Conditional distributions of the times of one-sided sequential approximations

Let $X_1,X_2,\dots$ be a sequence of independent random variables with common continuous distribution function $F(x)$. Let $a$ be an arbitrary point of the support of the underlying probability measure. The present paper deals with random indices of those $X$ which approach point $a$ from the left. Conditional distributions of such random variables are investigated and some limit laws for them are formulated.