On exact strong laws of large numbers for ratios of random variables and their applications

Abstract

We study the almost sure convergence of weighted sums of ratios of independent random variables satisfying some general, mild conditions. The obtained results are applied to exact laws for order statistics. An exact law for independent random variables which are nonidentically distributed is also proved and applied to ratios of adjacent order statistics for a sample of uniformly distributed random variables.

Keywords:

• Almost sure convergence
• strong law of large numbers
• exact laws
• order statistics

2010 Mathematics Subject Classification:

• 60F15
• 62G30