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Teoriya Veroyatnostei i ee Primeneniya, 2021, Volume 66, Issue 2, Pages 261–283
DOI: https://doi.org/10.4213/tvp5342
(Mi tvp5342)
 

This article is cited in 4 scientific papers (total in 4 papers)

Large deviations for a terminating compound renewal process

G. A. Bakay

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Full-text PDF (478 kB) Citations (4)
References:
Abstract: Let $(\xi(i),\eta(i)) \in \mathbf{R}^{d+1}$, $i \in \mathbf{N}$, be independent and identically distributed random vectors, let $\xi(i)\in \mathbf{R}^d$ be random vectors, let $\eta(i)$ be improper nonnegative random variables, and let $\mathbf{P}(\eta(i) = +\infty)\in(0,1)$. It is assumed that the distribution of the vector $(\xi(1),\eta(1))$ subject to $\{\eta(1)<+\infty\}$ satisfies the Cramér condition. By a terminating compound renewal process we mean the process $Z_T =\sum_{k=1}^{N_T}\xi(k)$, where $N_T = \max\{k \in \mathbf{N}\colon \eta(1)+\dots+\eta(k) \le T\}$ is the renewal process corresponding to improper random variables $\eta(1), \eta(2), \dotsc$. We find precise asymptotics of the probabilities $\mathbf{P}\bigl(Z_T\in I_{\Delta_T}(x)\bigr)$ and $\mathbf{P}(Z_T = x)$ in the nonlattice and strongly arithmetic cases, respectively; here $I_{\Delta_T}(x)=\{y\in\mathbf{R}^d\colon x_j\le y_j < x_j+\Delta_T$, $j=1,\dots,d\}$, and $\Delta_T$ is a positive function converging sufficiently slowly to zero.
Keywords: compound renewal process, large deviations, the Cramér condition, terminating renewal processes.
Funding agency Grant number
Russian Science Foundation 19-11-00111
This work was supported by the Russian Science Foundation (grant 19-11-00111) and carried out at the Steklov Mathematical Institute of Russian Academy of Sciences.
Received: 19.08.2019
Revised: 12.06.2020
Accepted: 26.07.2020
English version:
Theory of Probability and its Applications, 2021, Volume 66, Issue 2, Pages 209–227
DOI: https://doi.org/10.1137/S0040585X97T990356
Bibliographic databases:
Document Type: Article
Language: Russian
Citation: G. A. Bakay, “Large deviations for a terminating compound renewal process”, Teor. Veroyatnost. i Primenen., 66:2 (2021), 261–283; Theory Probab. Appl., 66:2 (2021), 209–227
Citation in format AMSBIB
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  • https://www.mathnet.ru/eng/tvp/v66/i2/p261
  • This publication is cited in the following 4 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Теория вероятностей и ее применения Theory of Probability and its Applications
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