A remark on the exact laws of large numbers for ratios of independent random variables
Let \(( X_{n}) _{n\in \mathbb{N}}\)
and \(( Y_{n}) _{n\in
\mathbb{N}}\) be two sequences of i.i.d. random variables which
are independent of each other and all have the distribution of a
positive random variable \(\xi\) with
density \(f_{\xi }.\) We study weighted
strong laws of large numbers for the ratios of the form \(\frac{1}{b_{n}}\sum_{k=1}^{n}a_{k}\frac{X_{k}}{Y_{k}}\)
in the cases when \(\mathbb{E}\xi=\infty\) or \(\lim_{x\rightarrow 0^{+}}f_{\xi }(x)=0\) or
\(f_{\xi }\) is unbounded. This
research complements some results known so far.
2010 AMS Mathematics Subject Classification: Primary 60F15; Secondary 60G50.
Keywords and phrases: almost sure convergence, law of large numbers, i.i.d. random variables.
