arXiv subjects
Publications and source records attributed to Min-Zhi Zhao.
Let $R_n$ be the number of distinct values of the $n$ simple samples from an infinite discrete distribution. In 1960 Bahadure proved $\displaystyle \lim_{n\to \infty} \frac{R_n}{\Enum R_n}=1$ in probability; Chen et al. proved the limit in the sense of almost sure convergence, along with other results. In this note we present results of CLT, MDP and LDP for $R_n$ under mild conditions.
Suppose that $\mathcal C$ is a finite collection of patterns. Observe a Markov chain until one of the patterns in $\mathcal C$ occurs as a run. This time is denoted by $τ$. In this paper, we aim to give an easy way to calculate the mean waiting time $E(τ)$ and the stopping probabilities $P(τ=τ_A)$ with $A\in\mathcal C$, where $τ_A$ is the waiting time until the pattern $A$ appears as a run.
Inspired by Benjamini et al (Ann. Inst. H. Poincaré Probab. Stat. 2010) and Windisch (Electron. J. Probab. 2010), we consider the entropy of the random walk range formed by a simple random walk on a discrete group. It is shown in this setting the existence of a quantity which we call the average entropy of the ranges. Some equivalent conditions for the vanishing of the average entropy of the ranges are given. Particularly, the average entropy of the ranges vanishes if and only if the random walk is recurrent or escaping to negative infinity without left jump. In order to characterize the recurrence further, we study the average entropy of the weighted digraphs formed by the random walk. We show that the random walk is recurrent if and only if the average entropy of the weighted digraphs vanishes.