scholarly journals Asymptotic Law of thejth Records in the Bivariate Exponential Case

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Grine Azedine
Keyword(s):  
Exponential Distribution ◽  
Stopping Time ◽  
Random Variables ◽  
Cumulative Distribution ◽  
Asymptotic Independence ◽  
Record Times ◽  
Bivariate Exponential ◽  
Exponential Case ◽  
Independent And Identically Distributed

We consider a sequence(Xi,Yi)1⩽i⩽nof independent and identically distributed random variables with joint cumulative distribution H(x,y), which has exponential marginalsF(x)andG(y)with parameterλ=1. We also assume thatXi(ω)≠Yi(ω),∀i∈N, andω∈Ω. We denoteRk(j)k⩾1andSk(j)k⩾1by the sequences of thejth records in the sequences(Xi)1⩽i⩽n,(Yi)1⩽i⩽n, respectively. The main result of of the paper is to prove the asymptotic independence ofRk(j)k⩾1andSk(j)k⩾1using the property of stopping time of thejth record times and that of the exponential distribution.

Prophet Inequalities for Independent and Identically Distributed Random Variables from an Unknown Distribution

2021 ◽  
Author(s):  
José Correa ◽  
Paul Dütting ◽  
Felix Fischer ◽  
Kevin Schewior
Keyword(s):  
Stopping Time ◽  
Random Variables ◽  
Optimal Solution ◽  
Random Order ◽  
Secretary Problem ◽  
Maximum Value ◽  
Optimal Stopping Theory ◽  
Long Time ◽  
Object Of Study ◽  
Independent And Identically Distributed

A central object of study in optimal stopping theory is the single-choice prophet inequality for independent and identically distributed random variables: given a sequence of random variables [Formula: see text] drawn independently from the same distribution, the goal is to choose a stopping time τ such that for the maximum value of α and for all distributions, [Formula: see text]. What makes this problem challenging is that the decision whether [Formula: see text] may only depend on the values of the random variables [Formula: see text] and on the distribution F. For a long time, the best known bound for the problem had been [Formula: see text], but recently a tight bound of [Formula: see text] was obtained. The case where F is unknown, such that the decision whether [Formula: see text] may depend only on the values of the random variables [Formula: see text], is equally well motivated but has received much less attention. A straightforward guarantee for this case of [Formula: see text] can be derived from the well-known optimal solution to the secretary problem, where an arbitrary set of values arrive in random order and the goal is to maximize the probability of selecting the largest value. We show that this bound is in fact tight. We then investigate the case where the stopping time may additionally depend on a limited number of samples from F, and we show that, even with o(n) samples, [Formula: see text]. On the other hand, n samples allow for a significant improvement, whereas [Formula: see text] samples are equivalent to knowledge of the distribution: specifically, with n samples, [Formula: see text] and [Formula: see text], and with [Formula: see text] samples, [Formula: see text] for any [Formula: see text].

scholarly journals Chaoticity for Multiclass Systems and Exchangeability Within Classes

2008 ◽  
Vol 45 (04) ◽  
pp. 1196-1203 ◽  
Author(s):  
Carl Graham
Keyword(s):  
Distributed System ◽  
Random Variables ◽  
Asymptotic Independence ◽  
Empirical Measure ◽  
Empirical Measures ◽  
Partial Exchangeability ◽  
Independent And Identically Distributed

Classical results for exchangeable systems of random variables are extended to multiclass systems satisfying a natural partial exchangeability assumption. It is proved that the conditional law of a finite multiclass system, given the value of the vector of the empirical measures of its classes, corresponds to independent uniform orderings of the samples within {each} class, and that a family of such systems converges in law {if and only if} the corresponding empirical measure vectors converge in law. As a corollary, convergence within {each} class to an infinite independent and identically distributed system implies asymptotic independence between {different} classes. A result implying the Hewitt-Savage 0-1 law is also extended.

scholarly journals Asymptotic Probabilities of an Exceedance Over Renewal Thresholds with an Application to Risk Theory

2005 ◽  
Vol 42 (01) ◽  
pp. 153-162 ◽  
Author(s):  
Christian Y. Robert
Keyword(s):  
Random Walk ◽  
Stopping Time ◽  
Random Variables ◽  
Risk Theory ◽  
Heavy Tailed Distribution ◽  
Negative Drift ◽  
Heavy Tailed ◽  
Independent And Identically Distributed

Let (Y n , N n ) n≥1 be independent and identically distributed bivariate random variables such that the N n are positive with finite mean ν and the Y n have a common heavy-tailed distribution F. We consider the process (Z n ) n≥1 defined by Z n = Y n - Σ n-1, where It is shown that the probability that the maximum M = max n≥1 Z n exceeds x is approximately as x → ∞, where F' := 1 - F. Then we study the integrated tail of the maximum of a random walk with long-tailed increments and negative drift over the interval [0, σ], defined by some stopping time σ, in the case in which the randomly stopped sum is negative. Finally, an application to risk theory is considered.

scholarly journals On the Joint Distribution of Stopping Times and Stopped Sums in Multistate Exchangeable Trials

2014 ◽  
Vol 51 (2) ◽  
pp. 483-491 ◽  
Author(s):  
M. V. Boutsikas ◽  
D. L. Antzoulakos ◽  
A. C. Rakitzis
Keyword(s):  
Joint Distribution ◽  
Stopping Time ◽  
Random Variables ◽  
Stopping Times ◽  
Independent Interest ◽  
Exchangeable Random Variables ◽  
Stopped Sums ◽  
Independent And Identically Distributed

Let T be a stopping time associated with a sequence of independent and identically distributed or exchangeable random variables taking values in {0, 1, 2, …, m}, and let ST,i be the stopped sum denoting the number of appearances of outcome 'i' in X1, …, XT, 0 ≤ i ≤ m. In this paper we present results revealing that, if the distribution of T is known, then we can also derive the joint distribution of (T, ST,0, ST,1, …, ST,m). Two applications, which have independent interest, are offered to illustrate the applicability and the usefulness of the main results.

