Related papers: Random linear recursions with dependent coefficien…

We obtain first decay rates of probabilities of tails of multivariate polynomials built on independent random variables with heavy tails. Then we derive stable limit theorems for nonconventional sums of the form $\sum_{Nt\geq n\geq…

We extend Goldie's (1991) Implicit Renewal Theorem to enable the analysis of recursions on weighted branching trees. We illustrate the developed method by deriving the power tail asymptotics of the distributions of the solutions R to: R =_D…

A random vector $X$ with representation $X=\sum_{j\geq0}A_jZ_j$ is considered. Here, $(Z_j)$ is a sequence of independent and identically distributed random vectors and $(A_j)$ is a sequence of random matrices, `predictable' with respect to…

We consider irreversible Markov chains on finite commutative rings randomly generated using both addition and multiplication. We restrict ourselves to the case where the addition is uniformly random and multiplication is arbitrary. We first…

Multivariate stochastic recurrence equations (SREs) are investigated when coefficients are triangular matrices. If coefficient matrices of SREs have all strictly positive elements, the Kesten's classical result yields solutions with…

Given a sequence $(M_{k}, Q_{k})_{k\ge 1}$ of independent, identically distributed ran\-dom vectors with nonnegative components, we consider the recursive Markov chain $(X_{n})_{n\ge 0}$, defined by the random difference equation…

We analyze certain stationary fields with linear regressions and quadratic conditional variances. This classic probabilistic problem leads somewhat unexpectedly to stationary Markov processes closely tied to non-commutative probability…

For i.i.d. random vectors $(M_{1},Q_{1}),(M_{2},Q_{2}),\ldots$ such that $M>0$ a.s., $Q\geq 0$ a.s. and $\mathbb{P}(Q=0)<1$, the random difference equation $X_{n}=M_{n}X_{n-1}+Q_{n}$, $n=1,2,\ldots$, is studied in the critical case when the…

We provide a new extension of Breiman's Theorem on computing tail probabilities of a product of random variables to a multivariate setting. In particular, we give a complete characterization of regular variation on cones in $[0,\infty)^d$…

Non-linear renewal theory is extended to include random walks perturbed by both a slowly changing sequence and a stationary one. Main results include a version of the Key Renewal Theorem, a derivation of the limiting distribution of the…

In the paper we consider the asymptotics of logarithmic tails of a perpetuity $$R \stackrel{d}{=}\sum_{j=1}^\infty Q_j \prod_{k=1}^{j-1}M_k,\qquad(M_n,Q_n)_{n=1}^\infty \mbox{ are i.i.d. copies of }(M,Q),$$ in the case when…

A multivariate, stationary time series is said to be jointly regularly varying if all its finite-dimensional distributions are multivariate regularly varying. This property is shown to be equivalent to weak convergence of the conditional…

For fixed $m>1$, we consider $m$ independent $n \times n$ non-Hermitian random matrices $X_1, ..., X_m$ with i.i.d. centered entries with a finite $(2+\eta)$-th moment, $ \eta>0.$ As $n$ tends to infinity, we show that the empirical…

Regular variation of distributional tails is known to be preserved by various linear transformations of some random structures. An inverse problem for regular variation aims at understanding whether the regular variation of a transformed…

We extend our previous study of Markov chains on finite commutative rings (arXiv:1605.05089) to arbitrary finite rings with identity. At each step, we either add or multiply by a randomly chosen element of the ring, where the addition…

Let $\{X_n\}_{n\ge0}$ be a sequence of real valued random variables such that $X_n=\rho_n X_{n-1}+\epsilon_n,~n=1,2,\ldots$, where $\{(\rho_n,\epsilon_n)\}_{n\ge1}$ are i.i.d. and independent of initial value (possibly random) $X_0$. In…

This paper presents an analysis of the stochastic recursion $W_{i+1} = [V_iW_i+Y_i]^+$ that can be interpreted as an autoregressive process of order 1, reflected at 0. We start our exposition by a discussion of the model's stability…

In this paper, we study Markov-modulated dependencies for the multiplicative Lindley's recursion $W_{n+1}=[V_{n}W_{n}+Y_{n}(V_{n})]^{+}$, where $Y_{n}(V_{n})$ may depend on $V_{n}$, and can be written as the difference of two nonnegative…

Let $\Psi_1,\Psi_2,...$ be a sequence of i.i.d. random Lipschitz functions on a complete separable metric space with unbounded metric $d$ and forward iterations $X_n$. Suppose that $X_n$ has a stationary distribution. We study the…

We study a non-linear Schroedinger equation with a Hartree-type nonlinearity and a localized random time-dependent external potential. Sharp dispersive estimates for the linear Schroedinger equation with a random time-dependent potential…