Geodesic
Small deviation probabilities for a weighted sum of independent positive random variables with common distribution function that can decrease at zero fast enough
Teoriâ veroâtnostej i ee primeneniâ, Tome 63 (2018) no. 1, pp. 191-202 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The present note puts forward the most general conditions to date under which the classical asymptotics of the small deviation probabilities for a weighted sum of independent positive random variables takes place. Regarding random variables it is assumed that their common distribution function can, in particular, have an exponential falloff at zero.
Keywords: small deviations, sums of independent positive random variables. 
@article{TVP_2018_63_1_a9,
     author = {L. V. Rozovskii},
     title = {Small deviation probabilities for a weighted sum of independent positive random variables with common distribution function that can decrease at zero fast enough},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {191--202},
     year = {2018},
     volume = {63},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2018_63_1_a9/}
}
TY  - JOUR
AU  - L. V. Rozovskii
TI  - Small deviation probabilities for a weighted sum of independent positive random variables with common distribution function that can decrease at zero fast enough
JO  - Teoriâ veroâtnostej i ee primeneniâ
PY  - 2018
SP  - 191
EP  - 202
VL  - 63
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TVP_2018_63_1_a9/
LA  - ru
ID  - TVP_2018_63_1_a9
ER  - 
%0 Journal Article
%A L. V. Rozovskii
%T Small deviation probabilities for a weighted sum of independent positive random variables with common distribution function that can decrease at zero fast enough
%J Teoriâ veroâtnostej i ee primeneniâ
%D 2018
%P 191-202
%V 63
%N 1
%U http://geodesic.mathdoc.fr/item/TVP_2018_63_1_a9/
%G ru
%F TVP_2018_63_1_a9
L. V. Rozovskii. Small deviation probabilities for a weighted sum of independent positive random variables with common distribution function that can decrease at zero fast enough. Teoriâ veroâtnostej i ee primeneniâ, Tome 63 (2018) no. 1, pp. 191-202. http://geodesic.mathdoc.fr/item/TVP_2018_63_1_a9/

[1] L. V. Rozovsky, “Small deviation probabilities for sums of independent positive random variables”, J. Math. Sci. (N. Y.), 147:4 (2007), 6935–6945 | DOI | MR | Zbl

[2] L. V. Rozovsky, “Small deviations of probabilities for weighted sum of independent positive random variables with a common distribution that decreases at zero not faster than a power”, Theory Probab. Appl., 60:1 (2016), 142–150 | DOI | DOI | MR | Zbl

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

[4] L. V. Rozovskii, “Veroyatnosti malykh uklonenii vzveshennoi summy nezavisimykh sluchainykh velichin s obschim raspredeleniem, imeyuschim stepennoe ubyvanie v nule, pri minimalnykh momentnykh predpolozheniyakh”, Teoriya veroyatn. i ee primen., 62:3 (2017), 610–616 | DOI

[5] L. V. Rozovsky, “Small deviation probabilities of weighted sums under minimal moment assumptions”, Statist. Probab. Lett., 86 (2014), 1–6 | DOI | MR | Zbl

[6] L. V. Rozovsky, “Probabilities of small deviations of the weighted sum of independent random variables with common distribution that decreases at zero not faster than a power”, J. Math. Sci. (N. Y.), 214:4 (2016), 540–545 | DOI | MR | Zbl

[7] L. V. Rozovsky, “Small deviation probabilities for sums of independent positive random variables with a distribution that slowly varies at zero”, J. Math. Sci. (N. Y.), 204:1 (2015), 155–164 | DOI | MR | Zbl

[8] V. M. Zolotarev, One-dimensional stable distributions, Transl. Math. Monogr., 65, Amer. Math. Soc., Providence, RI, 1986, x+284 pp. | MR | MR | Zbl | Zbl

[9] G. N. Sytaya, “O nekotorykh asimptoticheskikh predstavleniyakh gaussovskoi mery v gilbertovom prostranstve”, Teoriya sluchainykh protsessov, 2 (1974), 93–104 | MR | Zbl

[10] F. Aurzada, “On the lower tail probabilities of some random sequences in $l_p$”, J. Theoret. Probab., 20:4 (2007), 843–858 | DOI | MR | Zbl

[11] A. A. Borovkov, P. S. Ruzankin, “On small deviations of series of weighted random variables”, J. Theoret. Probab., 21:3 (2008), 628–649 | DOI | MR | Zbl

[12] L. V. Rozovsky, “On small deviations of series of weighted positive random variables”, J. Math. Sci. (N. Y.), 176:2 (2011), 224–231 | DOI | MR | Zbl

[13] L. V. Rozovsky, “Small deviations of series of independent nonnegative random variables with smooth weights”, Theory Probab. Appl., 58:1 (2014), 121–137 | DOI | DOI | MR | Zbl

[14] L. V. Rozovsky, “Small deviation probabilities for weighted sum of independent random variables with a common distribution that can decrease at zero fast enough”, Statist. Probab. Lett., 117 (2016), 192–200 | DOI | MR | Zbl

[15] V. V. Petrov, Predelnye teoremy dlya summ nezavisimykh sluchainykh velichin, Nauka, M., 1987, 318 pp. | MR | Zbl

[16] L. V. Rozovsky, “Superlarge deviation probabilities for sums of independent random variables with exponentially decreasing tails. II”, Theory Probab. Appl., 59:1 (2015), 168–177 | DOI | DOI | MR | Zbl