arXiv subjects
Publications and source records attributed to Chen-Xu Hao.
Consider a simple symmetric random walk on the integer lattice $\mathbb{Z}$. Let $E(n)$ denote a favorite edge of the random walk at time $n$. In this paper, we study the escape rate of $E(n)$, and show that almost surely $\liminf_{n\to\infty}\frac{|E(n)|}{\sqrt{n}\cdot(\log n)^{-γ}}$ equals 0 if $γ\le 1$, and is infinity otherwise. We also obtain a law of the iterated logarithm for $E(n)$.
Random walk is a very important Markov process and has important applications in many fields.For a one-dimensional simple symmetric random walk $(S_n)$, a site $x$ is called a favorite downcrossing site at time $n$ if its downcrossing local time at time $n$ achieves the maximum among all sites. In this paper, we study the cardinality of the favorite downcrossing site set, and will show that with probability 1 there are only finitely many times at which there are at least four favorite downcrossing sites and three favorite downcrossing sites occurs infinitely often. Some related open questions will be introduced.
For a one-dimensional simple symmetric random walk $(S_n)$, an edge $x$ (between points $x-1$ and $x$) is called a favorite edge at time $n$ if its local time at $n$ achieves the maximum among all edges. In this paper, we show that with probability 1 three favorite edges occurs infinitely often. Our work is inspired by Tóth and Werner [Combin. Probab. Comput. {\bf 6} (1997) 359-369], and Ding and Shen [Ann. Probab. {\bf 46} (2018) 2545-2561], disproves a conjecture mentioned in Remark 1 on page 368 of Tóth and Werner [Combin. Probab. Comput. {\bf 6} (1997) 359-369].
In this paper we mainly discuss sharp lower and upper bounds for the length of longest consecutive switches in IID Bernoulli sequences. This work is an extension of results in Erdős and Révész (1975) for longest head-run and Hao et al. (2021) for longest consecutive switches in unbiased coin-tossing, and might be applied to reliability theory, biology, quality control, pattern recognition, finance, etc.
An unbiased coin is tossed $n$ times independently and sequentially. In this paper, we will study the length of the longest consecutive switches, and prove that the limit behaviors are similar to that of the length of the longest head-run.