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Exact strong laws of large numbers for ratios of the smallest order statistics

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  • Matuła, Przemysław
  • Kurasiński, Paweł
  • Adler, André

Abstract

Let Xn,kn∈N,1≤k≤kn be an array of independent and identically distributed random variables. We take the smallest order statistics Xn,(1) and Xn,(2) in each row and consider the ratios Rn=Xn,(2)∕Xn,(1). Then we study the almost sure convergence of weighted sums of these ratios in two cases: kn→∞ and when kn=K is fixed. Our theorems are proved under very mild conditions and they extend the results of all the papers on this topic that only explored the uniform, exponential and Pareto distributions.

Suggested Citation

  • Matuła, Przemysław & Kurasiński, Paweł & Adler, André, 2019. "Exact strong laws of large numbers for ratios of the smallest order statistics," Statistics & Probability Letters, Elsevier, vol. 152(C), pages 69-73.
    • Handle: RePEc:eee:stapro:v:152:y:2019:i:c:p:69-73
    DOI: 10.1016/j.spl.2019.04.014
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    References listed on IDEAS

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    1. Miao, Yu & Wang, Rujun & Adler, Andre, 2016. "Limit theorems for order statistics from exponentials," Statistics & Probability Letters, Elsevier, vol. 110(C), pages 51-57.
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    1. Matuła, Przemysław & Adler, André, 2022. "A note on exact laws of large numbers for the range of a sample from Pareto-type distributions," Statistics & Probability Letters, Elsevier, vol. 182(C).

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