Characterization of Large Deviation Probabilities for Regenerative Sequences

G. A. Bakay

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Branching Processes and Related Topics, Tome 316 (2022), pp. 47-63

Résumé

Local theorems are considered for additive functionals of regenerative sequences, which are sequences of random vectors $\{S_n\}_{n\ge 0}$ of special form. Two cases of renewal are considered: proper and terminating renewal. Under the assumption that all renewal cycles satisfy the Cramér condition, in the case of proper renewal, A. A. Borovkov, A. A. Mogulskii and E. I. Prokopenko, as well as A. V. Shklyaev and G. A. Bakay, obtained exact asymptotics for large deviation probabilities $\mathbf P(S_n=x)\sim {D(x/n)}n^{-d/2}\exp (-L(x/n)n)$, $n\to \infty $, which are uniform with respect to $x/n=x(n)/n\in \mathbb R^d$ in compact sets, with certain functions $D$ and $L$. In the case of terminating renewal, similar results were obtained by Bakay; moreover, one more deviation zone was distinguished in which the result has the form $\mathbf P(S_n=x) \sim {D_0(x/n)}{n^{-(d-1)/2}}\exp (-L_0(x/n)n)$, $n\to \infty $, with certain functions $D_0$ and $L_0$. This relation holds uniformly with respect to $x/n=x(n)/n\in \mathbb R^d$ in compact sets. In the present paper, an alternative method is found for calculating the functions appearing in the asymptotics, and equivalent conditions are obtained for the theorems.

Détail

Keywords: local theorems, large deviations, random sequences with renewal, terminating renewal.