Using key tools such as Itô’s formula for general semimartingales, Kunita’s moment estimates for Lévy-type stochastic integrals, and the exponential martingale inequality, we find conditions under which the solutions to the stochastic differential equations (SDEs) driven by Lévy noise are stable in probability, almost surely and moment exponentially stable. Keywords; stochastic differential equ...

Abstract We prove that if $f(n)$ is a Steinhaus or Rademacher random multiplicative function, there almost surely exist arbitrarily large values of $x$ for which $|\sum _{n \leq x} f(n)| \geq \sqrt{x} (\log \log x)^{1/4+o(1)}$. This the first such bound grows faster than $\sqrt{x}$, answering question Halász and proving conjecture Erd?s. It plausible exponent $1/4$ sharp in this problem. The pr...

In last RTA, we introduced the notion of probabilistic rewrite systems and we gave some conditions entailing termination of those systems within a finite mean number of reduction steps. Termination was considered under arbitrary unrestricted policies. Policies correspond to strategies for non-probabilistic rewrite systems. This is often natural or more useful to restrict policies to a subclass....

The classical summability methods of Borel (B) and Euler (E(λ), λ > 0) play an important role in many areas of mathematics. For instance, in summability theory they are perhaps the most important methods other than the Cesàro (Cα) and Abel (A) methods, and two chapters of the classic book of Hardy (1949) are devoted to them. In probability, the distinction between methods of Cesàro-Abel and Eul...

In this paper we propose a notion of bisimulation for labelled Markov processes parameterised by negligible sets (LMPns). The point is to allow us to say things like two LMPs are “almost surely” bisimilar when they are bisimilar everywhere except on a negligible set. Usually negligible sets are sets of measure 0, but we work with abstract ideals of negligible sets and so do not introduce an ad-...