Small deviations of series of independent positive random variables with weights close to exponential

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 $\mathbb P(\xi<x)$ as $x\to0$, as well as of the coefficients $a(j)\ge0$ as $j\to\infty$. We study the asymptotics of $\mathbb P(S<x)$ 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 $\mathbb P(\xi<x)\sim bx^\alpha$ as $x\to 0$, $b>0$, $\alpha>0$, the asymptotics $\mathbb P(S<x)$ 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 [1].