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A bound for bivariate probability of large deviations

Published online by Cambridge University Press:  14 July 2016

Matthew Goldstein
Matthew Goldstein*
Affiliation:
Baruch College, City University of New York
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Abstract

Suppose (X1, Y1), (X2, Y2), …, (Xn, Yn) are independent random vectors such that a ≦ Xi ≦ b and a ≦ Yi ≦ b, i = 1, 2, …, n. An upper bound which exponentially converges to zero is derived for the probability Pr{Sx – nμx ≧ nt1;SY – nμY ≧ nt2} where Sx = Σ Xi, SY = Σ Yi,EYi = μY, EXi = μx and t1 > 0, t2 > 0. The bound is a function of the difference b — a, the correlation between Xi and Yi, μx and μY and t1 and t2.

Keywords

BIVARIATEUPPER BOUNDPROBABILITYEXPONENTIAL CONVERGENCE
Type
Short Communications
Information
Journal of Applied Probability , Volume 13 , Issue 2 , June 1976 , pp. 380 - 384
Copyright
Copyright © Applied Probability Trust 1976 

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References

[1] Agnew, R. A. (1972) Inequalities with application to economic risk analysis. J. Appl. Prob. 9, 441–444.Google Scholar
[2] Ben-Tal, A. and Hochman, E. (1972) More bounds on the expectation of a convex function of a random variable. J. Appl. Prob. 9, 803–812.Google Scholar
[3] Bernstein, S. N. (1924) Sur une modification de l'inequalité de Tchebichef. Ann. Sci. Inst. Math. Sav. Ukraine Ser. 3 1, 1–15.Google Scholar
[4] Brook, D. (1966) Bounds for moment generating functions and for extinction probabilities. J. Appl. Prob. 3, 171–178.Google Scholar
[5] Hoeffding, W. (1963) Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc. 58, 13–30.Google Scholar
[6] Mullen, K. (1973) Bernstein's inequality in the bivariate case. Canad. Math. Bull. 16, 83–86.CrossRefGoogle Scholar