Geodesic
Extension of the invariance principle for compound renewal processes
Teoriâ veroâtnostej i ee primeneniâ, Tome 65 (2020) no. 4, pp. 651-670 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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. 
@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
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/

[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