An extreme limit theorem for dependency bounds of normalized sums of random variables

Dependency bounds are lower and upper bounds on the probability distribution of functions of random variables when only their marginal distributions are known. Using the properties of T-conjugate transforms, we show that dependency bounds for normalized sums of random variables converge to step functions as the number of summands increases. The step functions are positioned at points, depending only on the extremes of the supports of the summands' distribution functions.

[1]  J. Glaz,et al.  Probability Inequalities for Multivariate Distributions with Dependence Structures , 1984 .

[2]  Richard A. Moynihan,et al.  Conjugate transforms for τT semigroups of probability distribution functions , 1980 .

[3]  J. D. Esary,et al.  Relationships Among Some Concepts of Bivariate Dependence , 1972 .

[4]  R. Bagnold,et al.  The nature and correlation of random distributions , 1983, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[5]  E. J. Gumbel,et al.  Statistics of Extremes. , 1960 .

[6]  Claudi Alsina,et al.  Some functional equations in the space of uniform distribution functions , 1981 .

[7]  Christian Genest,et al.  Copules archimédiennes et families de lois bidimensionnelles dont les marges sont données , 1986 .

[8]  G. Alefeld,et al.  Introduction to Interval Computation , 1983 .

[9]  A. Sklar,et al.  Inequalities Among Operations on Probability Distribution Functions , 1978 .

[10]  G. Kimeldorf,et al.  Positive dependence orderings , 1987 .

[11]  Richard Moynihan.  Conjugate transforms and limit theorems for $τ_{T}$ semigroups , 1981 .

[12]  G. D. Makarov.  Estimates for the Distribution Function of a Sum of Two Random Variables When the Marginal Distributions are Fixed , 1982 .

[13]  Abe Sklar,et al.  Random variables, joint distribution functions, and copulas , 1973, Kybernetika.

[14]  Robert C. Williamson,et al.  Probabilistic arithmetic. I. Numerical methods for calculating convolutions and dependency bounds , 1990, Int. J. Approx. Reason..

[15]  M. J. Frank.  On the simultaneous associativity ofF(x, y) andx+y−F(x, y) , 1978 .

[16]  E. Lehmann.  Some Concepts of Dependence , 1966 .

[17]  M. J. Frank.  On the simultaneous associativity of F(x, y) and x+y-F(x, y). (Short Communication). , 1978 .

[18]  M. J. Frank.  On the simultaneous associativity ofF(x,y) andx +y -F(x,y) , 1979 .

[19]  M. J. Frank,et al.  Best-possible bounds for the distribution of a sum — a problem of Kolmogorov , 1987 .

[20]  R. Moynihan,et al.  On τT semigroups of probability distribution functions II , 1977 .