Recently, in [11], the “coin-turning walk” was introduced on Z. It is a non-Markovian process where steps form (possibly) time-inhomogeneous Markov chain. In this article, we follow up investigation by introducing analogous processes Zd,d≥2: at time n direction of “updated” with probability pn; otherwise next step repeats previous one. We study some fundamental properties these walks, such as t...

Logarithmic Sobolev inequalities are a well-studied technique for estimating rates of convergence of Markov chains to their stationary distributions. In contrast to continuous state spaces, discrete settings admit several distinct log Sobolev inequalities, one of which is the subject of this paper. Here we derive modi!ed log Sobolev inequalities for some models of random walk, including the ran...

In this paper we give general criteria on tightness and weak convergence of discrete Markov chains to symmetric jump processes on metric measure spaces under mild conditions. As an application, we investigate discrete approximation for a large class of symmetric jump processes. We also discuss some application of our results to the scaling limit of random walk in random conductance. AMS 2010 Ma...

We consider a multivariate Langevin equation in discrete time, driven by a force induced by certain Gibbs' states. The main goal of the paper is to study the asymptotic behavior of a random walk with stationary increments (which are interpreted as discrete-time speed terms) satisfying the Langevin equation. We observe that (stable) functional limit theorems and laws of iterated logarithm for re...

Lookback options are popular in OTC markets for currency hedging. The payoff of a lookback option depends on the minimum or maximum price of the underlying asset over the life of the contract. When the extreme values are continuously monitored, these options can be valued analytically (Conze and Viswanathan, 1991; Goldman et al., 1979a,b). On the other hand, when the maximum or the minimum is o...

We introduce memory-dependent discrete-time quantum random walk models by adding uncorrelated memory terms and also by modifying the Hamiltonian of the walker to include couplings with memory-keeping agents. We next study numerically the correlation effects in these models. We also propose a correlation exponent as a relevant and promising tool for investigation of correlation or memory (hence ...