Geodesic
Large Deviations Probabilities for Generalized Renewal Processes with Regularly Varying Jump Distributions
Matematičeskie trudy, Tome 8 (2005) no. 2, pp. 69-136.

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

In the paper, we study the asymptotic behavior of the probabilities of crossing arbitrary “remote” curvilinear boundaries in the range of large deviations by generalized renewal processes with linear drift. We assume that the jump distributions vary regularly on the positive half-line. In a number of cases, we also assume that the tails of the distributions have regularly varying majorants on the negative half-axis. We impose weaker conditions on the distributions of the renewal intervals. The regular variation condition needs to be used only in the case when boundary crossing can occur due to a very long renewal interval. The results of the paper are obtained for the widest possible deviations zones. For one-dimensional distributions of the processes, more advanced results are obtained, including second order asymptotic expansions. 
@article{MT_2005_8_2_a3,
     author = {A. A. Borovkov and K. A. Borovkov},
     title = {Large {Deviations} {Probabilities} for {Generalized} {Renewal} {Processes} with {Regularly} {Varying} {Jump} {Distributions}},
     journal = {Matemati\v{c}eskie trudy},
     pages = {69--136},
     publisher = {mathdoc},
     volume = {8},
     number = {2},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MT_2005_8_2_a3/}
}
TY  - JOUR
AU  - A. A. Borovkov
AU  - K. A. Borovkov
TI  - Large Deviations Probabilities for Generalized Renewal Processes with Regularly Varying Jump Distributions
JO  - Matematičeskie trudy
PY  - 2005
SP  - 69
EP  - 136
VL  - 8
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MT_2005_8_2_a3/
LA  - ru
ID  - MT_2005_8_2_a3
ER  - 
%0 Journal Article
%A A. A. Borovkov
%A K. A. Borovkov
%T Large Deviations Probabilities for Generalized Renewal Processes with Regularly Varying Jump Distributions
%J Matematičeskie trudy
%D 2005
%P 69-136
%V 8
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MT_2005_8_2_a3/
%G ru
%F MT_2005_8_2_a3
A. A. Borovkov; K. A. Borovkov. Large Deviations Probabilities for Generalized Renewal Processes with Regularly Varying Jump Distributions. Matematičeskie trudy, Tome 8 (2005) no. 2, pp. 69-136. http://geodesic.mathdoc.fr/item/MT_2005_8_2_a3/

[1] Borovkov A. A., Veroyatnostnye protsessy v teorii massovogo obsluzhivaniya, Nauka, M., 1972 | MR

[2] Borovkov A. A., Teoriya veroyatnostei, izd. 2-e, Nauka, M., 1986 ; изд. 3-е., Эдиториал УРСС, М., 1999; Изд-во Ин-та математики СО РАН, Новосибирск, 1999 | MR | Zbl

[3] Borovkov A. A., “Otsenki dlya raspredeleniya summ i maksimumov summ sluchainykh velichin pri nevypolnenii usloviya Kramera”, Sib. mat. zhurn., 41:5 (2000), 997–1038 | MR | Zbl

[4] Borovkov A. A., “Veroyatnosti bolshikh uklonenii dlya sluchainykh bluzhdanii s semieksponentsialnymi raspredeleniyami”, Sib. mat. zhurn., 41:6 (2000), 1290–1324 | MR | Zbl

[5] Borovkov A. A., “Bolshie ukloneniya summ sluchainykh bluzhdanii dvukh tipov”, Mat. trudy, 4:2 (2001), 3–26 | MR | Zbl

[6] Borovkov A. A., “Integro-lokalnye i integralnye teoremy o bolshikh ukloneniyakh summ sluchainykh vektorov. Regulyarnye raspredeleniya”, Sib. mat. zhurn., 43:3 (2002), 508–525 | MR | Zbl

[7] Borovkov A. A., Borovkov K. A., “Veroyatnosti bolshikh uklonenii dlya sluchainykh bluzhdanii s regulyarnym raspredeleniem skachkov”, Dokl. RAN, 371 (2000), 14–16 | MR | Zbl

[8] Borovkov A. A., Borovkov K. A., “O veroyatnostyakh bolshikh uklonenii sluchainykh bluzhdanii. I: Raspredeleniya s pravilno menyayuschimisya khvostami”, Teoriya veroyatnostei i ee primeneniya, 46:2 (2001), 209–232 | MR | Zbl

[9] Borovkov A. A., Borovkov K. A., “O veroyatnosti bolshikh uklonenii dlya sluchainykh bluzhdanii. II: Regulyarnye eksponentsialno ubyvayuschie raspredeleniya”, Teoriya veroyatnostei i ee primeneniya, 49:2 (2004), 209–230 | MR | Zbl

