Summary
LetX 1,X 2,..., be i.i.d. random variables andS n=X 1+X 2+⋯. +X n. In this paper we simplify Rogozin's condition forS n/B n \(\xrightarrow{p}\) ±1for someB n→+∞, which generalises Hinčin's condition for relative stability ofS n. We also consider convergence of subsequences ofS n/B n. As an application of our methods, we extend a result of Chow and Robbins to show thatS n/B n→±1 a.s. for someB n→ + ∞ if and only if 0<¦EX¦≦E¦X¦<+ ∞.
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Chow, Y.S., Robbins, H.: On sums of independent random variables with infinite moments and “fair” games. Proc. Nat. Acad. Sci.47, 330–335 (1961)
Erickson, K.B., Kesten, H.: Strong and weak limit points of a normalised random walk. Ann. Probability2, 553–580 (1974)
Feller, W.: On regular variation and local limit theorems. Proc. 5th Berkeley Sympos. Math. Statist. Probab., Univ. Calif. Vol. i. pt 1, 373–388 (1965–66)
Feller, W.: An extension of the law of the iterated logarithm to variables without variance. J. Math. Mech.18, 343–355 (1968)
Feller, W.: An Introduction to Probability Theory and its Applications 2. 2nd ed. New York:Wiley 1971
Galambos, J., Seneta, E.: Regularly varying sequences. Proc. Amer. Math. Soc.41, 110–116 (1973)
Gnedenko, B.V., Kolmogorov, A.N.: Limit distributions for sums of independent random variables. 2nd ed. Reading Mass.: Addison-Wesley 1968
Gnedenko, B.V.: Limit theorems for sums of a random number of positive independent random variables. Proc. 6th Berkeley Sympos. Math. Statist. Probab. Univ. Calif.2, 537–549 (1970)
Hinčin, A.: Su una legge dei grandi numeri generalizzata. Giorn. Ist. Ital. Attuari, Anno VII, Vol.14, 365–377 (1936)
Lévy, P.: Théorie de l'Addition des Variables Aléatoires. Paris: Gautiers-Villars 1937
Rogozin, B.A.: Ladder moments and fluctuations. Theor. Probability Appl.16, 575–594 (1971)
Rogozin, B.A.: Relatively stable walks. Theor. Probability Appl.21, 375–379 (1976)
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Maller, R.A. Relative stability and the strong law of large numbers. Z. Wahrscheinlichkeitstheorie verw Gebiete 43, 141–148 (1978). https://doi.org/10.1007/BF00668456
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DOI: https://doi.org/10.1007/BF00668456