Many engineering applications require extracting a signal from observations corrupted by additive noise, possibly heavy-tailed. We assume that the observation noise is a Levy process, while the signal is Gaussian, and derive a non-linear recursive filter that minimizes the L error. A sub-optimal filter is proposed for numerical purposes, and simulations show that it out-performs the existing li...

We solve a problem of non-convex stochastic optimisation with help of simulated annealing of Lévy flights of a variable stability index. The search of the ground state of an unknown potential is non-local due to big jumps of the Levy flights process. The convergence to the ground state is fast due to a polynomial decrease rate of the temperature.

A Galton–Watson branching process with immigration evolving in a random environment is considered. Its associated walk assumed to be oscillating. We prove functional limit theorem which the under consideration normalized by coefficient depending on only. The distribution of limiting described terms strictly stable Levy and sequence independent identically distributed variables this process.

We study the stochastic growth process in discrete time $x_{i+1} = (1 + \mu_i) x_i$ with rate $\mu_i \rho e^{Z_i - \frac12 var(Z_i)}$ proportional to exponential of an Ornstein-Uhlenbeck (O-U) $dZ_t \gamma Z_t dt \sigma dW_t$ sampled on a grid uniformly spaced times $\{t_i\}_{i=0}^n$ step $\tau$. Using large deviation theory methods we compute asymptotic (Lyapunov exponent) $\lambda \lim_{n\to ...

Shortening of chromosome ends, known as telomeres, is one of the supposed mechanisms of cellular aging and death. We provide a probabilistic analysis of the process of loss of telomere ends. The first work concerned with that issue is the paper by Levy et al. [J. Molec. Biol. 225 (1992) 951-960]. Their deterministic model reproduced the observed frequencies of viable cells in the in vitro exper...

A critical branching process {Z k , k = 0, 1, 2, ...} in a random environment is considered. A conditional functional limit theorem for the properly scaled process {log Zpu, 0 ≤ u < ∞} is established under the assumptions Zn > 0 and p ≪ n. It is shown that the limiting process is a Levy process conditioned to stay nonnegative. The proof of this result is based on a limit theorem describing the ...