[10] Korshunov D. A., “Veroyatnosti bolshikh uklonenii maksimumov summ nezavisimykh slagaemykh s otritsatelnym srednim i subeksponentsialnym raspredeleniem”, Teoriya veroyatnostei i ee primeneniya, 46:2 (2001), 387–397 | MR | Zbl

[11] Malinovskii V. K., “Predelnye teoremy dlya ostanovlennykh sluchainykh posledovatelnostei. II: Veroyatnosti bolshikh uklonenii”, Teoriya veroyatnostei i ee primeneniya, 41:1 (1996), 107–132 | MR | Zbl

[12] Nagaev A. V., “Predelnye teoremy s uchetom bolshikh uklonenii pri narushenii uslovii Kramera”, Izv. AN Uz SSR. Ser. fiz. mat. nauk, 6 (1969), 17–22 | MR

[13] Pinelis I. F., “Odna zadacha o bolshikh ukloneniyakh v prostranstve traektorii”, Teoriya veroyatnostei i ee primeneniya, 26:1 (1981), 73–87 | MR | Zbl

[14] Pinelis I. F., “Ob asimptoticheskoi ekvivalentnosti veroyatnostei bolshikh uklonenii summy i maksimuma sluchainykh velichin”, Predelnye teoremy teorii veroyatnostei, Tr. IM SO AN SSSR, 5, 1985, 144–173 | MR | Zbl

[15] Rozovskii L. V., “Veroyatnosti bolshikh uklonenii summ nezavisimykh sluchainykh velichin s obschei funktsiei raspredeleniya iz oblasti prityazheniya normalnogo zakona”, Teoriya veroyatnostei i ee primeneniya, 34:4 (1989), 686–705 | MR

[16] Feller V., Vvedenie v teoriyu veroyatnostei i ee prilozheniya, t. 1, Mir, M., 1984

[17] Feller V., Vvedenie v teoriyu veroyatnostei i ee prilozheniya, t. 2, Mir, M., 1984

[18] Asmussen S., Ruin Probabilities, World Scientific, Singapore, 2000 | MR

[19] Bingham N. H., Goldie C. M., and Teugels J. L., Regular Variation, Encyclopedia of Mathematics and Its Applications, 27, Cambridge University Press, Cambridge, etc., 1987 | MR | Zbl

[20] Borovkov A. A. and Borovkov K. A., On Large Deviation Probabilities for Random Walks. I: Regularly Varying Distribution Tails. II: Regular Exponential Tails, Preprint, N 62, Sobolev Institute Press, Novosibirsk, 1999 | MR

[21] Borovkov A. A. and Boxma O. J., “Large deviation probabilities for random walks with heavy tails”, Siberian Adv. Math., 13:1 (2003), 1–31 | MR | Zbl

[22] Embrechts P., Klüppelberg C., and Mikosch T., Modelling Extremal Events for Insurance and Finance, Springer-Verlag, Berlin, etc., 1997 | MR | Zbl

[23] Embrecbts P. and Veraverbeke N., “Estimates for the probability of ruin with special emphasis on the possibility of large claims”, Insurance Math. Econom., 1 (1982), 55–72 | DOI | MR

[24] Goldie C. M. and Kliippelberg C., “Subexponential distributions”, A Practical Guide to Heavy Tails: Statistical Techniques for Analyzing Heavy-Tailed Distributions, eds. Adler R. et al., Birkhäuser, Boston, 1998, 435–459 | MR | Zbl

[25] Gut A., Stopped Random Walks. Limit Theorems and Applications, Applied Probability, 5, Springer-Verlag, Berlin, etc., 1988 | MR | Zbl

[26] Heyde S. S., “A contribution to the theory of large deviations for sums of independent random variables”, Z. Wahrsch. Verw. Gebiete., 7 (1967), 303–308 | DOI | MR | Zbl

[27] Karamata J., “Sur un mode de croissance régulière des functions”, Mathematica (Cluj), 4 (1930), 38–53 | Zbl

[28] Kliippelberg C. and Mikosch T., “Large deviations of heavy-tailed random sums with applications in insurance and finance”, J. Appl. Probab., 34:2 (1997), 293–308 | DOI | MR

[29] Nagaev S. V., “Large deviations of sums of independent random variables”, Ann. Probab., 7:5 (1979), 745–789 | DOI | MR | Zbl

[30] Sparre A. E., “On the collective theory of risk in the case of contagion between the claims”, Transactions of XV th International Congress of Actuaries, V. II, New York, 1957, 219–229

[31] Tang Q., Su C., Jiang T., and Zhang J., “Large deviations for heavy-tailed random sums in compound renewal model”, Statist. Probab. Lett., 52:1 (2001), 91–100 | DOI | MR | Zbl