gamma) mu, k >= 1, for some it mu > 0 and 0 < gamma <= 1.

  • 4.
    Gut, Allan
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    The gambler's ruin problem with delays2013In: Statistics and Probability Letters, ISSN 0167-7152, E-ISSN 1879-2103, Vol. 83, no 11, p. 2549-2552Article in journal (Refereed)
    Abstract [en]

    A recent paper by Pozdnyakov and Steele (2010) is devoted to the so-called binary-plus-passive design. Two problems that the authors do not consider can be identified with the classical gambler's ruin problem in which delays are allowed.

  • 5.
    Gut, Allan
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.
    Martin-Lof, Anders
    Extreme-trimmed St. Petersburg games2015In: Statistics and Probability Letters, ISSN 0167-7152, E-ISSN 1879-2103, Vol. 96, p. 341-345Article in journal (Refereed)
    Abstract [en]

    Let S-n, n >= 1, describe the successive sums of the payoffs in the classical St. Petersburg game. Feller's famous weak law, Feller (1945), states that s(n)/n log(2) n (sic) 1 as n -> infinity. However, almost sure convergence fails, more precisely, lim supn ->infinity S-n/n log(2) n = +infinity a.s. and lim inf(n ->infinity) S-n/n log(2) n = 1 a.s. as n -> infinity. Csorgo and Simons (1996) have shown that almost sure convergence holds for trimmed sums, that is, for S-n - max(1 <= k <= n) X-k and, moreover, that this remains true if the sums are trimmed by an arbitrary fixed number of maximal sums. A predecessor of the present paper was devoted to sums trimmed by the random number of maximal summands. The present paper concerns analogs for the random number of summands equal to the minimum, as well as analogs for joint trimmings. (C) 2014 Elsevier B.V. All rights reserved.

  • 6.
    Gut, Allan
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Mathematical Statistics.
    Stadtmueller, Ulrich
    An intermediate Baum-Katz theorem2011In: Statistics and Probability Letters, ISSN 0167-7152, E-ISSN 1879-2103, Vol. 81, no 10, p. 1486-1492Article in journal (Refereed)
    Abstract [en]

    We extend the classical Hsu–Robbins–Erdős theorem to the case when all moments exist, but the moment generating function does not, viz., we assume that Eexp{(log+|X|)α}< for some α>1. We also present multi-index versions of the same and of a related result due to Lanzinger in which the assumption is that Eexp{|X|α}< for some α(0,1).

  • 7.
    Gut, Allan
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.
    Stadtmueller, Ulrich
    Univ Ulm, Dept Number Theory & Probabil Theory, D-89069 Ulm, Germany..
    Strong laws for sequences in the vicinity of the LIL2017In: Statistics and Probability Letters, ISSN 0167-7152, E-ISSN 1879-2103, Vol. 122, p. 63-72Article in journal (Refereed)
    Abstract [en]

    The present paper is devoted to strong laws of large numbers under moment conditions near those of the law of the iterated logarithm (LIL) for i.i.d sequences. More precisely, we wish to investigate possible limit theorems under moment conditions which are stronger than p for any p < 2, in which case we know that there is a.s. convergence to 0, and weaker than E X-2 < infinity, in which case the LIL holds.

  • 8.
    Gut, Allan
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Mathematical Statistics.
    Stadtmüller, U.
    Records in subsets of a random field2013In: Statistics and Probability Letters, ISSN 0167-7152, E-ISSN 1879-2103, Vol. 83, no 3, p. 689-699Article in journal (Refereed)
    Abstract [en]

    Consider an i.i.d.random field {Xk:k∈Z+d}, together with a sequence of unboundedly increasing nested sets Sj={n-ary union}k=1jHk,j≥1, where the sets H j are disjoint. The canonical example consists of the hyperbolas Hj={k∈Z+d:{pipe}k{pipe}=j}. We are interested in the number of "hyperbolas" H j that contain at least one record, and, furthermore, the number of records on the "next hyperbola", that is, the number of observations on H j that exceed max { Xk : k ∈ S j -1 } Various limit theorems under mild conditions on the size of the sets H j are presented.

