Abstract
Double arrays of random variables obtained by normalizing a sequence that is asymptotically close to a martingale difference sequence are considered, and conditions ensuring that the row sums converge in distribution to a mixture of normal distributions are found. The main condition is that the sums of squares in each row converge in probability to a random variable.
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Chatterji, S.D.: A principle of subsequences in probability theory: The central limit theorem. Advances in Math. 13, 31–54 (1974)
Billingsley, P.: Convergence of Probability Measures. New York: Wiley 1968
Doob, J.L.: Stochastic Processes. New York: Wiley 1953
Dvoretsky, A.: Central limit theorems for dependent random variables. Proc. Sixth Berkeley Sympos. Math. Statist. Probab. Berkeley: Univ. of Calif. Press 1972.
Eagleson, G.K.: Martingale convergence to mixtures of infinitely divisible laws. Ann. Probability 3, 557–562 (1975)
McLeish, D.L.: Dependent central limit theorems and invariance principles. Ann. Probability 2, 620–628 (1974)
Rootzén, H.: On the functional central limit theorem for martingales. Z. Wahrscheinlichkeitstheorie verw. Gebiete 38, 199–210 (1977)
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Research sponsored in part by the Office of Naval Research, Contract # N00014-75-C-0809
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Rootzén, H. A note on convergence to mixtures of normal distributions. Z. Wahrscheinlichkeitstheorie verw Gebiete 38, 211–216 (1977). https://doi.org/10.1007/BF00537264
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DOI: https://doi.org/10.1007/BF00537264