Some remarks on probability inequalities for sums of bounded convex random variables

1975 ◽  
Vol 12 (1) ◽  
pp. 155-158 ◽  
Author(s):  
M. Goldstein
Keyword(s):  
Convex Function ◽  
Exponential Convergence ◽  
Random Variables ◽  
Upper Bounds ◽  
Independent Random Variables ◽  
Probability Inequalities ◽  
Continuous Convex Function ◽  
Bounded Convex

Let X1, X2, · ··, Xn be independent random variables such that ai ≦ Xi ≦ bi, i = 1,2,…n. A class of upper bounds on the probability P(S−ES ≧ nδ) is derived where S = Σf(Xi), δ > 0 and f is a continuous convex function. Conditions for the exponential convergence of the bounds are discussed.

Some remarks on probability inequalities for sums of bounded convex random variables

1975 ◽  
Vol 12 (01) ◽  
pp. 155-158
Author(s):  
M. Goldstein
Keyword(s):  
Convex Function ◽  
Exponential Convergence ◽  
Random Variables ◽  
Upper Bounds ◽  
Independent Random Variables ◽  
Probability Inequalities ◽  
Continuous Convex Function ◽  
Bounded Convex

Let X 1, X 2, · ··, Xn be independent random variables such that ai ≦ Xi ≦ bi , i = 1,2,…n. A class of upper bounds on the probability P(S−ES ≧ nδ) is derived where S = Σf(Xi ), δ > 0 and f is a continuous convex function. Conditions for the exponential convergence of the bounds are discussed.

Some Inequalities of Partial Sums for Negatively Dependent Random Variables

2012 ◽  
Vol 195-196 ◽  
pp. 694-700
Author(s):  
Hai Wu Huang ◽  
Qun Ying Wu ◽  
Guang Ming Deng
Keyword(s):  
Random Variables ◽  
Independent Random Variables ◽  
Partial Sums ◽  
Exponential Inequalities ◽  
Dependent Random Variables ◽  
Probability Inequalities ◽  
Associated Random Variables ◽  
Negatively Dependent Random Variables

The main purpose of this paper is to investigate some properties of partial sums for negatively dependent random variables. By using some special numerical functions, and we get some probability inequalities and exponential inequalities of partial sums, which generalize the corresponding results for independent random variables and associated random variables. At last, exponential inequalities and Bernsteins inequality for negatively dependent random variables are presented.

A unification of some approaches to Poisson approximation

1990 ◽  
Vol 27 (03) ◽  
pp. 611-621 ◽  
Author(s):  
Hans-Jürgen Witte
Keyword(s):  
Complex Analysis ◽  
Random Variables ◽  
Random Variable ◽  
Poisson Approximation ◽  
Upper Bounds ◽  
Secretary Problem ◽  
Independent Random Variables ◽  
Poisson Random Variable ◽  
Semigroup Approach ◽  
Metric Distances

Let Sn be a sum of independent random variables. For the approximation of Sn by a Poisson random variable Y with the same mean, the complex analysis approaches based on generating functions and the semigroup approach are presented in a unified setting which permits us to refine Kerstan's complex analysis approach obtaining considerably sharper upper bounds for some metric distances of Sn and Y. These results are applied to some special Sn counting the records of an i.i.d. sequence of random variables which is important to various applied problems, for instance the secretary problem.

A unification of some approaches to Poisson approximation

1990 ◽  
Vol 27 (3) ◽  
pp. 611-621 ◽  
Author(s):  
Hans-Jürgen Witte
Keyword(s):  
Complex Analysis ◽  
Random Variables ◽  
Random Variable ◽  
Poisson Approximation ◽  
Upper Bounds ◽  
Secretary Problem ◽  
Independent Random Variables ◽  
Poisson Random Variable ◽  
Semigroup Approach ◽  
Metric Distances

Let Sn be a sum of independent random variables. For the approximation of Sn by a Poisson random variable Y with the same mean, the complex analysis approaches based on generating functions and the semigroup approach are presented in a unified setting which permits us to refine Kerstan's complex analysis approach obtaining considerably sharper upper bounds for some metric distances of Sn and Y. These results are applied to some special Sn counting the records of an i.i.d. sequence of random variables which is important to various applied problems, for instance the secretary problem.

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