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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A tool in establishing total variation convergence

Authors: K. R. Parthasarathy and Ton Steerneman
Journal: Proc. Amer. Math. Soc. 95 (1985), 626-630
MSC: Primary 60B10
Corrigendum: Proc. Amer. Math. Soc. 99 (1987), 600.
MathSciNet review: 810175
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Abstract: Let $ {X_0},{X_1},{X_2}, \ldots {\text{ and }}{Y_0},{Y_1},{Y_2}, \ldots $ be sequences of random variables where $ {X_n}$ and $ {Y_n}$ are independent, $ L{X_n} \to L{X_0}$ in total variation and $ L{Y_n} \to L{Y_0}$ in distribution. For certain mappings $ T$ sufficient conditions are given in order that $ LT\left( {{X_n},{Y_n}} \right) \to LT\left( {{X_0},{Y_0}} \right)$ in total variation. For example, if $ \left( {{{\mathbf{R}}^k},{B_k}} \right)$ is the outcome space of the $ {X_n}$ and $ {Y_n}$, and if $ L{X_0}$ is absolutely continuous (with respect to Lebesgue measure), then $ L\left( {{X_n} + {Y_n}} \right) \to L\left( {{X_0} + {Y_0}} \right)$ in total variation.

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Keywords: Total variation norm, weak convergence
Article copyright: © Copyright 1985 American Mathematical Society