scholarly journals Stop rule inequalities for uniformly bounded sequences of random variables

1983 ◽  
Vol 278 (1) ◽  
pp. 197-197 ◽  
Author(s):  
Theodore P. Hill ◽  
Robert P. Kertz
Keyword(s):  
Random Variables ◽  
Stop Rule ◽  
Uniformly Bounded ◽  
Bounded Sequences

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 A note on H-convergence

2021 ◽  
Vol 17 ◽  
pp. 150
Author(s):  
O.P. Kogut ◽  
P.I. Kogut ◽  
T.N. Rudyanova
Keyword(s):  
Convergence Property ◽  
Sufficient Conditions ◽  
Weak Limit ◽  
Uniformly Bounded ◽  
Bounded Sequences

In this paper we study the H-convergence property for the uniformly bounded sequences of square matrices $\left\{ A_{\varepsilon} \in L^{\infty} (D; \mathbb{R}^{n \times n}) \right\}_{\varepsilon > 0}$. We derive the sufficient conditions, which guarantee the coincidence of $H$-limit with the weak-* limit of such sequences in $L^{\infty} (D; \mathbb{R}^{n \times n})$.

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).

scholarly journals A Stone-Weierstrass theorem for random functions

1970 ◽  
Vol 2 (2) ◽  
pp. 233-236 ◽  
Author(s):  
A. Mukherjea
Keyword(s):  
Random Function ◽  
Random Variables ◽  
Compact Set ◽  
Mean Square ◽  
Weierstrass Theorem ◽  
Random Functions ◽  
The Given ◽  
Uniformly Bounded

It is shown in this note that if Q is an algebra of uniformly bounded mean-square continuous real-valued random functions indexed in a compact set T, containing all bounded random variables and separating points of T (i.e., given t1 and t2 in T, there is a random function Xt in Q such that , then given any mean square continuous random function, there is a sequence in Q converging in mean square to the given random function uniformly on T.

Large correlated families of positive random variables

1988 ◽  
Vol 103 (1) ◽  
pp. 147-162 ◽  
Author(s):  
D. H. Fremlin
Keyword(s):  
Probability Space ◽  
Random Variables ◽  
Characteristic Functions ◽  
Measurable Functions ◽  
Uniformly Bounded

S. Argyros and N. Kalamidas([l], repeated in [2], Theorem 6·15) proved the following. If κ is a cardinal of uncountable cofinality, and 〈Eξ〉ξ<κ is a family of measurable sets in a probability space (X, μ) such that infξ<κμEξ = δ, and if n ≥ 1, , then there is a set Γ ⊆ κ such that #(Γ) = κ and μ(∩ξ∈IEξ) ≥ γ whenever I ⊆ ξ has n members. In Proposition 7 below I refine this result by (i) taking any γ < δn (which is best possible) and (ii) extending the result to infinite cardinals of countable cofinality, thereby removing what turns out to be an irrelevant restriction. The proof makes it natural to perform a further extension to general uniformly bounded families of non-negative measurable functions in place of characteristic functions.

scholarly journals A Note on the Large Deviation Principle for Discrete Associated Random Variables

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Przemysław Matuła ◽  
Maciej Ziemba
Keyword(s):  
Large Deviation Principle ◽  
Large Deviation ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Stationary Sequence ◽  
Arithmetic Means ◽  
Associated Random Variables ◽  
Partial Sum ◽  
Deviation Principle ◽  
Uniformly Bounded

We present sufficient conditions under which the sequence of arithmetic means Sn/n, where Sn=X1+⋯+Xn, is the partial sum built on a stationary sequence {Xn}n≥1 of associated integer-valued and uniformly bounded random variables, which satisfy the large deviation principle.

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