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Consider a finite irreducible Markov chain with invariant distribution $\pi$. We use the inner product induced by $\pi$ and the associated heat operator to simplify and generalize some results related to graph partitioning and the small-set…

Let $\{X_{v}:v\in\mathbb{Z}^d\}$ be i.i.d. random variables. Let $S(\pi)=\sum_{v\in\pi}X_v$ be the weight of a self-avoiding lattice path $\pi$. Let \[M_n=\max\{S(\pi):\pi\text{ has length }n\text{ and starts from the origin}\}.\] We are…

Let $P$ be the transition matrix of a finite, irreducible and reversible Markov chain. We say the continuous time Markov chain $X$ has transition matrix $P$ and speed $\lambda$ if it jumps at rate $\lambda$ according to the matrix $P$. Fix…

Asymptotics deviation probabilities of the sum S n = X 1 + $\times$ $\times$ $\times$ + X n of independent and identically distributed real-valued random variables have been extensively investigated, in particular when X 1 is not…

The paper is devoted to studies of perturbed Markov chains commonly used for description of information networks. In such models, the matrix of transition probabilities for the corresponding Markov chain is usually regularised by adding a…

An (upward) skip-free Markov chain with the set of nonnegative integers as state space is a chain for which upward jumps may be only of unit size; there is no restriction on downward jumps. In a 1987 paper, Brown and Shao determined, for an…

The purpose of this paper is to ensure the conditions of G\"artner-Ellis Theorem for evaluations of the empirical measure. We show that up-to-date conditions for ensuring the convergence to a quasi-stationary distribution can be applied…

The classical isomorphism theorems for reversible Markov chains have played an important role in studying the properties of local time processes of strongly symmetric Markov processes~\cite{mr06}, bounding the cover time of a graph by a…

Many networking-related settings can be modeled by Markov-modulated infinite-server systems. In such models, the customers' arrival rates and service rates are modulated by a Markovian background process, additionally, there are infinitely…

Exponential inequalities are main tools in machine learning theory. To prove exponential inequalities for non i.i.d random variables allows to extend many learning techniques to these variables. Indeed, much work has been done both on…

We consider evaluation of proper posterior distributions obtained from improper prior distributions. Our context is estimating a bounded function $\phi$ of a parameter when the loss is quadratic. If the posterior mean of $\phi$ is…

Our object of study is the asymptotic growth of heaviest paths in a charged (weighted with signed weights) complete directed acyclic graph. Edge charges are i.i.d. random variables with common distribution $F$ supported on $[-\infty,1]$…

Consider the partial sums {S_t} of a real-valued functional F(Phi(t)) of a Markov chain {Phi(t)} with values in a general state space. Assuming only that the Markov chain is geometrically ergodic and that the functional F is bounded, the…

In the first part of this thesis, we study a Markov chain on $\mathbb{R}_+ \times S$, where $\mathbb{R}_+$ is the non-negative real numbers and $S$ is a finite set, in which when the $\mathbb{R}_+$-coordinate is large, the $S$-coordinate of…

We consider a continuous time Markov chain on a countable state space. We prove a joint large deviation principle (LDP) of the empirical measure and current in the limit of large time interval. The proof is based on results on the joint…

Perturbation theory for Markov chains addresses the question how small differences in the transitions of Markov chains are reflected in differences between their distributions. We prove powerful and flexible bounds on the distance of the…

A time-dependent finite-state Markov chain that uses doubly stochastic transition matrices, is considered. Entropic quantities that describe the randomness of the probability vectors, and also the randomness of the discrete paths, are…

We prove tail estimates for variables $\sum_i f(X_i)$, where $(X_i)_i$ is the trajectory of a random walk on an undirected graph (or, equivalently, a reversible Markov chain). The estimates are in terms of the maximum of the function $f$,…

We consider a Markov chain $(M_{n})_{n\ge 0}$ on the set $\mathbb{N}_{0}$ of nonnegative integers which is eventually decreasing, i.e. $\mathbb{P}\{M_{n+1}<M_{n}|M_{n}\ge a\}=1$ for some $a\in\mathbb{N}$ and all $n\ge 0$. We are interested…

We consider Markov chains with random transition probabilities which, moreover, fluctuate randomly with time. We describe such a system by a product of stochastic matrices, $U(t)=M_t\cdots M_1$, with the factors $M_i$ drawn independently…