arXiv subjects

Aklilu Zeleke

Publications and source records attributed to Aklilu Zeleke.

4 recordsLinked to original sources

arXiv2018

Strong laws of large numbers for arrays of random variables and stable random fields

Strong laws of large numbers are established for random fields with weak or strong dependence. These limit theorems are applicable to random fields with heavy-tailed distributions including fractional stable random fields. The conditions for SLLN are described in terms of the $p$-th moments of the partial sums of the random fields, which are convenient to verify. The main technical tool in this paper is a maximal inequality for the moments of partial sums of random fields that extends the technique of Levental, Chobanyan and Salehi \cite{chobanyan-l-s} for a sequence of random variables indexed by a one-parameter.

math.PR↗

arXiv2008

A Strong Law of Large Numbers with Applications to Self-Similar Stable Processes

Let $p \in (0, \infty)$ be a constant and let $\{ξ_n\} \subset L^p(Ω, {\mathcal F}, ¶)$ be a sequence of random variables. For any integers $m, n \ge 0$, denote $S_{m, n} = \sum_{k=m}^{m + n} ξ_k$. It is proved that, if there exist a nondecreasing function $φ: \R_+\to \R_+$ (which satisfies a mild regularity condition) and an appropriately chosen integer $a\ge 2$ such that $$ \sum_{n=0}^\infty \sup_{k \ge 0} \E\bigg|\frac{S_{k, a^n}} {φ(a^n)} \bigg|^p < \infty,$$ Then $$ \lim_{n \to \infty} \frac{S_{0, n}} {φ(n)} = 0\qquad \hbox{a.s.} $$ This extends Theorem 1 in Levental, Chobanyan and Salehi \cite{chobanyan-l-s} and can be applied conveniently to a wide class of self-similar processes with stationary increments including stable processes.

math.PR↗

arXiv2007

On proofs of certain combinatorial identities

In this paper we formulate combinatorial identities that give representation of positive integers as linear combination of even powers of 2 with binomial coefficients. We present side by side combinatorial as well as computer generated proofs using the Wilf-Zeilberger(WZ) method.

math.NT↗

arXiv2004

The Effect of Finite Memory Cutoff on Loop Erased Walk in Z^3

Let ζbe the intersection exponent of random walks in Z^3 and αbe a positive real number. We construct a stochastic process from a simple random walk by erasing loops of length at most N^α. We will prove that for α< \frac{1}{1+2ζ}, the limiting distribution is Gaussian. For α> 2 the limiting distribution will be shown to be equal to the limiting distribution of the loop erased walk.

math.PR↗