## On the total variation and Hellinger distance between signed measures; an application to product measures

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- by Ton Steerneman PDF
- Proc. Amer. Math. Soc.
**88**(1983), 684-688 Request permission

## Abstract:

Firstly, the Hellinger metric on the set of probability measures on a measurable space is extended to the set of signed measures. An inequality between total variation and Hellinger metric due to Kraft is generalized to the case of signed measures. The inequality is used in order to derive a lower estimate concerning the total variation distance between products of signed measures. The lower bound depends on the total variation norms of the signed measures and the total variation distances between the total variation measures of the single components.## References

- K. Behnen and G. Neuhaus,
*A central limit theorem under contiguous alternatives*, Ann. Statist.**3**(1975), no. 6, 1349–1353. MR**388611** - 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 - Wassily Hoeffding and J. Wolfowitz,
*Distinguishability of sets of distributions. (The case of independent and identically distributed chance variables)*, Ann. Math. Statist.**29**(1958), 700–718. MR**95555**, DOI 10.1214/aoms/1177706531 - Charles Kraft,
*Some conditions for consistency and uniform consistency of statistical procedures*, Univ. California Publ. Statist.**2**(1955), 125–141. MR**73896** - R.-D. Reiss,
*Approximation of product measures with an application to order statistics*, Ann. Probab.**9**(1981), no. 2, 335–341. MR**606998** - 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 - Kôsaku Yosida,
*Functional analysis*, 5th ed., Grundlehren der Mathematischen Wissenschaften, Band 123, Springer-Verlag, Berlin-New York, 1978. MR**0500055**

## Additional Information

- © Copyright 1983 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**88**(1983), 684-688 - MSC: Primary 28A33; Secondary 46E27
- DOI: https://doi.org/10.1090/S0002-9939-1983-0702299-0
- MathSciNet review: 702299