scholarly journals On the continuity of the vector valued and set valued conditional expectations

1989 ◽  
Vol 12 (3) ◽  
pp. 477-486 ◽  
Author(s):  
Nikolaos S. Papageorgiou
Keyword(s):  
Conditional Expectation ◽  
Random Variables ◽  
Random Variable ◽  
Sufficient Condition ◽  
Conditional Expectations ◽  
Vector Valued

In this paper we study the dependence of the vector valued conditional expectation (for both single valued and set valued random variables), on the σ–field and random variable that determine it. So we prove that it is continuous for theL1(X)convergence of the sub–σ–fields and of the random variables. We also present a sufficient condition for theL1(X)–convergence of the sub–σ–fields. Then we extend the work to the set valued conditional expectation using the Kuratowski–Mosco (K–M) convergence and the convergence in the Δ–metric. We also prove a property of the set valued conditional expectation.

scholarly journals A relation between positive dependence of signal and the variability of conditional expectation given signal

2006 ◽  
Vol 43 (04) ◽  
pp. 1181-1185 ◽  
Author(s):  
Toshihide Mizuno
Keyword(s):  
Conditional Expectation ◽  
Marginal Distribution ◽  
Stochastic Order ◽  
Random Variable ◽  
Sufficient Condition ◽  
Conditional Distributions ◽  
Convex Order ◽  
Positive Dependence ◽  
Dispersive Order ◽  
Conditional Expectations

Let S 1 and S 2 be two signals of a random variable X, where G 1(s 1 ∣ x) and G 2(s 2 ∣ x) are their conditional distributions given X = x. If, for all s 1 and s 2, G 1(s 1 ∣ x) - G 2(s 2 ∣ x) changes sign at most once from negative to positive as x increases, then the conditional expectation of X given S 1 is greater than the conditional expectation of X given S 2 in the convex order, provided that both conditional expectations are increasing. The stochastic order of the sufficient condition is equivalent to the more stochastically increasing order when S 1 and S 2 have the same marginal distribution and, when S 1 and S 2 are sums of X and independent noises, it is equivalent to the dispersive order of the noises.

scholarly journals A relation between positive dependence of signal and the variability of conditional expectation given signal

2006 ◽  
Vol 43 (4) ◽  
pp. 1181-1185 ◽  
Author(s):  
Toshihide Mizuno
Keyword(s):  
Conditional Expectation ◽  
Marginal Distribution ◽  
Stochastic Order ◽  
Random Variable ◽  
Sufficient Condition ◽  
Conditional Distributions ◽  
Convex Order ◽  
Positive Dependence ◽  
Dispersive Order ◽  
Conditional Expectations

Let S1 and S2 be two signals of a random variable X, where G1(s1 ∣ x) and G2(s2 ∣ x) are their conditional distributions given X = x. If, for all s1 and s2, G1(s1 ∣ x) - G2(s2 ∣ x) changes sign at most once from negative to positive as x increases, then the conditional expectation of X given S1 is greater than the conditional expectation of X given S2 in the convex order, provided that both conditional expectations are increasing. The stochastic order of the sufficient condition is equivalent to the more stochastically increasing order when S1 and S2 have the same marginal distribution and, when S1 and S2 are sums of X and independent noises, it is equivalent to the dispersive order of the noises.

Some Reliability and Other Properties of Beta-Transformed Random Variables

2018 ◽  
Vol 33 (2) ◽  
pp. 83-92
Author(s):  
M. Sreehari ◽  
E. Sandhya ◽  
V. K. Mohamed Akbar
Keyword(s):  
Power Series ◽  
Geometric Distribution ◽  
Random Variables ◽  
Random Variable ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Power Series Distribution ◽  
Geometric Random Variable ◽  
Necessary And Sufficient

Abstract The reliability properties of beta-transformed random variables are discussed. A necessary and sufficient condition for a beta-transformed geometric random variable to follow a power series distribution is derived. It is shown that a beta-transformed member of the Katz family does not belong to the Katz family unless it is a geometric distribution, thereby getting a characterization.

scholarly journals Positive Dependence of Signals

2010 ◽  
Vol 47 (3) ◽  
pp. 893-897 ◽  
Author(s):  
Michel Denuit
Keyword(s):  
Conditional Expectation ◽  
Conditional Variance ◽  
Sufficient Condition ◽  
Convex Order ◽  
Positive Dependence ◽  
Increasing Convex Order ◽  
Quantity Of Interest ◽  
Conditional Expectations ◽  
Special Case

In this paper we further investigate the problem considered by Mizuno (2006) in the special case of identically distributed signals. Specifically, we first propose an alternative sufficient condition of crossing type for the convex order to hold between the conditional expectations given signal. Then, we prove that the bivariate (2,1)-increasing convex order ensures that the conditional expectations are ordered in the convex sense. Finally, the L2 distance between the quantity of interest and its conditional expectation given signal (or expected conditional variance) is shown to decrease when the strength of the dependence increases (as measured by the (2,1)-increasing convex order).

