Related papers: Random linear recursions with dependent coefficien…

As a motivating problem, we aim to study some special aspects of the marginal distributions of the order statistics for exchangeable and (more generally) for minimally stable non-negative random variables $T_{1},...,T_{r}$. In any case, we…

At high levels, the asymptotic distribution of a stationary, regularly varying Markov chain is conveniently given by its tail process. The latter takes the form of a geometric random walk, the increment distribution depending on the sign of…

By the Lyapunov-Perron method,we prove the existence of random inertial manifolds for a class of equations driven simultaneously by non-autonomous deterministic and stochastic forcing. These invariant manifolds contain tempered pullback…

We study the distributions of waiting times in variations of the $q$-sooner and later waiting time problem. One variation imposes length and frequency quotas on the runs of successes and failures. Another case considers binary trials for…

In this note we consider several kind of partition functions of one-dimensional models with nearest - neighbor interactions $I_n, n\in \mathbf{Z}$ and spin values $\pm 1$. We derive systems of recursive equations for each kind of such…

Ulam has defined a history-dependent random sequence of integers by the recursion $X_{n+1}$ $= X_{U(n)}+X_{V(n)}, n \geqslant r$ where $U(n)$ and $V(n)$ are independently and uniformly distributed on $\{1,\dots,n\}$, and the initial…

We study the minimal/endogenous solution $R$ to the maximum recursion on weighted branching trees given by $$R\stackrel{\mathcal{D}}{=}\left(\bigvee_{i=1}^NC_iR_i \right)\vee Q,$$ where $(Q,N,C_1,C_2,\dots)$ is a random vector with $N\in…

Let $X$ be an integrable discrete random variable over $\{0, 1, 2, \ldots\}$ with $\mathbb{P}(X = i + 1) \leq \mathbb{P}(X = i)$ for all $i$. Then for any integer $a \geq 1$, $\mathbb{P}(X \leq a) \leq \mathbb{E}[X] / (2a - 1)$. Let $W$ be…

Let F be a distribution function with negative mean and regularly varying right tail. Under a mild smoothness condition we derive higher order asymptotic expansions for the tail distribution of the maxima of the random walk generated by F.…

We apply Tsallis's q-indexed nonextensive entropy to formulate a random matrix theory (RMT), which may be suitable for systems with mixed regular-chaotic dynamics. We consider the super-extensive regime of q < 1. We obtain analytical…

We study the distribution of cycle lengths in models of nonuniform random permutations with cycle weights. We identify several regimes. Depending on the weights, the length of typical cycles grows like the total number $n$ of elements, or a…

In this paper, we consider the sums of non-negative integer valued $m$-dependent random variables, and its approximation to the power series distribution. We first discuss some relevant results for power series distribution such as Stein…

This note examines linear combinations of multi-indexed sequences and derives the multivariate generating function of such a linear combination in terms of the original sequence's m.g.f. Applications include finding distributions and…

In this paper we consider a linear homogeneous system of $m$ equations in $n$ unknowns with integer coefficients over the reals. Assume that the sum of the absolute values of the coefficients of each equation does not exceed $k+1$ for some…

We obtain necessary and sufficient conditions for the existence of strictly stationary solutions of multivariate ARMA equations with independent and identically distributed noise. For general ARMA$(p,q)$ equations these conditions are…

We give the cumulative distribution function of $M_n$, the maximum of a sequence of $n$ observations from an ARMA(1, 1) process. Solutions are first given in terms of repeated integrals and then for the case, where the underlying random…

Given a linear recurrence of the form $c_n=a_1c_{n-1}+\cdots+a_j c_{n-j}$, it is well-known that $c_n=\sum_{r}p_r(n)r^n$, where the sum is taken over the set of characteristic roots and each $p_r(n)$ is some polynomial. We give a closed…

Linear combinations of independent random variables have been extensively studied in the literature. However, most of the work is based on some specific distribution assumptions. In this paper, a companion of (J. Appl. Probab. 48 (2011)…

In this paper, using inverse integral transforms, we derive the exact distribution of the random variable $X$ that is involved in the ratio $Z \stackrel{d}{=} X/(X+Y)$ where $X$ and $Y$ are independent random variables having the same…

Let $\mathcal{F}_n$ be the set of unitary polynomials of degree $n \ge 2$ that have their roots in $\mathbb{Z}^*$. We note $$ Q(x) := x^n+a_{1}x^{n-1}+\dots+a_{n}. $$ We show that any two fixed consecutive coefficients $(a_{j},a_{j+1})$ ($j…