Large deviations for a terminating compound renewal process. (English. Russian original) Zbl 1470.60081

Bakay, G. A.

Theory Probab. Appl. 66, No. 2, 209-227 (2021); translation from Teor. Veroyatn. Primen. 66, No. 2, 261-283 (2021).

Summary: Let \((\xi(i),\eta(i)) \in\mathbb{R}^{d+1}\), \(i \in\mathbb{N}\), be independent and identically distributed random vectors, let \(\xi(i)\in\mathbb{R}^d\) 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 \(N_T = \max\{k \in\mathbb{N}\colon \eta(1)+\dots+\eta(k) \le T\}\) is the renewal process corresponding to improper random variables \(\eta(1), \eta(2), \dots \). 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 \(I_{\Delta_T}(x)=\{y\in\mathbb{R}^d\colon x_j\le y_j < x_j+\Delta_T, j=1,\dots,d\} \), and \(\Delta_T\) is a positive function converging sufficiently slowly to zero.

MSC:

60F10	Large deviations
60K05	Renewal theory

Keywords:

compound renewal process; large deviations; Cramér condition; terminating renewal processes

References:

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