The Extended Large Deviation Principle for the Trajectories of a Compound Renewal Process

Siberian Advances in Mathematics Pub Date : 2022-03-17 DOI:10.1134/s1055134422010047

A. A. Mogul’skiĭ

引用次数: 0

Abstract

We study a homogeneous compound renewal process (c.r.p.) $Z(t) $. It is assumed that the elements of the sequence that rules the process satisfy Cramér’s moment condition $[{\bf C}_0] $. We consider the family of processes

$$ z_T(t):=\frac 1xZ(tT),\enspace \enspace 0\le t\le 1,$$

where $x=x_T\sim T $ as $T\to \infty $. Conditions are proposed under which the extended large deviation principle holds for the trajectories $ z_T$ in the space $(\mathbb {V},\rho B) $ of functions with bounded variation, endowed with Borovkov’s metric. If the trajectories of the process $Z(t) $ are monotone with probability 1 then, under the same condition, we prove the classical trajectory large deviation principle.

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复合更新过程轨迹的扩展大偏差原理

摘要研究了一种均相化合物更新过程(c.r.p) $Z(t) $。假设控制过程的序列元素满足cramsamr矩条件$[{\bf C}_0] $。我们考虑了过程族$$ z_T(t):=\frac 1xZ(tT),\enspace\enspace 0\le t\le 1,$$，其中$x=x_T\sim T $为$T\to \infty $，给出了具有borovkov度规的有界变分函数空间$(\mathbb {V},\rho B) $中轨迹$ z_T$的扩展大偏差原理成立的条件。如果过程$Z(t) $的轨迹是概率为1的单调，那么在相同的条件下，我们证明了经典轨迹大偏差原理。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Siberian Advances in Mathematics Mathematics-Mathematics (all)

CiteScore

0.70

自引率

0.00%

发文量

期刊介绍： Siberian Advances in Mathematics is a journal that publishes articles on fundamental and applied mathematics. It covers a broad spectrum of subjects: algebra and logic, real and complex analysis, functional analysis, differential equations, mathematical physics, geometry and topology, probability and mathematical statistics, mathematical cybernetics, mathematical economics, mathematical problems of geophysics and tomography, numerical methods, and optimization theory.