Subsequence principles for vector-valued random variables

1979 ◽  
Vol 86 (2) ◽  
pp. 301-312 ◽  
Author(s):  
D. J. H. Garling
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Principal Result ◽  
Moment Condition ◽  
Special Cases ◽  
Cesaro Averages ◽  
Image Position ◽  
Vector Valued ◽  
Subsequence Principle ◽  
Independent Identically Distributed

1. Introduction. Révész(8) has shown that if (fn) is a sequence of random variables, bounded in L2, there exists a subsequence (fnk) and a random variable f in L2 such that converges almost surely whenever . Komlós(5) has shown that if (fn) is a sequence of random variables, bounded in L1, then there is a subsequence (A*) with the property that the Cesàro averages of any subsequence converge almost surely. Subsequently Chatterji(2) showed that if (fn) is bounded in LP (where 0 < p ≤ 2) then there is a subsequence (gk) = (fnk) and f in Lp such thatalmost surely for every sub-subsequence. All of these results are examples of subsequence principles: a sequence of random variables, satisfying an appropriate moment condition, has a subsequence which satisfies some property enjoyed by sequences of independent identically distributed random variables. Recently Aldous(1), using tightness arguments, has shown that for a general class of properties such a subsequence principle holds: in particular, the results listed above are all special cases of Aldous' principal result.

scholarly journals Recovering a Random Variable from Conditional Expectations Using Reconstruction Algorithms for the Gauss Radon Transform

2019 ◽  
pp. 1-31
Author(s):  
Jeremy Becnel ◽  
Daniel Riser-Espinoza
Keyword(s):  
Euclidean Space ◽  
Computerized Tomography ◽  
Radon Transform ◽  
Gaussian Measure ◽  
Random Variables ◽  
Random Variable ◽  
Dimensional Euclidean Space ◽  
Computer Implementation ◽  
Reconstruction Algorithms ◽  
Conditional Expectations

The Radon transform maps a function on n-dimensional Euclidean space onto its integral over a hyperplane. The fields of modern computerized tomography and medical imaging are fundamentally based on the Radon transform and the computer implementation of the inversion, or reconstruction, techniques of the Radon transform. In this work we use the Radon transform with a Gaussian measure to recover random variables from their conditional expectations. We derive reconstruction algorithms for random variables of unbounded support from samples of conditional expectations and discuss the error inherent in each algorithm.

scholarly journals Positive Dependence of Signals

2010 ◽  
Vol 47 (03) ◽  
pp. 893-897 ◽  
Author(s):  
Michel Denuit
Keyword(s):  
Conditional Expectation ◽  
Conditional Variance ◽  
Sufficient Condition ◽  
Convex Order ◽  
Positive Dependence ◽  
Increasing Convex Order ◽  
Quantity Of Interest ◽  
Conditional Expectations ◽  
Special Case

In this paper we further investigate the problem considered by Mizuno (2006) in the special case of identically distributed signals. Specifically, we first propose an alternative sufficient condition of crossing type for the convex order to hold between the conditional expectations given signal. Then, we prove that the bivariate (2,1)-increasing convex order ensures that the conditional expectations are ordered in the convex sense. Finally, the L 2 distance between the quantity of interest and its conditional expectation given signal (or expected conditional variance) is shown to decrease when the strength of the dependence increases (as measured by the (2,1)-increasing convex order).

scholarly journals Some Characterization and Relations Based on K-th Lower Record Values

2016 ◽  
Vol 7 (1) ◽  
pp. 36-44
Author(s):  
Ali A. Al-Shomrani
Keyword(s):  
Conditional Expectation ◽  
Recurrence Relations ◽  
Sufficient Conditions ◽  
Random Variable ◽  
Record Values ◽  
Distribution Functions ◽  
Conditional Expectations ◽  
Lower Record ◽  
Lower Record Values ◽  
Necessary And Sufficient

In this paper, we obtain certain expressions and recurrence relations for two general classes of distributions based on some conditional expectations of k-th lower record values. We consider the necessary and sufficient conditions such that these conditional expectations hold for some distribution functions. Furthermore, an expression of conditional expectation of other general class of distributions through truncated moments of some random variable is considered. Some distributions as examples of these general classes are shown in Tables1and2accordingly.

Optimality of myopic stopping times for geometric discounting

1988 ◽  
Vol 25 (02) ◽  
pp. 437-443 ◽  
Author(s):  
Bert Fristedt ◽  
Donald A. Berry
Keyword(s):  
Beta Distribution ◽  
Conditional Expectation ◽  
Random Variables ◽  
Random Variable ◽  
Discount Factor ◽  
Stopping Times ◽  
Bernoulli Random Variables

Consider a sequence of conditionally independent Bernoulli random variables taking on the values 1 and − 1. The objective is to stop the sequence in order to maximize the discounted sum. Suppose the Bernoulli parameter has a beta distribution with integral parameters. It is optimal to stop when the conditional expectation of the next random variable is negative provided the discount factor is less than or equal to . Moreover, is best possible. The case where the parameters of the beta distribution are arbitrary positive numbers is also treated.

Optimality of myopic stopping times for geometric discounting

1988 ◽  
Vol 25 (2) ◽  
pp. 437-443 ◽  
Author(s):  
Bert Fristedt ◽  
Donald A. Berry
Keyword(s):  
Beta Distribution ◽  
Conditional Expectation ◽  
Random Variables ◽  
Random Variable ◽  
Discount Factor ◽  
Stopping Times ◽  
Bernoulli Random Variables

Consider a sequence of conditionally independent Bernoulli random variables taking on the values 1 and − 1. The objective is to stop the sequence in order to maximize the discounted sum. Suppose the Bernoulli parameter has a beta distribution with integral parameters. It is optimal to stop when the conditional expectation of the next random variable is negative provided the discount factor is less than or equal to . Moreover, is best possible. The case where the parameters of the beta distribution are arbitrary positive numbers is also treated.

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