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Contents of PMS, Vol. 42, Fasc. 2,
pages 219 - 225
DOI: 10.37190/0208-4147.00071
Published online 20.9.2022
 

A remark on the exact laws of large numbers for ratios of independent random variables

P. Matuła

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.

A. Adler, Exact strong laws, Bull. Inst. Math. Acad. Sinica 28 (2000), 141-166.

A. Adler, Laws of large numbers for ratios of uniform random variables, Open Math. 13 (2015), 571-576.

P. Kurasiński and P. Matuła, Exact weak laws of large numbers with applications to ratios of random variables, Appl. Math. (Warsaw) 47 (2020), 59-66.

P. Kurasiński, Exact laws of large numbers with applications, PhD thesis, 2022.

P. Matuła, A. Adler and P. Kurasiński, On exact strong laws of large numbers for ratios of random variables and their applications, Comm. Statist. Theory Methods 49 (2020), 3153-3167.

P. Matuła, P. Kurasiński and A. Adler, Exact strong laws of large numbers for ratios of the smallest order statistics, Statist. Probab. Lett. 152 (2019), 69-73.

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