Conditional Distributions of the Times of One-Sided Sequential Approximations

Abstract

Let $X_1,X_2,\ldots$ 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.