A Nonconventional Local Limit Theorem
A Nonconventional Local Limit Theorem
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非常规局部极限定理
DOI：
10.1007/s10959-015-0625-9
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发表时间：
2014
影响因子：
0.8
通讯作者：
Y. Kifer
中科院分区：
文献类型：
--
作者：
Y. Hafouta;Y. Kifer
局部极限定理起源于经典的De Moivre-Laplace定理，它们研究的渐近行为是如下形式的概率的→∞：$P S_N=k$P{SN=k}其中$$S_N=^N_n=1}F(Xi_N)$SN=∑n=1nF(ξn)是一整数值函数F在I.I.D.上取的和。或马尔可夫相关的随机变量序列$$\xi_j\$${ξj}。对于格值函数和一般函数F，也得到了相应的结果。在这里，我们将这类结果推广到形式为$$S_N=\SUM_{n=1}^Nf(\Xi_n，\Xi_{2n}，\ldots，\xi_\ell n})$$SN=∑n=1Nf(ξn，ξ2n，…，ξℓn)，它延续了最近研究这类表达式的各种极限定理的研究路线。
Local limit theorems have their origin in the classical De Moivre–Laplace theorem, and they study the asymptotic behavior as $$N\rightarrow \infty $$N→∞ of probabilities having the form $$P\{ S_N=k\}$$P{SN=k} where $$S_N=\sum ^N_{n=1}F(\xi _n)$$SN=∑n=1NF(ξn) is a sum of an integer-valued function F taken on i.i.d. or Markov-dependent sequence of random variables $$\{\xi _j\}$${ξj}. Corresponding results for lattice-valued and general functions F were obtained, as well. We extend here this type of results to nonconventional sums of the form $$S_N=\sum _{n=1}^NF(\xi _n,\xi _{2n}, \ldots ,\xi _{\ell n})$$SN=∑n=1NF(ξn,ξ2n,…,ξℓn) which continues the recent line of research studying various limit theorems for such expressions.