×
Liese, Friedrich

Estimates of Hellinger integrals of infinitely divisible distributions. (English) Zbl 0638.60002

Kybernetika 23, 227-238 (1987).
A functional called the f-divergence of two sigma-finite measures was introduced by J. Csiszar in Publ. Math. Inst. Hungar. Acad. Sci., Ser. A 8, 85-108 (1963; Zbl 0124.087). The present author shows that the f-divergence is a lower semicontinuous functional on the space of probability measures if this space is equipped with a topology which includes as special cases both the setwise convergence and the usual weak convergence on metric spaces. The author then uses this result to estimate Hellinger integrals and the variational distance of infinitely divisible distributions.
Reviewer: A.Mukherjea

Cited in 5 Documents

MSC:

60A10 Probabilistic measure theory
60B10 Convergence of probability measures
60E07 Infinitely divisible distributions; stable distributions

Keywords:

f-divergence of two sigma-finite measures; weak convergence on metric spaces; Hellinger integrals; infinitely divisible distributions

Citations:

Zbl 0124.087
PDFBibTeX XMLCite
\textit{F. Liese}, Kybernetika 23, 227--238 (1987; Zbl 0638.60002)
Full Text: EuDML

References:

[1] H. Chernoff: A measure of efficiency for tests of a hypothesis based on sum of observations. Ann. Math. Statist. 23 (1952), 493-507. · Zbl 0048.11804 · doi:10.1214/aoms/1177729330
[2] J. Csiszar: Eine informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizitat von Markoffschen Ketten. Magyar Tud. Akad. Mat. Kutato Int. Közl. 8 (1964), 85-108. · Zbl 0124.08703
[3] J. Csiszar: Information type measures of difference of probability distributions. Stud. Sci. Math. Hungar. 2 (1967), 229-318. · Zbl 0157.25802
[4] I. I. Gichman, A. B. Skorochod: Theory of Stochastic Processes (in Russian). Nauka, Moscow 1973.
[5] P. Groneboom J. Ooslerhoff, F. H. Ruymgaart: Large deviation theorems for empirical probability measures. Ann. Probab. 7 (1979), 553 - 586. · Zbl 0425.60021
[6] J. Kerstan K. Matthes, J. Mecke: Unbegrenzt teilbare Punktprozesse. Akademie Verlag, Berlin 1974. · Zbl 0287.60057
[7] O. Krafft, D. Plachky: Bounds for the power of likelihood ratio tests and their asymptotic properties. Ann. Math. Statist. 41 (1970), 5, 1646-1654. · Zbl 0214.18003 · doi:10.1214/aoms/1177696808
[8] C. M. Newman: On the orthogonality of independent increment processes. Topics in Probability Theory (D. W. Strock, S. R. S. Varadhan. Courant Institute of Mathematical Sciences, N. Y. U. 1973, pp. 93-111. · Zbl 0269.60055
[9] J. V. Prokhorov: Convergence of stochastic processes and limit theorems of probability theory (in Russian). Prob. Theory and Applic. 1, (1956) 177-237.
[10] A. Renyi: On measures of entropy and information. Proc. 4th Berkeley Symposium Vol. 1, Berkeley, California 1961, pp. 547-561. · Zbl 0106.33001
[11] I. Vajda: Limit theorems for total variation of Cartesian product measures. Stud. Sci. Math. Hungar. 6 (1973), 317-333. · Zbl 0243.62034
[12] I. Vajda: On the \(f\)-divergence and singularity of probability measures. Period. Math. Hungar. 2 (1972), 223-234. · Zbl 0248.62001 · doi:10.1007/BF02018663
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.