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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Limit theorems for variational sums

Authors: William N. Hudson and Howard G. Tucker
Journal: Trans. Amer. Math. Soc. 191 (1974), 405-426
MSC: Primary 60F05
MathSciNet review: 0358928
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Abstract: Limit theorems in the sense of a.s. convergence, convergence in $ {L_1}$-norm and convergence in distribution are proved for variational series. In the first two cases, if g is a bounded, nonnegative continuous function satisfying an additional assumption at zero, and if $ \{ X(t),0 \leq t \leq T\} $ is a stochastically continuous stochastic process with independent increments, with no Gaussian component and whose trend term is of bounded variation, then the sequence of variational sums of the form $ \Sigma _{k = 1}^ng(X({t_{nk}}) - X({t_{n,k - 1}}))$ is shown to converge with probability one and in $ {L_1}$-norm. Also, under the basic assumption that the distribution of the centered sum of independent random variables from an infinitesimal system converges to a (necessarily) infinitely divisible limit distribution, necessary and sufficient conditions are obtained for the joint distribution of the appropriately centered sums of the positive parts and of the negative parts of these random variables to converge to a bivariate infinitely divisible distribution. A characterization of all such limit distributions is obtained. An application is made of this result, using the first theorem, to stochastic processes with (not necessarily stationary) independent increments and with a Gaussian component.

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  • [1] S. M. Berman, Sign-invariant random variables and stochastic processes with sign-invariant increments, Trans. Amer. Math. Soc. 119 (1965), 216-243. MR 32 #3113. MR 0185651 (32:3113)
  • [2] R. M. Blumenthal and R. K. Getoor, Sample functions of stochastic processes with stationary independent increments, J. Math. Mech. 10 (1961), 493-516. MR 23 #A689. MR 0123362 (23:A689)
  • [3] R. Cogburn and H. G. Tucker, A limit theorem for a function of the increments of a decomposable process, Trans. Amer. Math. Soc. 99 (1961), 278-284. MR 23 #A681. MR 0123353 (23:A681)
  • [4] P. Greenwood and B. Fristedt, Variations of processes with stationary independent increments, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 23 (1972), 171-186. MR 0321197 (47:9730)
  • [5] M. Loève, A l'intérieur du problème central, Publ. Inst. Statist. Univ. Paris (homage à M. Paul Lévy) 6 (1957), 313-325. MR 20 #7336. MR 0100911 (20:7336)
  • [6] -, Probability theory. Foundations. Random sequences, University Series in Higher Math., Van Nostrand, Princeton, N. J., 1963. MR 34 #3596. MR 0203748 (34:3596)
  • [7] P. W. Millar, Path behavior of processes with stationary independent increments, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 17 (1971), 53-73. MR 0324781 (48:3130)
  • [8] E. L. Rvačeva, On domains of attraction of multi-dimensional distributions, L'vov. Gos. Univ. Uč. Zap. 29, Ser. Meh.-Mat. No. 6 (1954), 5-44; English transl., Selected Transl. Math Statist. and Probability, vol. 2, Amer. Math. Soc., Providence, R. I., 1962, pp. 183-205. MR 17, 864; 27 #782. MR 0076211 (17:864e)
  • [9] H. G. Tucker, On asymptotic independence of the partial sums of the positive parts and negative parts of independent random variables, Advances in Appl. Probability 3 (1971), 404-425. MR 44 #6017. MR 0288822 (44:6017)
  • [10] -, A graduate course in probability, Probability and Math. Statist., vol. 2, Academic Press, New York and London, 1967. MR 36 #4593. MR 0221541 (36:4593)
  • [11] S. J. Wolfe, On moments of infinitely divisible distributions, Ann. Math. Statist. 42 (1971), 2036-2043. MR 0300319 (45:9365)

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Keywords: Variational sums, infinitely divisible distributions, stochastic processes with independent increments
Article copyright: © Copyright 1974 American Mathematical Society

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