Geodesic
Extension of the invariance principle for compound renewal processes
Teoriâ veroâtnostej i ee primeneniâ, Tome 65 (2020) no. 4, pp. 651-670

Voir la notice de l'article provenant de la source Math-Net.Ru

The invariance principle for compound renewal processes is extended (in the sense of asymptotic equivalence) to the zone of moderately large and small deviations. It is assumed that the vector $(\tau,\zeta)$, which “governs” the process, satisfies certain moment conditions (for example, the Cramér condition), and its components $\tau$ and $\zeta$ are either independent or linearly dependent. This extension holds, in particular, for random walks.
Keywords: compound renewal process, invariance principle, large deviations, small deviations, random walk. 
A. A. Borovkov. Extension of the invariance principle for compound renewal processes. Teoriâ veroâtnostej i ee primeneniâ, Tome 65 (2020) no. 4, pp. 651-670. http://geodesic.mathdoc.fr/item/TVP_2020_65_4_a0/
@article{TVP_2020_65_4_a0,
     author = {A. A. Borovkov},
     title = {Extension of the invariance principle for compound renewal processes},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {651--670},
     year = {2020},
     volume = {65},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2020_65_4_a0/}
}
TY  - JOUR
AU  - A. A. Borovkov
TI  - Extension of the invariance principle for compound renewal processes
JO  - Teoriâ veroâtnostej i ee primeneniâ
PY  - 2020
SP  - 651
EP  - 670
VL  - 65
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/TVP_2020_65_4_a0/
LA  - ru
ID  - TVP_2020_65_4_a0
ER  - 
%0 Journal Article
%A A. A. Borovkov
%T Extension of the invariance principle for compound renewal processes
%J Teoriâ veroâtnostej i ee primeneniâ
%D 2020
%P 651-670
%V 65
%N 4
%U http://geodesic.mathdoc.fr/item/TVP_2020_65_4_a0/
%G ru
%F TVP_2020_65_4_a0

[1] M. Csörgő, L. Horváth, J. Steinebach, “Invariance principles for renewal processes”, Ann. Probab., 15:4 (1987), 1441–1460 | DOI | MR | Zbl

[2] J. Steinebach, “Invariance principles for renewal processes when only moments of low order exist”, J. Multivariate Anal., 26:2 (1988), 169–183 | DOI | MR | Zbl

[3] A. A. Borovkov, “Functional limit theorems for compound renewal processes”, Siberian Math. J., 60:1 (2019), 27–40 | DOI | DOI | MR | Zbl

[4] J. Komlós, P. Major, G. Tusnády, “An approximation of partial sums of independent RV's and the sample DF. I”, Z. Wahrscheinlichkeitstheorie und verw. Gebiete, 32 (1975), 111–131 | DOI | MR | Zbl

[5] J. Komlós, P. Major, G. Tusnády, “An approximation of partial sums of independent RV's and the sample DF. II”, Z. Wahrscheinlichkeitstheorie und verw. Gebiete, 34 (1976), 33–58 | DOI | MR | Zbl

[6] M. Csörgő, P. Deheuvels, L. Horváth, “An approximation of stopped sums with applications in queueing theory”, Adv. in Appl. Probab., 19:3 (1987), 674–690 | DOI | MR | Zbl

[7] J. Steinebach, “On the optimality of strong approximation rates for compound renewal processes”, Statist. Probab. Lett., 6:4 (1988), 263–267 | DOI | MR | Zbl

[8] A. A. Borovkov, “Moderate large deviations principles for trajectories of compound renewal processes”, Theory Probab. Appl., 64:2 (2019), 324–333 | DOI | DOI | MR | Zbl

[9] A. A. Borovkov, Probability theory, Universitext, Springer, London, 2013, xxviii+733 pp. | DOI | MR | Zbl

[10] A. A. Borovkov, “Large deviation principles in boundary problems for compound renewal processes”, Siberian Math. J., 57:3 (2016), 442–469 | DOI | DOI | MR | Zbl

[11] A. A. Borovkov, “Analysis of large deviations in boundary-value problems with arbitrary boundaries. I, II”, Select. Transl. Math. Statist. and Probability, 6, Amer. Math. Soc., Providence, RI, 1966, 218–256, 257–274 | MR | MR | Zbl

[12] A. A. Borovkov, Asymptotic analysis of random walks. Light-tailed distributions, Encyclopedia Math. Appl., 176, Cambridge Univ. Press, Cambridge, 2020, 450 pp. | Zbl | Zbl

[13] B. A. Rogozin, “Distribution of the maximum of a process with independent increments”, Siberian Math. J., 10:6 (1969), 989–1010 | DOI | MR | Zbl

[14] A. A. Mogul'skii, “Small deviations in a space of trajectories”, Theory Probab. Appl., 19:1 (1975), 726–736 | DOI | MR | Zbl

[15] A. A. Mogul'skii, “Fourier method for determining the asymptotic behavior of small deviations of a Wiener process”, Siberian Math. J., 23:3 (1983), 420–431 | DOI | MR | Zbl