Let ξ, ξ0, ξ1, ... be independent identically distributed (i.i.d.) positive random variables. The present paper is a continuation of the article  in which the asymptotics of probabilities of small deviations of series S = Σ ∞ j=0 a(j)ξ j was studied under different assumptions on the rate of decrease of the probability ℙ(ξ < x) as x → 0, as well as of the coefficients a(j) ≥ 0 as j → ∞. We study the asymptotics of ℙ(S < x) as x → 0 under the condition that the coefficients a(j) are close to exponential. In the case when the coefficients a(j) are exponential and ℙ(ξ < x) ∼ bx α as x → 0, b > 0, a > 0, the asymptotics ℙ(S < x) 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 .
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Original Russian Text © A. A. Borovkov and P. S. Ruzankin, 2008, published in Matematicheskie Trudy, 2008, Vol. 11, No. 1, pp. 49–67.
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Borovkov, A.A., Ruzankin, P.S. Small deviations of series of independent positive random variables with weights close to exponential. Sib. Adv. Math. 18, 163–175 (2008). https://doi.org/10.3103/S1055134408030024
- small deviations
- series of independent random variables
- delayed differential equations