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.References
- Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0233396
- Julius R. Blum and Pramod K. Pathak, A note on the zero-one law, Ann. Math. Statist. 43 (1972), 1008–1009. MR 300314, DOI 10.1214/aoms/1177692564
- Alain Hillion, Sur l’intégrale d’Hellinger et la séparation asymptotique, C. R. Acad. Sci. Paris Sér. A-B 283 (1976), no. 2, Aii, A61–A64. MR 410897
- Shizuo Kakutani, On equivalence of infinite product measures, Ann. of Math. (2) 49 (1948), 214–224. MR 23331, DOI 10.2307/1969123
- T. Nemetz, Equivalence-orthogonality dichotomies of probability measures, Limit theorems of probability theory (Colloq., Keszthely, 1974) Colloq. Math. Soc. János Bolyai, Vol. 11, North-Holland, Amsterdam, 1975, pp. 183–191. MR 0394852
- Wolfgang Sendler, A note on the proof of the zero-one law of J. R. Blum and P. K. Pathak: “A note on the zero-one law” (Ann. Math. Statist. 43 (1972), 1008–1009), Ann. Probability 3 (1975), no. 6, 1055–1058. MR 380953, DOI 10.1214/aop/1176996234
- Ton Steerneman, On the total variation and Hellinger distance between signed measures; an application to product measures, Proc. Amer. Math. Soc. 88 (1983), no. 4, 684–688. MR 702299, DOI 10.1090/S0002-9939-1983-0702299-0
Additional Information
- © Copyright 1985 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 95 (1985), 626-630
- MSC: Primary 60B10
- DOI: https://doi.org/10.1090/S0002-9939-1985-0810175-X
- MathSciNet review: 810175