Probability Density Function of the Product and Quotient of Two Correlated Exponential Random Variables

1986 ◽  
Vol 29 (4) ◽  
pp. 413-418 ◽  
Author(s):  
Henrick J. Malik ◽  
Roger Trudel
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Exponential Distribution ◽  
Random Variables ◽  
Type I ◽  
Type Ii ◽  
Exponential Distributions ◽  
Bivariate Exponential Distribution ◽  
Bivariate Exponential ◽  
Exponential Random Variables

AbstractThis article deals with the distributions of the product and the quotient of two correlated exponential random variables. We consider here three types of bivariate exponential distributions: Marshall-Olkin's bivariate exponential distribution, Gumbel's Type I bivariate exponential distribution, and Gumbel's Type II bivariate exponential distribution.

scholarly journals Chaoticity for Multiclass Systems and Exchangeability Within Classes

2008 ◽  
Vol 45 (4) ◽  
pp. 1196-1203 ◽  
Author(s):  
Carl Graham
Keyword(s):  
Distributed System ◽  
Random Variables ◽  
Asymptotic Independence ◽  
Empirical Measure ◽  
Empirical Measures ◽  
Partial Exchangeability ◽  
Independent And Identically Distributed

Classical results for exchangeable systems of random variables are extended to multiclass systems satisfying a natural partial exchangeability assumption. It is proved that the conditional law of a finite multiclass system, given the value of the vector of the empirical measures of its classes, corresponds to independent uniform orderings of the samples within {each} class, and that a family of such systems converges in law {if and only if} the corresponding empirical measure vectors converge in law. As a corollary, convergence within {each} class to an infinite independent and identically distributed system implies asymptotic independence between {different} classes. A result implying the Hewitt-Savage 0-1 law is also extended.

A note on record times

1977 ◽  
Vol 14 (3) ◽  
pp. 637-639 ◽  
Author(s):  
Mark Westcott
Keyword(s):  
Continuous Time ◽  
Random Variables ◽  
Record Times ◽  
Independent And Identically Distributed

This note indicates two alternative proofs for a result of Galambos and Seneta [1] on the distribution of ratios of successive record times in a sequence of independent and identically distributed random variables. Some consequences of one method of proof are discussed, and its analogues in a continuous time setting are described.

scholarly journals On the Joint Distribution of Stopping Times and Stopped Sums in Multistate Exchangeable Trials

2014 ◽  
Vol 51 (02) ◽  
pp. 483-491
Author(s):  
M. V. Boutsikas ◽  
D. L. Antzoulakos ◽  
A. C. Rakitzis
Keyword(s):  
Joint Distribution ◽  
Stopping Time ◽  
Random Variables ◽  
Stopping Times ◽  
Independent Interest ◽  
Exchangeable Random Variables ◽  
Stopped Sums ◽  
Independent And Identically Distributed

Let T be a stopping time associated with a sequence of independent and identically distributed or exchangeable random variables taking values in {0, 1, 2, …, m}, and let S T,i be the stopped sum denoting the number of appearances of outcome 'i' in X 1, …, X T , 0 ≤ i ≤ m. In this paper we present results revealing that, if the distribution of T is known, then we can also derive the joint distribution of (T, S T,0, S T,1, …, S T,m ). Two applications, which have independent interest, are offered to illustrate the applicability and the usefulness of the main results.

scholarly journals Some Characterizations of Exponential Distribution

2017 ◽  
Vol 6 (5) ◽  
pp. 132
Author(s):  
M. Ahsanullah ◽  
M. Z. Anis
Keyword(s):  
Distribution Function ◽  
Linear Combination ◽  
Lebesgue Measure ◽  
Exponential Distribution ◽  
Random Variables ◽  
Absolutely Continuous ◽  
Measure Distribution ◽  
Independent And Identically Distributed

There are some characterizations of the exponential distribution based on the relation of the maximum of two observations expressed as linear combination of the two observations. In this paper some generalizations of this known characterization of the exponential distribution using the relations between the maximum and minimum of  independent and identically distributed random variables having absolutely continuous (with respect to Lebesgue measure) distribution function will be presented.

scholarly journals Finiteness of Record values and Alternative Asymptotic Theory of Records with Atom Endpoints

2020 ◽  
Vol 15 (3) ◽  
pp. 2371-2385
Author(s):  
Gane Samb Lo ◽  
Harouna Sangaré ◽  
Cherif Mamadou Moctar Traoré ◽  
Mohammad Ahsanullah
Keyword(s):  
Negative Binomial ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Random Variable ◽  
Record Values ◽  
Cumulative Distribution ◽  
Hitting Times ◽  
Record Times ◽  
Binomial Random Variable ◽  
Upper Record Values

Asymptotic theories on record values and times, including central limit theorems, make sense only if the sequence of records values (and of record times) is infinite. If not, such theories could not even be an option. In this paper, we give necessary and/or sufficient conditions for the finiteness of the number of records. We prove, for example for iid real valued random variable, that strong upper record values are finite if and only if the upper endpoint is finite and is an atom of the common cumulative distribution function. The only asymptotic study left to us concerns the infinite sequence of hitting times of that upper endpoints, which by the way, is the sequence of weak record times. The asymptotic characterizations are made using negative binomial random variables and the dimensional multinomial random variables. Asymptotic comparison in terms of consistency bounds and confidence intervals on the different sequences of hitting times are provided. The example of a binomial random variable is given.

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