A local limit theorem for random walks in random scenery and on randomly oriented lattices
A local limit theorem for random walks in random scenery and on randomly oriented lattices
复制标题
随机场景和随机方向格上随机游动的局部极限定理
DOI：
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发表时间：
2010
期刊：
影响因子：
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通讯作者：
Bruno Schapira
中科院分区：
文献类型：
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作者：
F. Castell;N. Guillotin;Franccoise Pene;Bruno Schapira
随机场景中的随机游动是由$Z_n：=sum_{k=1}^nxi_{X_1+.+ X_k}$，其中$（X_k，kge 1）$和$（xi_y，yinmathbb Z）$是独立同分布的两个序列。随机变量我们假定它们的分布属于指数为α in（0，2]$和$的稳定律的正态吸引域埃坦（0，2]$。这些过程首先由H. Kesten和F.斯皮策，谁证明了收敛分布时，
等式1$和as $n o infty$，of $n^{-delta}Z_n$，for some suitable $delta>0$ depending on $alpha$ and $eta$。这里我们对收敛性感兴趣，因为$n o infty$，of $n^delta{mathbb P}（Z_n=lfloor n^{delta} x
floor）$，当$xin RR$固定时。我们还考虑了随机定向格上的随机游动的情况，我们得到了类似的结果。
Random walks in random scenery are processes defined by $Z_n:=sum_{k=1}^nxi_{X_1+...+X_k}$, where $(X_k,kge 1)$ and $(xi_y,yinmathbb Z)$ are two independent sequences of i.i.d. random variables. We assume here that their distributions belong to the normal domain of attraction of stable laws with index $alphain (0,2]$ and $etain (0,2]$ respectively. These processes were first studied by H. Kesten and F. Spitzer, who proved the convergence in distribution when $alpha
eq 1$ and as $n o infty$, of $n^{-delta}Z_n$, for some suitable $delta>0$ depending on $alpha$ and $eta$. Here we are interested in the convergence, as $n o infty$, of $n^delta{mathbb P}(Z_n=lfloor n^{delta} x
floor)$, when $xin RR$ is fixed. We also consider the case of random walks on randomly oriented lattices for which we obtain similar results.