期刊论文详细信息
Electronic Communications in Probability
On the ladder heights of random walks attracted to stable laws of exponent 1
Kôhei Uchiyama1 
关键词: random walk;    stable law of exponent 1;    domain of attraction;    slowly varying;    ladder height;    large deviation;   
DOI  :  10.1214/18-ECP122
学科分类:统计和概率
来源: Institute of Mathematical Statistics
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【 摘 要 】

Let $Z$ be the first ladder height of a one dimensional random walk $S_n=X_1+\cdots + X_n$ with i.i.d. increments $X_j$ which are in the domain of attraction of a stable law of exponent $\alpha $, $0x]$ is slowly varying at infinity if and only if $\lim _{n\to \infty } n^{-1}\sum _1^n P[S_k>0]=0$. By a known result this provides a criterion for $S_{T(R)} /R \stackrel{{\rm P}} \longrightarrow \infty $ as $R\to \infty $, where $T(R)$ is the time when $S_n$ crosses over the level $R$ for the first time. The proof mostly concerns the case $\alpha =1$.

【 授权许可】

CC BY   

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