Inequalities For Characteristic Functions Research Articles

Some inequalities for absolute moments

In this note, we obtain several inequalities for absolute moments of sums and differences of independent random variables, using one representation of absolute moments in terms of the characteristic function and inequalities for characteristic functions.

Some inequalities for Laplace transforms

Recently we established Matysiak and Szablowski's conjecture [V. Matysiak, P.J. Szablowski, Some inequalities for characteristic functions, Theory Probab. Appl. 45 (2001) 711–713] about a lower bound of real-valued characteristic functions. In this paper, we investigate the counterparts for Laplace transforms of non-negative random variables. Surprisingly, the resulting inequalities hold true on the right half-line. Besides, we show some more inequalities by applying the convex/concave properties of the remainder in Taylor's expansion for the exponential function.

Some inequalities for characteristic functions. II

Recently we established Matysiak and Szablowski's conjecture [V. Matysiak, P.J. Szablowski, Theory Probab. Appl. 45 (2001) 711–713] about a lower bound of real-valued characteristic functions. In this paper, applying an alternative approach we are able to give explicitly the ranges of argument for which the obtained inequalities hold true for general characteristic functions.

Some inequalities for characteristic functions

In a recent paper, Matysiak and Szablowski [V. Matysiak, P.J. Szablowski, Theory Probab. Appl. 45 (2001) 711–713] posed an interesting conjecture about a lower bound of real-valued characteristic functions. Under a suitable moment condition on distributions, we prove the conjecture to be true. The unified approach proposed here enables us to obtain new inequalities for characteristic functions. We also show by example that the improvement in the bounds is significant if more information about the distribution is available.

Some Inequalities for Characteristic Functions

Some Inequalities for Characteristic Functions

Lower and upper bounds for characteristic functions

In the paper, several known inequalities for characteristic functions are sharpened and several new ones are derived.

Weak stability of the theorem of I. Marcinkiewicz and some inequalities for characteristic functions

Weak stability is investigated for the well-known theorem of I. Marcinkiewicz which states that if P(t) is an algebraic polynomial, then exp (P(t)) can be a characteristic function only when P is a polynomial of degree no higher than two. Some other inequalities for characteristic functions are also established.

Asymptotic Expansions for Distributions of Sums of Identically Distributed Independent Lattice Random Variables

Previous article Next article Asymptotic Expansions for Distributions of Sums of Identically Distributed Independent Lattice Random VariablesA. BikyalisA. Bikyalishttps://doi.org/10.1137/1114060PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Carl-Gustav Esseen, Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian law, Acta Math., 77 (1945), 1–125 MR0014626 0060.28705 CrossrefGoogle Scholar[2] Harald Bergström, On the central limit theorem in the space $R\sb k, k>1$, Skand. Aktuarietidskr., 28 (1945), 106–127 MR0015704 Google Scholar[3] R. Ranga Rao, On the central limit theorem in $R\sb{k}$, Bull. Amer. Math. Soc., 67 (1961), 359–361 MR0133150 CrossrefGoogle Scholar[4] S. M. Sadikova, Some inequalities for characteristic functions, Theory Prob. Applications, 11 (1966), 441–447 10.1137/1111044 LinkGoogle Scholar[5] S. M. Sadikova, Two-dimensional analogues of an inequality of Esseen with applications to the central limit theorem, Theory Prob. Applications, 11 (1966), 325–335 10.1137/1111035 LinkGoogle Scholar[6] V. M. Zolotarev, A multi-dimensional analogue of the Berry-Esseen inequality for sets with bounded diameter, Theory Prob. Applications, 11 (1966), 447–454, 13 (1968), p. 192 10.1137/1111045 LinkGoogle Scholar[7] V. V. Sazanov, Masters Thesis, Studies on multi-dimensional and infinite-dimensional limit theorems of probability theory, Doctoral dissertation, Moscow, 1968, (In Russian.) Google Scholar[8] B. Bahr, On the central limit theorem in $R_{k}$, Ark. Mat., 7 (1966), 61–69 0221.60013 CrossrefGoogle Scholar[9] A. Bikjalis, Asymptotic expansions for the densities and distributions of sums of independent identically distributed random vectors, Litovsk. Mat. Sb., 8 (1968), 405–422, (In Russian.) MR0245069 Google Scholar[10] A. Bikyalis, Multi-dimensional characteristic functions, Lit. Mat. Sb., 8 (1968), 21–39, (In Russian.) Google Scholar[11] A. Bikjalis, The Euler-Maclaurin summation formula for functions of several variables, Litovsk. Mat. Sb., 8 (1968), 681–684, (In Russian.) MR0273262 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails A precise local limit theorem for the multinomial distribution and some applicationsJournal of Statistical Planning and Inference, Vol. 215 Cross Ref On the Le Cam distance between Poisson and Gaussian experiments and the asymptotic properties of Szasz estimatorsJournal of Mathematical Analysis and Applications, Vol. 499, No. 1 Cross Ref Asymptotic Cumulants of Functions of Multinomial Sample Proportions with Adjustment for Empty Cells15 July 2012 | Behaviormetrika, Vol. 39, No. 2 Cross Ref A Measure of “smoothness” of Multidimensional Distributions of Integer-Valued Random Vectors17 July 2006 | Theory of Probability & Its Applications, Vol. 30, No. 2AbstractPDF (503 KB)On approximating the distributions ofFriedman's 75-175-175-1and related statisticsMetrika, Vol. 24, No. 1 Cross Ref Volume 14, Issue 3| 1969Theory of Probability & Its Applications History Submitted:16 April 1968Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1114060Article page range:pp. 481-489ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Addendum: Some Inequalities for Characteristic Functions

Addendum: Some Inequalities for Characteristic Functions

Two-Dimensional Analogues of an Inequality of Esseen with Applications to the Central Limit Theorem

Two-Dimensional Analogues of an Inequality of Esseen with Applications to the Central Limit Theorem

Some Inequalities for Characteristic Functions

Some Inequalities for Characteristic Functions