scholarly journals Probability Inequalities for the Sum of Random Variables When Sampling Without Replacement

2013 ◽  
Vol 2 (4) ◽  
Author(s):  
Kent Riggs ◽  
Dean Young ◽  
Jeremy J Becnel
Keyword(s):  
Random Variables ◽  
Sampling Without Replacement ◽  
Probability Inequalities ◽  
Sum Of Random Variables

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.

A central limit theorem for exchangeable variates with geometric applications

1973 ◽  
Vol 10 (4) ◽  
pp. 837-846 ◽  
Author(s):  
P. A. P. Moran
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Asymptotic Normality ◽  
Central Limit ◽  
Random Variables ◽  
Multinomial Distribution ◽  
Sum Of Random Variables ◽  
Th Cell ◽  
Occupancy Problems

A central limit theorem is proved for the sum of random variables Xr all having the same form of distribution and each of which depends on a parameter which is the number occurring in the rth cell of a multinomial distribution with equal probabilities in N cells and a total n, where nN–1 tends to a non-zero constant. This result is used to prove the asymptotic normality of the distribution of the fractional volume of a large cube which is not covered by N interpenetrating spheres whose centres are at random, and for which NV–1 tends to a non-zero constant. The same theorem can be used to prove asymptotic normality for a large number of occupancy problems.

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