Large deviations of generalized renewal process

Gavriil A. Bakay; Aleksandr V. Shklyaev

doi:10.1515/dma-2020-0020

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Large deviations of generalized renewal process

Gavriil A. Bakay and Aleksandr V. Shklyaev

Published/Copyright: August 14, 2020

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From the journal Discrete Mathematics and Applications Volume 30 Issue 4

Abstract

Let (ξ(i), η(i)) ∈ ℝ^d+1, 1 ≤ i < ∞, be independent identically distributed random vectors, η(i) be nonnegative random variables, the vector (ξ(1), η(1)) satisfy the Cramer condition. On the base of renewal process, N_T = max{k : η(1) + … + η(k) ≤ T} we define the generalized renewal process Z_T = ∑i=1NTξ(i). Put I_{Δ_T}(x) = {y ∈ ℝ^d : x_j ≤ y_j < x_j + Δ_T, j = 1, …, d}. We find asymptotic formulas for the probabilities P(Z_T ∈ I_{Δ_T}(x)) as Δ_T → 0 and P(Z_T = x) in non-lattice and arithmetic cases, respectively, in a wide range of x values, including normal, moderate, and large deviations. The analogous results were obtained for a process with delay in which the distribution of (ξ(1), η(1)) differs from the distribution on the other steps. Using these results, we prove local limit theorems for processes with regeneration and for additive functionals of finite Markov chains, including normal, moderate, and large deviations.

Keywords: generalized renewal process; Cramer condition; large deviations; local limit theorems; integro-local limit theorems

Originally published in Diskretnaya Matematika (2019) 31, №2, 21–55 (in Russian).

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Received: 2017-12-01

Revised: 2018-07-24

Published Online: 2020-08-14

Published in Print: 2020-08-26

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Articles in the same Issue

https://doi.org/10.1515/dma-2020-0020

Keywords for this article

generalized renewal process; Cramer condition; large deviations; local limit theorems; integro-local limit theorems