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April, 1972 An Invariance Principle for Martingales
Richard Drogin
Ann. Math. Statist. 43(2): 602-620 (April, 1972). DOI: 10.1214/aoms/1177692640


Many discrete martingales with increments in $L_2$ can be normalized so that the resulting trajectory is distributed approximately like Brownian motion. This paper will find all such martingales, subject to a natural side condition. Two techniques of normalization are possible: The usual one involving the partial sums of conditional variances of the increments given the past, and the analogous method using the partial sums of squares of the increments. This result is applied to obtain a central limit theorem and an arc sin law for dependent random variables.


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Richard Drogin. "An Invariance Principle for Martingales." Ann. Math. Statist. 43 (2) 602 - 620, April, 1972.


Published: April, 1972
First available in Project Euclid: 27 April 2007

zbMATH: 0243.60029
MathSciNet: MR303595
Digital Object Identifier: 10.1214/aoms/1177692640

Rights: Copyright © 1972 Institute of Mathematical Statistics


Vol.43 • No. 2 • April, 1972
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