Leonid V. Bogachev

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Consider the set L n of convex polygons Γ with vertices on the integer lattice Z 2 , non-negative inclination of the edges and fixed endpoints 0 = (0, 0) and n = (n 1 , n 2). We study the asymptotic properties of the ensemble L n , as n 1 , n 2 → ∞, with respect to a certain parametric class of probability distributions P n = P (r) n (0 < r < ∞) on the(More)
We consider a continuous-time branching random walk on the integer lattice Z d (d 1) with a finite number of branching sources, or catalysts. The random walk is assumed to be spatially homogeneous and irreducible. The branching mechanism at each catalyst, being independent of the random walk, is governed by a Markov branching process. The quantities of(More)
We study limiting distributions of exponential sums SN (t) = N i=1 e tX i as t → ∞, N → ∞, where (Xi) are i.i.d. random variables. Two cases are considered: (A) ess sup Xi = 0 and (B) ess sup Xi = ∞. We assume that the function h(x) = − log P{Xi > x} (case B) or h(x) = − log P{Xi > −1/x} (case A) is regularly varying at ∞ with index 1 < < ∞ (case B) or 0 <(More)
We consider the occupancy problem where balls are thrown independently at infinitely many boxes with fixed positive frequencies. It is well known that the random number of boxes occupied by the first n balls is asymptotically normal if its variance Vn tends to infinity. In this work, we mainly focus on the opposite case where Vn is bounded, and derive a(More)
The distribution µ of a Gibbs cluster point process in X = R d (with n-point clusters) is studied via the projection of an auxiliary Gibbs measure defined on the space of configurations in X × X n. We show that µ is quasi-invariant with respect to the group Diff 0 (X) of compactly supported diffeomorphisms of X and prove an integration-by-parts formula for(More)
The distribution µ of a Poisson cluster process in X = R d (with n-point clusters) is studied via the projection of an auxiliary Poisson measure in the space of configurations in X n , with the intensity measure being the convolution of the background intensity (of cluster centres) with the probability distribution of a generic cluster. We show that µ is(More)
Let Π n be the set of planar convex lattice polygons Γ (i.e., with vertices on Z 2 + and non-negative inclination of all edges) with fixed endpoints 0 = (0, 0) and n = (n 1 , n 2). We are concerned with the limit shape of a typical polygon Γ ∈ Π n as n → ∞ with respect to a certain parametric family of probability measures {P r n } (0 < r < ∞) on the space(More)
The distribution µ cl of a Poisson cluster process in X = R d (with i.i.d. clusters) is studied via an auxiliary Poisson measure on the space of configurations in X = n X n , with intensity measure defined as a convolution of the background intensity of cluster centres and the probability distribution of a generic cluster. We show that the measure µ cl is(More)