Abstract

We continue the study of compound renewal processes (c.r.p.) under Cramér’s moment condition initiated in [2, 3, 6, 7, 8, 4, 5, 9, 10, 12, 16, 13, 14, 15]. We examine two types of arithmetic multidimensional c.r.p. \({\mathbf {Z}}(n)\) and \({\mathbf {Y}}(n) \), for which the random vector \({\xi }=({\tau },{\boldsymbol {\zeta }})\) controlling these processes ( \(\tau >0 \) defines the distance between jumps, \(\boldsymbol {\zeta }\) defines the value of jumps of the c.r.p.) has an arithmetic distribution and satisfies Cramér’s moment condition. For these processes, we find the exact asymptotics in the local limit theorems for the probabilities

$$ {\mathbb {P}}\left ({{\mathbf {Z}}(n)={\boldsymbol {x}}}\right ), \quad {\mathbb {P}}\left ({{\mathbf {Y}}(n)={\boldsymbol {x}}}\right )$$

in the Cramér zone of deviations for \({\boldsymbol {x}}\in {\mathbb Z}^d \) (in [9, 10, 13, 14, 15], the analogous problem was solved for nonlattice c.r.p., where the vector \( {\boldsymbol {\xi }}=(\tau ,{\boldsymbol {\zeta }}) \) has a nonlattice distribution).

中文翻译：

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Cramér条件下的多维多维化合物更新过程的局部定理

摘要

我们继续研究在[2、3、6、7、8、4、5、9、10、12、16、13、14、15]中启动的克拉美尔矩条件下的化合物更新过程（crp）。我们研究了两种算术多维crp \（{\ mathbf {Z}}（n）\）和\（{\ mathbf {Y}}（n）\），对于它们，随机向量\（{\ xi} = （{\ tau}，{\ boldsymbol {\ zeta}}）\）控制这些过程（ \（\ tau> 0 \）定义跳转之间的距离， \（\ boldsymbol {\ zeta} \）定义跳转的值的crp）具有算术分布，并满足Cramér矩条件。对于这些过程，我们在概率的局部极限定理中找到了精确的渐近性

$$ {\ mathbb {P}} \ left（{{\ mathbf {Z}}（n）= {\ boldsymbol {x}}} \ right），\ quad {\ mathbb {P}} \ left（{{ \ mathbf {Y}}（n）= {\ boldsymbol {x}}} \ right）$$

在\（{\ boldsymbol {x}} \ in {\ mathbb Z} ^ d \）的Cramér偏差区域中（在[9，10，13，14，15，15]中，对于非晶格crp解决了类似的问题，向量\（{\ boldsymbol {\ xi}} =（\ tau，{\ boldsymbol {\ zeta}}）\）具有非晶格分布）。