  • 9.
    Gut, Allan
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Stadtmüller, Ulrich
    An asymmetric Marcinkiewicz-Zygmund LLN for random fields2009In: Statistics and Probability Letters, ISSN 0167-7152, E-ISSN 1879-2103, Vol. 79, no 8, p. 1016-1020Article in journal (Refereed)
    Abstract [en]

    The classical Marcinkiewicz-Zygmund law for i.i.d. random variables has been generalized by Gut [Gut, A., 1978. Marcinkiewicz laws and convergence rates in the law of large numbers for random variables with multidimensional indices. Ann. Probab. 6, 469-482] to random fields.   Therein all indices have the same power in the normalization. Looking into some weighted means of random fields. such as Cesaro summation, it is of interest to generalize these laws to the case where different indices have different powers in the normalization. In this paper we give precise moment conditions for such laws.

  • 10.
    Gut, Allan
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Mathematical Statistics.
    Steinebach, Josef
    Asymptotics for increments of stopped renewal processes2010In: Statistics and Probability Letters, ISSN 0167-7152, E-ISSN 1879-2103, Vol. 80, no 7-8, p. 558-565Article in journal (Refereed)
    Abstract [en]

    Motivated by our earlier work on change-point analysis we prove a number of limit theorems for increments of renewal counting processes, or the corresponding first passage times. The starting point of the increments is deterministic as well as random, a typical example being the first stopping time to detect a change-point of some (continuously) observed process.

  • 11.
    Görgens, Maik
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Thulin, Måns
    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)
    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.

  • 12.
    Janson, Svante
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Tail bounds for sums of geometric and exponential variables2018In: Statistics and Probability Letters, ISSN 0167-7152, E-ISSN 1879-2103, Vol. 135, p. 1-6Article in journal (Refereed)
    Abstract [en]

    We give explicit bounds for the tail probabilities for sums of independent geometric or exponential variables, possibly with different parameters.

  • 13.
    Jonsson, Fredrik
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Mathematical Statistics.
    On the quadratic moment of self-normalized sums2010In: Statistics and Probability Letters, ISSN 0167-7152, E-ISSN 1879-2103, Vol. 80, no 17-18, p. 1289-1296Article in journal (Refereed)
    Abstract [en]

    Let an integer n >= 2 and a vector of independent, identically distributed random variables X-1, ..., X-n be given with P(X = 0) = 0 and define the self-normalized sum Z(n) = (Sigma(n)(i=1) X-i)/(Sigma(n)(i=1) X-i(2))(1/2). With a formula for EZ(n)(2) we prove that EZ(n)(2) >= 1 and that EZ(n)(2) = 1 if and only if the summands are symmetrically distributed. We also construct examples where Z(n) converges to the standard normal distribution as n tends to infinity while EZ(n)(2) tends to infinity (the distribution of the summands varies with n).

  • 14.
    Kaj, Ingemar
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Tahir, Daniah
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Stochastic equations and limit results for some two-type branching models2019In: Statistics and Probability Letters, ISSN 0167-7152, E-ISSN 1879-2103, Vol. 150, p. 35-46Article in journal (Refereed)
    Abstract [en]

    A class of binary state, asymmetric, continuous time Markov branching processes are analyzed under supercritical conditions. Stochastic equations are provided, and limit results for the long time asymptotics as well as for the behavior of the model under rescaling are reviewed. Extensions are presented for model variations, such as population size dependence, with the purpose of promoting further use of these models for applications.

