Comparison of optimal value and constrained maxima expectations for independent random variables

1986 ◽  
Vol 18 (02) ◽  
pp. 311-340
Author(s):  
Robert P. Kertz
Keyword(s):  
Random Variables ◽  
Poisson Approximation ◽  
Independent Random Variables ◽  
Problem Analysis ◽  
Universal Inequalities ◽  
Optimal Value ◽  
Approximation Type ◽  
Stop Rule ◽  
Uniformly Bounded ◽  
Bounded Sequences

For all uniformly bounded sequences of independent random variablesX1, X2,···, a complete comparison is made between the optimal valueV(X1, X2, ···) = sup {EXt:tis an (a.e.) finite stop rule forX1,X2, ···} and, whereMi(X1,X2, ···) is theith largest order statistic forX1, X2, ··· In particular, fork>1, the set of ordered pairs {(x,y):x=V(X1, X2,···) andfor some independent random variablesX1, X2, ··· taking values in [0, 1]} is precisely the set, whereBk(0) = 0,Bk(1) = 1, and forThe result yields sharp, universal inequalities for independent random variables comparing two choice mechanisms, the mortal&s value of the gameV(X1, X2,···) and the prophet&s constrained maxima expectation of the game. Techniques of proof include probability- and convexity-based reductions; calculus-based, multivariate, extremal problem analysis; and limit theorems of Poisson-approximation type. Precise results are also given for finite sequences of independent random variables.

Comparison of optimal value and constrained maxima expectations for independent random variables

1986 ◽  
Vol 18 (2) ◽  
pp. 311-340 ◽  
Author(s):  
Robert P. Kertz
Keyword(s):  
Random Variables ◽  
Poisson Approximation ◽  
Independent Random Variables ◽  
Problem Analysis ◽  
Universal Inequalities ◽  
Optimal Value ◽  
Approximation Type ◽  
Stop Rule ◽  
Uniformly Bounded ◽  
Bounded Sequences

For all uniformly bounded sequences of independent random variables X1, X2, ···, a complete comparison is made between the optimal value V(X1, X2, ···) = sup {EXt:t is an (a.e.) finite stop rule for X1,X2, ···} and , where Mi(X1,X2, ···) is the ith largest order statistic for X1, X2, ··· In particular, for k> 1, the set of ordered pairs {(x, y):x = V(X1, X2, ···) and for some independent random variables X1, X2, ··· taking values in [0, 1]} is precisely the set , where Bk(0) = 0, Bk(1) = 1, and for The result yields sharp, universal inequalities for independent random variables comparing two choice mechanisms, the mortal&s value of the game V(X1, X2, ···) and the prophet&s constrained maxima expectation of the game . Techniques of proof include probability- and convexity-based reductions; calculus-based, multivariate, extremal problem analysis; and limit theorems of Poisson-approximation type. Precise results are also given for finite sequences of independent random variables.

scholarly journals An Efficient Time-Variant Reliability Analysis Method with Mixed Uncertainties

Algorithms ◽  
2021 ◽  
Vol 14 (8) ◽  
pp. 229
Author(s):  
Fangyi Li ◽  
Yufei Yan ◽  
Jianhua Rong ◽  
Houyao Zhu
Keyword(s):  
Reliability Analysis ◽  
Limit State ◽  
Random Variables ◽  
Equivalent Model ◽  
Independent Random Variables ◽  
Problem Analysis ◽  
Analysis Method ◽  
Calculation Algorithm ◽  
Reliability Problem ◽  
Interval Variables

In practical engineering, due to the lack of information, it is impossible to accurately determine the distribution of all variables. Therefore, time-variant reliability problems with both random and interval variables may be encountered. However, this kind of problem usually involves a complex multilevel nested optimization problem, which leads to a substantial computational burden, and it is difficult to meet the requirements of complex engineering problem analysis. This study proposes a decoupling strategy to efficiently analyze the time-variant reliability based on the mixed uncertainty model. The interval variables are treated with independent random variables that are uniformly distributed in their respective intervals. Then the time-variant reliability-equivalent model, containing only random variables, is established, to avoid multi-layer nesting optimization. The stochastic process is first discretized to obtain several static limit state functions at different times. The time-variant reliability problem is changed into the conventional time-invariant system reliability problem. First order reliability analysis method (FORM) is used to analyze the reliability of each time. Thus, an efficient and robust convergence hybrid time-variant reliability calculation algorithm is proposed based on the equivalent model. Finally, numerical examples shows the effectiveness of the proposed method.

Convergence rates in the law of large numbers. II

1968 ◽  
Vol 64 (2) ◽  
pp. 485-488 ◽  
Author(s):  
V. K. Rohatgi
Keyword(s):  
Convergence Rates ◽  
Random Variables ◽  
Almost Sure Convergence ◽  
Random Variable ◽  
Rates Of Convergence ◽  
Independent Random Variables ◽  
Large Numbers ◽  
Present Context ◽  
Image Position ◽  
Uniformly Bounded

Let {Xn: n ≥ 1} be a sequence of independent random variables and write Suppose that the random vairables Xn are uniformly bounded by a random variable X in the sense thatSet qn(x) = Pr(|Xn| > x) and q(x) = Pr(|Xn| > x). If qn ≤ q and E|X|r < ∞ with 0 < r < 2 then we have (see Loève(4), 242)where ak = 0, if 0 < r < 1, and = EXk if 1 ≤ r < 2 and ‘a.s.’ stands for almost sure convergence. the purpose of this paper is to study the rates of convergence ofto zero for arbitrary ε > 0. We shall extend to the present context, results of (3) where the case of identically distributed random variables was treated. The techniques used here are strongly related to those of (3).

Convergence of weighted sums of independent random variables

1971 ◽  
Vol 69 (2) ◽  
pp. 305-307 ◽  
Author(s):  
V. K. Rohatgi
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Independent Random Variables ◽  
Weighted Sums ◽  
Uniformly Bounded

1. Introduction. Let {Xk: k ≥ 1} be a sequence of independent, but not necessarily identically distributed, random variables. Suppose that the random variables Xn are uniformly bounded by a random variable X in the sense that(1) P(|Xn| ≥x) ≤ P(|X| ≥ x)for all x > 0. Write qn(x) = P(|Xn| ≥ x) and q(x) = P(|X| ≥ x).

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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