A tool in establishing total variation convergence

Parthasarathy, K. R.; Steerneman, Ton

doi:10.1090/S0002-9939-1985-0810175-X

A tool in establishing total variation convergence
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by K. R. Parthasarathy and Ton Steerneman PDF

Proc. Amer. Math. Soc. 95 (1985), 626-630 Request permission

Corrigendum: Proc. Amer. Math. Soc. 99 (1987), 600.

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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