The Limiting Behaviour of Randomly Indexed Sequences

Andrzej Krajka ()
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Andrzej Krajka: Maria Curie-Skłodowska University

Journal of Theoretical Probability, 2001, vol. 14, issue 3, 797-811

Abstract: Abstract Let {N n , n≥1} be an arbitrary sequence of positive integer-valued random variables and let F and G be given distribution functions. We present necessary and sufficient conditions under which there exists an array {X n, k , 1≤k≤k n , n≥1} of random variables such that $$X_{n,k} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\mathcal{D}} F,{\text{ }}1 \leqslant k \leqslant k_n ,{\text{ }}n \geqslant 1$$ , and $$X_{n,N_n } \xrightarrow{\mathcal{D}}G$$ , as n→∞. Furthermore, we consider the speed of weak convergence of $$X_{n,N_n }$$ to G, as n→∞.

Keywords: randomly indexed sequence of random elements; weak limit law; the construction of random elements; the transportation problem (search for similar items in EconPapers)
Date: 2001
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DOI: 10.1023/A:1017549224382

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