Small deviations of series of weighted positive random variables

Let $\{X_j\}$ be i.i.d. positive random variables and let $\{\lambda_j\}$ be a sequence of nonnegative nonincreasing numbers. We continue to examine the conditions under which asymptotics of the log Laplace transform of $\sum_{j\ge1}\lambda_jX_j$ has an explicit form at infinity. A behavior of $\sup_{j\ge1}\lambda_jX_j$ is also under consideration. Bibl. 14 titles.