Large deviations of generalized renewal process

G. A. Bakai ; A. V. Shklyaev

Diskretnaya Matematika, Tome 31 (2019) no. 1, pp. 21-55.

Résumé

Let $(\xi(i),\eta(i))\in\mathbb{R}^{d+1}, 1 \le i \infty$, be independent identically distributed random vectors, $\eta(i)$ be nonnegative random variables, the vector $(\xi(1),\eta(1))$ satisfy the Cramer condition. On the base of renewal process $N_T = \max\{k:\eta(1)+\ldots+\eta(k)~\le~T\}$ we define the generalized renewal process $Z_T=\sum_{i=1}^{N_T} \xi(i)$. Put $I_{\Delta_T}(x)=\{y\in\mathbb{R}^d\colon x_j\le y_j$. We find asymptotic formulas for the probabilities ${\mathbf P}\left(Z_T \in I_{\Delta_T}(x)\right)$ as $\Delta_T\to 0$ and ${\mathbf P}\left(Z_T = x \right)$ 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 $(\xi(1),\eta(1))$ 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.

Détail

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

Bibliographie

[1] Embrechts P., Kl{ü}ppelberg, C., “Some aspects of insurance mathematics”, Theory Probab. Appl., 38:2 (1993), 262–295 | DOI | MR | Zbl

[2] Anscombe, F, “Large-sample theory of sequential estimation”, Biometrika, 36:3-4 (1949), 455-458 | DOI | MR

[3] Borovkov A.A., “Integro-Local Limit Theorems for Compound Renewal Processes”, Theory Probab. Appl., 62:2, 175-195 | DOI | DOI | MR

[4] Borovkov A. A., Mogulskii A. A., “Large deviation principles for trajectories of compound renewal processes. I”, Theory Probab. Appl., 60:2 (2016), 207–221 | DOI | DOI | MR

[5] Frolov A. N., Martikainen A. I., Steinebach J., “On probabilities of small deviations for compound renewal processes”, Theory Probab. Appl., 52:2, 328–337 | DOI | DOI | MR

[6] Mogulskii, A., “Integro-local limit theorems for multidimensional compound renewal processes”, Analiticheskie i vychislitelnye metody v teorii veroyatnostei i ee prilozheniyakh (AVMTV-2017), RUDN, M., 2017, 495–499

[7] Mogulskii A.A., Prokopenko E.I., “Integro-lokalnye teoremy dlya mnogomernykh obobschennykh protsessov vosstanovleniya pri momentnom uslovii Kramera. I”, Teoriya veroyatnostei i ee primeneniya, 2018, no. 15, 503-527 | Zbl

[8] Fenchel W., “On conjugate convex functions”, Canad. J. Math., 1 (1949), 73–77 | DOI | MR | Zbl

[9] Borovkov A.A., Mogulskii A.A., “On large and superlarge deviations of sums of independent random vectors under Cramér's condition. I”, Theory Probab. Appl., 51:2, 227–255 | DOI | DOI | MR

[10] Dembo A., Zeitouni O., Large Deviations Techniques and Applications, v. XVI, Springer, Berlin-Heidelberg, 2010, 396 pp. | MR | Zbl

[11] Kirkland S., Neumann M., Group inverses of M-matrices and their applications, Chapman and Hall/CRC Press, New York, 2013, 311 pp. | MR

[12] Kolmogorov A.N., “Lokalnaya predelnaya teorema dlya klassicheskikh tsepei Markova”, Izv. AN SSSR, 13:4 (1949), 281–300 | Zbl

[13] Iscoe I., Ney P., Nummelin E., “Large deviations of uniformly recurrent Markov additive processes”, Adv. Appl. Math., 1:4, 12 (1985), 373–412 | DOI | MR

[14] Miller H., “A convexity property in the theory of random variables defined on a finite markov chain”, Ann. Math. Statist., 32:4 (1961), 1260–1272 | DOI | MR

[15] Meyer D., Matrix analysis and applied linear algebra, SIAM, Philadelphia, 2000, xii + 718 pp. | MR | Zbl