Geodesic
Small deviations of series of independent positive random variables with weights close to exponential
Matematičeskie trudy, Tome 11 (2008) no. 1, pp. 49-67 
A. A. Borovkov; P. S. Ruzankin. Small deviations of series of independent positive random variables with weights close to exponential. Matematičeskie trudy, Tome 11 (2008) no. 1, pp. 49-67. http://geodesic.mathdoc.fr/item/MT_2008_11_1_a2/
@article{MT_2008_11_1_a2,
     author = {A. A. Borovkov and P. S. Ruzankin},
     title = {Small deviations of series of independent positive random variables with weights close to exponential},
     journal = {Matemati\v{c}eskie trudy},
     pages = {49--67},
     year = {2008},
     volume = {11},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MT_2008_11_1_a2/}
}
TY  - JOUR
AU  - A. A. Borovkov
AU  - P. S. Ruzankin
TI  - Small deviations of series of independent positive random variables with weights close to exponential
JO  - Matematičeskie trudy
PY  - 2008
SP  - 49
EP  - 67
VL  - 11
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/MT_2008_11_1_a2/
LA  - ru
ID  - MT_2008_11_1_a2
ER  - 
%0 Journal Article
%A A. A. Borovkov
%A P. S. Ruzankin
%T Small deviations of series of independent positive random variables with weights close to exponential
%J Matematičeskie trudy
%D 2008
%P 49-67
%V 11
%N 1
%U http://geodesic.mathdoc.fr/item/MT_2008_11_1_a2/
%G ru
%F MT_2008_11_1_a2

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

Let $\xi,\xi_0,\xi_1,\dots$ be independent identically distributed (i.i.d.) positive random variables. The present paper is a continuation of the article [1] in which the asymptotics of probabilities of small deviations of series $S=\sum_{j=0}^{\infty}a(j)\xi_j$ was studied under different assumptions on the rate of decrease of the probability $\mathbb P(\xi$ as $x\to0$, as well as of the coefficients $a(j)\ge0$ as $j\to\infty$. We study the asymptotics of $\mathbb P(S$ as $x\to 0$ under the condition that the coefficients $a(j)$ are close to exponential. In the case when the coefficients $a(j)$ are exponential and $\mathbb P(\xi$ as $x\to 0$, $b>0$, $\alpha>0$, the asymptotics $\mathbb P(S$ is obtained in an explicit form up to the factor $x^{o(1)}$. Originality of the approach of the present paper consists in employing the theory of delayed differential equations. This approach differs significantly from that in [1].

[1] Myshkis A. D., Lineinye differentsialnye uravneniya s zapazdyvayuschim argumentom, Nauka, M., 1972 | MR | Zbl

[2] Rozovskii L. V., “O veroyatnostyakh malykh uklonenii nekotorykh sluchainykh summ”, XII Vserossiiskaya shkola-kollokvium po statisticheskim metodam: Tez. dokl. (Sochi, 1–7 oktyabrya 2005), Obozrenie prikladnoi i promyshlennoi matematiki, 12:4, 865–866

[3] Sytaya G. N., “O nekotorykh asimptoticheskikh predstavleniyakh dlya gaussovskoi mery v gilbertovom prostranstve”, Teoriya sluchainykh protsessov, Vyp. 2, Naukova dumka, Kiev, 1974, 93–104 | MR

[4] Borovkov A. A., Ruzankin P. S., “On small deviations of series of weighted random variables”, J. Theoret. Probab., 2008 (to appear) | DOI | MR

[5] Davis R., Resnick S., “Extremes of moving averages of random variables with finite endpoint”, Ann. Probab., 19:1 (1991), 312–328 | DOI | MR | Zbl

[6] Dunker T., Lifshits M. A., Linde W., “Small deviations of sums of independent variables”, High Dimensional Probability (Oberwolfach, 1996), Progr. Probab., 43, Birkhäuser, Basel, 1998, 59–74 | MR | Zbl

[7] Lifshits M. A., “On the lower tail probabilities of some random series”, Ann. Probab., 25:1 (1997), 424–442 | DOI | MR | Zbl