A tool in establishing total variation convergence
Authors:
K. R. Parthasarathy and Ton Steerneman
Journal:
Proc. Amer. Math. Soc. 95 (1985), 626630
MSC:
Primary 60B10
Corrigendum:
Proc. Amer. Math. Soc. 99 (1987), 600.
MathSciNet review:
810175
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Abstract: Let be sequences of random variables where and are independent, in total variation and in distribution. For certain mappings sufficient conditions are given in order that in total variation. For example, if is the outcome space of the and , and if is absolutely continuous (with respect to Lebesgue measure), then in total variation.
 [P]
Patrick
Billingsley, Convergence of probability measures, John Wiley
& Sons, Inc., New YorkLondonSydney, 1968. MR 0233396
(38 #1718)
 [J]
Julius
R. Blum and Pramod
K. Pathak, A note on the zeroone law, Ann. Math. Statist.
43 (1972), 1008–1009. MR 0300314
(45 #9360)
 [A]
Alain
Hillion, Sur l’intégrale d’Hellinger et la
séparation asymptotique, C. R. Acad. Sci. Paris Sér. AB
283 (1976), no. 2, Aii, A61–A64. MR 0410897
(53 #14639)
 [S]
Shizuo
Kakutani, On equivalence of infinite product measures, Ann. of
Math. (2) 49 (1948), 214–224. MR 0023331
(9,340e)
 [T]
T.
Nemetz, Equivalenceorthogonality dichotomies of probability
measures, Limit theorems of probability theory (Colloq., Keszthely,
1974) NorthHolland, Amsterdam, 1975, pp. 183–191. Colloq.
Math. Soc. János Bolyai, Vol. 11. MR 0394852
(52 #15651)
 [W]
Wolfgang
Sendler, A note on the proof of the zeroone law of J. R. Blum and
P. K. Pathak: “A note on the zeroone law” (Ann. Math. Statist.
43 (1972), 1008–1009), Ann. Probability 3
(1975), no. 6, 1055–1058. MR 0380953
(52 #1850)
 [A]
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 (84h:28007), http://dx.doi.org/10.1090/S00029939198307022990
 [P]
 Billingsley (1968), Convergence of probability measures, Wiley, New York. MR 0233396 (38:1718)
 [J]
 R. Blum and P. K. Pathak (1972), A note on the zeroone law, Ann. Math. Statist. 43, 10081009. MR 0300314 (45:9360)
 [A]
 Hillion (1976), Sur l'intégrale Hellinger et la separation asymptotique, C. R. Acad. Sci. Paris Sér. A 283, 6164. MR 0410897 (53:14639)
 [S]
 Kakutani (1948), On equivalence of infinite product measures, Ann. of Math. (2) 49, 214224. MR 0023331 (9:340e)
 [T]
 Nemetz (1975), Equivalenceorthogonality dichotomies of probability measures, Colloquium Mathematica Societatis János Bolyai, 11, Limit Theorems in Probability Theory, Keszthely, 1974, Hungary; NorthHolland, Amsterdam. MR 0394852 (52:15651)
 [W]
 Sendler (1975), A note on the proof of the zeroone law of Blum and Pathak, Ann. Probab. 3, 10551058. MR 0380953 (52:1850)
 [A]
 G. M. Steerneman (1983), On the total variation and Hellinger distance between signed measures; an application to product measures, Proc. Amer. Math. Soc. 88, 684688. MR 702299 (84h:28007)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002993919850810175X
PII:
S 00029939(1985)0810175X
Keywords:
Total variation norm,
weak convergence
Article copyright:
© Copyright 1985
American Mathematical Society