  • 15.
    Lindholm, Mathias
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Mathematical Statistics.
    Vallier, Thomas
    On the degree evolution of a fixed vertex in some growing networks2011In: Statistics and Probability Letters, ISSN 0167-7152, E-ISSN 1879-2103, Vol. 81, no 6, p. 673-677Article in journal (Refereed)
    Abstract [en]

    Two preferential attachment-type graph models which allow for dynamic addition/deletion of edges/vertices are considered. The focus of this paper is on the limiting expected degree of a fixed vertex. For both models a phase transition is seen to occur, i.e. if the probability with which edges are deleted is below a model-specific threshold value, the limiting expected degree is infinite, but if the probability is higher than the threshold value, the limiting expected degree is finite. In the regime above the critical threshold probability, however, the behaviour of the two models may differ. For one of the models a non-zero (as well as zero) limiting expected degree can be obtained whilst the other only has a zero limit. Furthermore, this phase transition is seen to occur for the same critical threshold probability of removing edges as the one which determines whether the degree sequence is of power-law type or if the tails decays exponentially fast.

  • 16.
    Lyhagen, Johan
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Social Sciences, Department of Information Science.
    The Exact Covariance Matrix of Dynamic Models with Latent Variables2005In: Statistics and Probability Letters, ISSN 0167-7152, E-ISSN 1879-2103, Vol. 75, no 2, p. 133-139Article in journal (Refereed)
  • 17.
    Pingel, Ronnie
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Social Sciences, Department of Statistics.
    Some approximations of the logistic distribution with application to the covariance matrix of logistic regression2014In: Statistics and Probability Letters, ISSN 0167-7152, E-ISSN 1879-2103, Vol. 85, p. 63-68Article in journal (Refereed)
    Abstract [en]

    In this paper, we show that a two-component normal mixture model provides a good approximation to the logistic distribution. This approximation is an improvement over using the normal distribution and is comparable to using the t-distribution as approximating distributions. The results from using the mixture model is exemplified by finding an approximative analytic expression for the covariance matrix of logistic regression using normally distributed random regressors.

  • 18.
    Stoica, Peter
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Automatic control.
    Xu, Luzhou
    Li, Jian
    Xie, Yao
    Optimal correction of an indefinite estimated MA spectral density matrix2007In: Statistics and Probability Letters, ISSN 0167-7152, E-ISSN 1879-2103, Vol. 77, no 10, p. 973-980Article in journal (Refereed)
    Abstract [en]

    Consider a vector moving-average sequence of order n, MA (n), and let Φ (ω) = ∑k = - nn Rk e- j ω k denote its spectral density matrix, where { Rk }k = - nn are the covariance matrices and ω stands for the frequency variable. A nonparametric estimate over(Φ, ^) (ω) = ∑k = - nn over(R, ^)k e- j ω k of Φ (ω) can easily become indefinite at some frequencies, and thus invalid, due to the estimation errors. In this paper, we provide a computationally efficient procedure that obtains the optimal (in a least-squares sense) valid approximation Φ (ω) to over(Φ, ^) (ω) in a polynomial time, by means of a semidefinite programming (SDP) algorithm.

  • 19.
    Thulin, Måns
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.
    On split sample and randomized confidence intervals for binomial proportions2014In: Statistics and Probability Letters, ISSN 0167-7152, E-ISSN 1879-2103, Vol. 92, p. 65-71Article in journal (Refereed)
    Abstract [en]

    We study randomized confidence intervals for binomial proportions, comparing coverage, length and the impact of the randomization. It is seen that the recently proposed split sample intervals can be improved upon in various ways. Criticisms of randomized intervals are discussed.

  • 20.
    Önskog, Thomas
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Existence of pathwise unique Langevin processes on polytopes with perfect reflection at the boundary2013In: Statistics and Probability Letters, ISSN 0167-7152, E-ISSN 1879-2103, Vol. 83, no 10, p. 2211-2219Article in journal (Refereed)
    Abstract [en]

    Exploiting an explicit projection from the real line into an interval, we prove existence and pathwise uniqueness of one-dimensional Langevin processes confined to an interval with perfect reflection at the boundary. This result is subsequently generalized to multidimensional Langevin processes confined to box domains or general polytopes.

1 - 20 of 20
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