Conditioned local limit theorems for random walks on the real line
数学专题报告
报告题目(Title)：Conditioned local limit theorems for random walks on the real line
报告人(Speaker)：肖惠 (Universität Hildesheim, Germany)
地点(Place)：腾讯会议 539-115-956
时间(Time)：2022年5月18日(周三) 下午4:00-5:00
邀请人(Inviter)：何辉
报告摘要
Consider a random walk $S_n=\sum_{i=1}^n X_i$ with independent and identically distributed real-valued increments $X_i$ of zero mean and finite variance. Assume that $X_i$ is non-lattice and has a moment of order $2 + \delta$. For any $x\geq 0$, let $\tau_x = \inf \left\{ k\geq 1: x+S_{k} < 0 \right\}$ be the first time when the random walk $x+S_n$ leaves the half-line $[0,\infty)$. We study the asymptotic behavior of the probability $\mathbb P (\tau_x >n)$ and that of the expectation $\mathbb{E} \left( f(x + S_n - y), \tau_x > n \right)$ for a large class of target function $f$ and various values of $x$, $y$ possibly depending on $n$. This general setting implies limit theorems for the joint distribution $\mathbb{P} \left( x + S_n \in y+ [0, \Delta], \tau_x > n \right)$ where $\Delta>0$ may also depend on $n$.
In particular, the case of moderate deviations $y=\sigma \sqrt{q n\log n}$ is considered. We also deduce some new asymptotics for random walks with drift and give explicit constants in the asymptotic of the probability ${\mathbb P} (\tau_x =n)$.
For the proofs we establish new conditioned integral limit theorems with precise error terms. Joint work with I. Grama.
