Small deviation probabilities for weighted sum of independent random variables with a common distribution, decreasing at zero not faster than a power

In the note we give estimates of small deviation probabilities of a sum $\sum_{j\ge1}\lambda_jX_j$, where $\{\lambda_j\}$ are positive numbers and $\{X_j\}$ are i.i.d. positive random variables, satisfying mild assumptions at zero and infinity.