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Previous article Next article Limit Theorems for Sequences of Random Variables with Sequences of Random IndecesWolfgang RichterWolfgang Richterhttps://doi.org/10.1137/1110007PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] F. J. Anscombe, Large-sample theory of sequential estimation, Proc. Cambridge Philos. Soc., 48 (1952), 600–607 MR0051486 0047.13401 CrossrefGoogle Scholar[2] Simeon M. Berman, Limiting distribution of the maximum term in sequences of dependent random variables, Ann. Math. Statist., 33 (1962), 894–908 MR0142142 0109.11804 CrossrefGoogle Scholar[3] Patrick Billingsley, Limit theorems for randomly selected partial sums, Ann. Math. Statist., 33 (1962), 85–92 MR0137138 0113.12402 CrossrefGoogle Scholar[4] David Blackwell, On an equation of Wald, Ann. Math. Statistics, 17 (1946), 84–87 MR0019902 0063.00422 CrossrefGoogle Scholar[5] J. R. Blum, , D. L. Hanson and , J. I. Rosenblatt, On the central limit theorem for the sum of a random number of independent random variables, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 1 (1962/1963), 389–393 10.1007/BF00533414 MR0155349 0109.36402 CrossrefGoogle Scholar[6] R. L. Dobrušin, Lemma on the limit of compound random functions, Uspehi Mat. Nauk, 10 (1955), 157–159, (In Russian.) MR0070870 Google Scholar[7] B. Gnedenko, Sur la distribution limité du terme maximum d'une série aléatoire, Ann. of Math. (2), 44 (1943), 423–453 MR0008655 0063.01643 CrossrefGoogle Scholar[8] S. Guiaşu, Asymptotic distribution of the random sequences of random variables, Rev. Math. Pures Appl. (Bucarest), 8 (1963), 661–671 MR0198527 0232.60021 Google Scholar[9] A. N. Kolmogorov, On the logarithmic law of the distribution of the size of particles undergoing splitting, DAN SSSR, 31 (1941), 99–101, (In Russian.) Google Scholar[10] Casimir Kuratowski, Topologie. I. Espaces Métrisables, Espaces Complets, Monografie Matematyczne, vol. 20, Warszawa-Wrocław, 1948xi+450 MR0028007 0041.09603 Google Scholar[11] J. Mogyoródi, On limiting distributions for sums of a random number of independent random variables, Magyar Tud. Akad. Mat. Kutató Int. Közl., 6 (1961), 365–371 MR0148098 Google Scholar[12] J. Mogyoródi, A central limit theorem for the sum of a random number of independent random variables, Magyar Tud. Akad. Mat. Kutató Int. Közl., 7 (1962), 409–424 MR0151998 Google Scholar[13] A. Rényi, On the asymptotic distribution of the sum of a random number of independent random variables, Acta Math. Acad. Sci. Hungar., 8 (1957), 193–199 MR0088093 0080.34301 CrossrefGoogle Scholar[14] A. Rényi, On mixing sequences of sets, Acta Math. Acad. Sci. Hungar., 9 (1958), 215–228 MR0098161 0089.13201 CrossrefGoogle Scholar[15] A. Rényi, On the central limit theorem for the sum of a random number of independent random variables, Acta Math. Acad. Sci. Hungar., 11 (1960), 97–102 (unbound insert) MR0115204 0091.14205 CrossrefGoogle Scholar[16] Herbert Robbins, The asymptotic distribution of the sum of a random number of random variables, Bull. Amer. Math. Soc., 54 (1948), 1151–1161 MR0027974 0034.22503 CrossrefGoogle Scholar[17] D. V. Lindley, The theory of queues with a single server, Proc. Cambridge Philos Soc., 48 (1952), 277–289 MR0046597 0046.35501 CrossrefGoogle Scholar[18] W. Richter, Zu einigen Konvergenzeigenschaften von Folgen zufälliger Elemente, Studia Math., 25 (1964/1965), 231–243 MR0174684 0138.12001 CrossrefGoogle Scholar[19] Wolfgang Richter, Das Null-Eins-Gesetz und ein Grenzwertsatz für zufällige Prozesse mit diskreter zufälliger Zeit, Wiss. Z. Techn. Univ. Dresden, 14 (1965), 497–504, Heft 2. 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W. Richter (1965) studied this question.
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