We study generalized random fields which arise as rescaling limits of spatial configurations of uniformly scattered random balls as the mean radius of the balls tends to 0 or infinity. Assuming that the radius distribution has a power-law behavior, we prove that the centered and renormalized random balls field admits a limit with self-similarity properties. Our main result states that all self-similar, translation-and rotation-invariant Gaussian fields can be obtained through a unified zooming procedure starting from a random balls model. This approach has to be understood as a microscopic description of macroscopic properties. Under specific assumptions, we also get a Poisson-type asymptotic field. In addition to investigating stationarity and self-similarity properties, we give L-2-representations of the asymptotic generalized random fields viewed as continuous random linear functionals.

We consider Galton-Watson trees with Geom(p) offspring distribution. We let T-infinity (p) denote such a tree conditioned on being infinite. We prove that for any 1/2 <= p(1) <= p2 <= 1, there exists a coupling between T-infinity (p(1)) and T-infinity (p(2)) such that P(T-infinity(p(1)) subset of T-infinity(p(2))) = 1.

We relate various concepts of fractal dimension of the limiting set in fractal percolation to the dimensions of the set consisting of connected components larger than one point and its complement in (the "dust"). In two dimensions, we also show that the set consisting of connected components larger than one point is almost surely the union of non-trivial Holder continuous curves, all with the same exponent. Finally, we give a short proof of the fact that in two dimensions, any curve in the limiting set must have Hausdorff dimension strictly larger than 1.

Let B-1, B-2, aEuro broken vertical bar be independent one-dimensional Brownian motions parameterized by the whole real line such that B (i) (0)=0 for every ia parts per thousand yen1. We consider the nth iterated Brownian motion W (n) (t)=B (n) (B (n-1)(a <-(B (2)(B (1)(t)))a <-)). Although the sequence of processes (W (n) ) (na parts per thousand yen1) does not converge in a functional sense, we prove that the finite-dimensional marginals converge. As a consequence, we deduce that the random occupation measures of W (n) converge to a random probability measure mu (a). We then prove that mu (a) almost surely has a continuous density which should be thought of as the local time process of the infinite iteration W (a) of independent Brownian motions. We also prove that the collection of random variables (W (a)(t),taa"ea-{0}) is exchangeable with directing measure mu(infinity).

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.

Let S-n, n >= 1, describe the successive sums of the payoffs in the classical St. Petersburg game. The celebrated Feller weak law states that s(n)/n log(2)n ->(p) 1 as n -> infinity It is also known that almost sure convergence fails. However, Csorgo and Simons (Stat Probab Lett 26: 65-73, 1996) have shown that almost sure convergence holds for trimmed sums, that is, for S-n - max(1 <= k <= n) X-k. Since our actual distribution is discrete there may be ties. Our main focus in this paper is on the "maxtrimmed sum", that is, the sum trimmed by the random number of observations that are equal to the largest one. We prove an analog of Martin-Lof's (J Appl Probab 22: 634-643, 1985) distributional limit theorem for maxtrimmed sums, but also for the simply trimmed ones, as well as for the "total maximum". In a final section, we interpret these findings in terms of sums of (truncated) Poisson random variables.

In some earlier work, we have considered extensions of Lai's (Ann. Probab. 2:432-440, 1974) law of the single logarithm for delayed sums to a multi-index setting with the same as well as different expansion rates in the various dimensions. A further generalization concerns window sizes that are regularly varying with index 1 (on the line). In the present paper, we establish multi-index versions of the latter as well as for some mixtures of expansion rates. In order to keep things within reasonable size, we confine ourselves to some special cases for the index set Z(+.)(2).

Various methods of summation for divergent series of real numbers have been generalized to analogous results for sums of i.i.d. random variables. The natural extension of results corresponding to CesA ro summation amounts to proving almost sure convergence of the CesA ro means. In the present paper we extend such results as well as weak laws and results on complete convergence to random fields, more specifically to random variables indexed by a"currency sign (+) (2) , the positive two-dimensional integer lattice points.