A compound Poisson process is considered. We estimate the current position of the stochastic process based on past discrete-time observations (non-linear discrete filtering problem) in Bayesian setting. We obtain bounds for the asymptotic rate of the expected square error of…

We show that an asymmetric Bernoulli random variable is symmetry resistant in the sense that any independent random variable, which when added to it produces a symmetric sum, must have a variance at least as much as itself. The main…

It was proved by Michel Talagrand in [10] that the Parisi formula for the free energy in the Sherrington-Kirkpatrick model is differentiable with respect to inverse temperature parameter. We present a simpler proof of this result by using approximate solutions…

We give a new proof of a theorem by Le Gall and Paulin, showing that scaling limits of random planar quadrangulations are homeomorphic to the 2-sphere. The main geometric tool is a reinforcement of the notion of Gromov-Hausdorff convergence, called…

The stochastic growth model Internal Diffusion Limited Aggregation was defined in 1991 by Diaconis and Fulton. Several shape results are known when the underlying state space is the d-dimensional lattice, or a discrete group with exponential growth. We prove an…

We give a characterization of the family of all probability measures on the extended line $(-\infty,+\infty]$, which may be obtained as the distribution of the maximum of some linear Bernoulli process.

We prove an almost sure limit theorem on the exact convergence rate of the maximum of standardized gaussian random walk increments. This gives a more precise version of Shao's theorem ( Shao, Q.-M., 1995. On a conjecture of R&eacutev&eacutesz. Proc.…

This note continues investigation of randomness-type properties emerging in idealized financial markets with continuous price processes. It is shown, without making any probabilistic assumptions, that the strong variation exponent of non-constant price processes has to be 2, as in the…

We prove a non-standard functional limit theorem for a two dimensional simple random walk on some randomly oriented lattices. This random walk, already known to be transient, has different horizontal and vertical fluctuations leading to different normalizations in the functional…

We study the precise relationship between the subordinate killed and killed subordinate processes in the case of an underlying Hunt process, and show that, under minimal conditions, the former is a subprocess of the latter obtained by killing at a…