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The weak law of large numbers for nonnegative summands

Published online by Cambridge University Press:  01 February 2019

Eugene Seneta*
Affiliation:
University of Sydney
*
School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia. Email address: [email protected]
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Abstract

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Khintchine's (necessary and sufficient) slowly varying function condition for the weak law of large numbers (WLLN) for the sum of n nonnegative, independent and identically distributed random variables is used as an overarching (sufficient) condition for the case that the number of summands is more generally [cn],cn→∞. Either the norming sequence {an},an→∞, or the number of summands sequence {cn}, can be chosen arbitrarily. This theorem generalizes results from a motivating branching process setting in which Khintchine's sufficient condition is automatically satisfied. A second theorem shows that Khintchine's condition is necessary for the generalized WLLN when it holds with cn→∞ and an→∞. Theorem 3, which is known, gives a necessary and sufficient condition for Khintchine's WLLN to hold with cn=n and an a specific function of n; it is extended to general cn subject to a growth restriction in Theorem 4. Section 6 returns to the branching process setting.

Type
Original Article
Copyright
Copyright © Applied Probability Trust 2018 

References

Athreya, K. B. (1971).A note on a functional equation arising in Galton‒Watson branching processes.J. Appl. Prob. 8,589598.Google Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987).Regular Variation.Cambridge University Press.Google Scholar
Bojanić, R. and Seneta, E. (1971).Slowly varying functions and asymptotic relations.J. Math. Anal. Appl. 34,302315.Google Scholar
Cohn, H. (1982).Another look at the finite mean supercritical Bienaymé‒Galton‒Watson process. In Essays in Statistical Science (J. Appl. Prob. Spec. Vol. 19A), eds J. Gani and E. J. Hannan, Applied Probability Trust,Sheffield, pp. 307312.Google Scholar
Csörgö, S. and Simons, G. (2008).Weak laws of large numbers for cooperative gamblers.Periodica Math. Hung. 57,3160.Google Scholar
Feller, W. (1966).An Introduction to Probability Theory and Its Applications, Vol. II.John Wiley,New York.Google Scholar
Feller, W. (1971).An Introduction to Probability Theory and Its Applications, Vol. II., 2nd edn.John Wiley,New York.Google Scholar
Gnedenko, B. V. and Kolmogorov, A. N. (1954).Limit Distributions for Sums of Independent Random Variables.Addison-Wesley,Cambridge, MA.Google Scholar
Jagers, P. (1975).Branching Processes with Biological Applications.John Wiley,London.Google Scholar
Khintchine, A. (1936).Su una legge dei grandi numeri generalizzata.Giornale dell'Istituto Italiano degli Attuari 7,365377.Google Scholar
Krasnoselskii, M. A. and Rutickii, Ya. B. (1961).Convex Functions and Orlicz Spaces.Noordhoff,Groningen.Google Scholar
Maller, R. A. (1978).Relative stability and the strong law of large numbers.Z. Wahrscheinlichkeitsth. 43,141148.Google Scholar
Rogozin, B. A. (1971).The distribution of the first ladder moment and height and fluctuation of a random walk.Theory Prob. Appl. 16,575595.Google Scholar
Seneta, E. (1970).On invariant measures for simple branching processes (summary).Bull. Austral. Math. Soc. 2,359362.Google Scholar
Seneta, E. (1971).Sequential criteria for regular variation.Quart. J. Math. Oxford Ser. 22,565570.Google Scholar
Seneta, E. (1974).Regularly varying functions in the theory of simple branching processes.Adv. Appl. Prob. 6,408420.Google Scholar
Seneta, E. (1975).Characterization by functional equations of branching process limit laws. In Statistical Distributions in Scientific Work: Characterizations and Applications, Vol. 3, eds G. P. Patil, S. Kotz and J. K. Ord, D. Reidel,Dordrecht, pp. 249254.Google Scholar
Seneta, E. (1976).Regularly Varying Functions (Lecture Notes Math. 508).Springer,Berlin.Google Scholar
Seneta, E. (2013).A tricentenary history of the law of large numbers.Bernoulli 19,10881121.Google Scholar