Abstract
Let {X n F n ,n = 0, 1, 2, …} be a martingale with X 0 = 0 a.s., \(X_n = \sum {_{i = 1}^n Y_i,n\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \geqslant } 1} \), n ≧ 1, and F n the σ-field generated by X 0, X 1, …, X n . Write
and suppose that there is a constant δ, with \(0 < \delta \leqq 1\), such that \(E|Y_n|^{2+2\delta} < \infty, n = 1, 2, \cdots \). It is the object of this paper to establish the following theorem on departure from normality.
Received January 19, 1970.
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References
Brown, B. M. (1970). Martingale central limit theorems. To appear Ann. Math. Statist. 42
Burkholder, D. L. (1966). Martingale transforms. Ann. Math. Statist. 37 1494–1504.
Strassen, V. (1967). Almost sure behaviour of sums of independent random variables and martingales. Proc. Fifth Berkeley Symp. Math. Statist. Prob. 2 315–343. Univ. of California Press.
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Heyde, C.C., Brown, B.M. (2010). On the Departure from Normality of a Certain Class of Martingales. In: Maller, R., Basawa, I., Hall, P., Seneta, E. (eds) Selected Works of C.C. Heyde. Selected Works in Probability and Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-5823-5_22
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DOI: https://doi.org/10.1007/978-1-4419-5823-5_22
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