Limit theorems for threshold-stopped random variables with applications to optimal stopping

1990 ◽  
Vol 22 (2) ◽  
pp. 396-411 ◽  
Author(s):  
Douglas P. Kennedy ◽  
Robert P. Kertz
Keyword(s):  
Asymptotic Behaviour ◽  
Optimal Stopping ◽  
Limit Theorems ◽  
Random Variables ◽  
Limit Distributions ◽  
Joint Distributions ◽  
Normalizing Constants

The extremal types theorem identifies asymptotic behaviour for the maxima of sequences of i.i.d. random variables. A parallel theorem is given which identifies the asymptotic behaviour of sequences of threshold-stopped random variables. Three new types of limit distributions arise, but normalizing constants remain the same as in the maxima case. Limiting joint distributions are also given for maxima and threshold-stopped random variables. Applications to the optimal stopping of i.i.d. random variables are given.

Limit theorems for threshold-stopped random variables with applications to optimal stopping

1990 ◽  
Vol 22 (02) ◽  
pp. 396-411 ◽  
Author(s):  
Douglas P. Kennedy ◽  
Robert P. Kertz
Keyword(s):  
Asymptotic Behaviour ◽  
Optimal Stopping ◽  
Limit Theorems ◽  
Random Variables ◽  
Limit Distributions ◽  
Joint Distributions ◽  
Normalizing Constants

The extremal types theorem identifies asymptotic behaviour for the maxima of sequences of i.i.d. random variables. A parallel theorem is given which identifies the asymptotic behaviour of sequences of threshold-stopped random variables. Three new types of limit distributions arise, but normalizing constants remain the same as in the maxima case. Limiting joint distributions are also given for maxima and threshold-stopped random variables. Applications to the optimal stopping of i.i.d. random variables are given.

Limit theorems for suprema, threshold-stopped random variables and last exits of i.i.d. random variables with costs and discounting, with applications to optimal stopping

1992 ◽  
Vol 24 (02) ◽  
pp. 241-266
Author(s):  
Douglas P. Kennedy ◽  
Robert P. Kertz
Keyword(s):  
Optimal Stopping ◽  
Limit Theorems ◽  
Continuous Mapping ◽  
Random Variables ◽  
Exit Times ◽  
Convergence Results ◽  
Process Convergence ◽  
Optimal Stopping Problems ◽  
The Cost ◽  
Linear Cost

For linear-cost-adjusted and geometric-discounted infinite sequences of i.i.d. random variables, point process convergence results are proved as the cost or discounting effect diminishes. These process convergence results are combined with continuous-mapping principles to obtain results on joint convergence of suprema and threshold-stopped random variables, and last-exit times and locations. Applications are made to several classical optimal stopping problems in these settings.

Limit theorems for suprema, threshold-stopped random variables and last exits of i.i.d. random variables with costs and discounting, with applications to optimal stopping

1992 ◽  
Vol 24 (2) ◽  
pp. 241-266 ◽  
Author(s):  
Douglas P. Kennedy ◽  
Robert P. Kertz
Keyword(s):  
Optimal Stopping ◽  
Limit Theorems ◽  
Continuous Mapping ◽  
Random Variables ◽  
Exit Times ◽  
Convergence Results ◽  
Process Convergence ◽  
Optimal Stopping Problems ◽  
The Cost ◽  
Linear Cost

For linear-cost-adjusted and geometric-discounted infinite sequences of i.i.d. random variables, point process convergence results are proved as the cost or discounting effect diminishes. These process convergence results are combined with continuous-mapping principles to obtain results on joint convergence of suprema and threshold-stopped random variables, and last-exit times and locations. Applications are made to several classical optimal stopping problems in these settings.

scholarly journals Two-choice optimal stopping

2004 ◽  
Vol 36 (04) ◽  
pp. 1116-1147 ◽  
Author(s):  
David Assaf ◽  
Larry Goldstein ◽  
Ester Samuel-Cahn
Keyword(s):  
Distribution Function ◽  
Asymptotic Behaviour ◽  
Large Class ◽  
Optimal Stopping ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Indicator Function ◽  
Stopping Rules ◽  
A Value ◽  
Independent Identically Distributed

Let X n ,…,X 1 be independent, identically distributed (i.i.d.) random variables with distribution function F. A statistician, knowing F, observes the X values sequentially and is given two chances to choose Xs using stopping rules. The statistician's goal is to stop at a value of X as small as possible. Let equal the expectation of the smaller of the two values chosen by the statistician when proceeding optimally. We obtain the asymptotic behaviour of the sequence for a large class of Fs belonging to the domain of attraction (for the minimum) 𝒟(G α), where G α(x) = [1 - exp(-x α)]1(x ≥ 0) (with 1(·) the indicator function). The results are compared with those for the asymptotic behaviour of the classical one-choice value sequence , as well as with the ‘prophet value’ sequence

A Poisson limit theorem for weakly exchangeable events

1979 ◽  
Vol 16 (04) ◽  
pp. 794-802 ◽  
Author(s):  
G. K. Eagleson
Keyword(s):  
Limit Theorem ◽  
Spatial Data ◽  
Random Variables ◽  
Random Variable ◽  
Poisson Random Variable ◽  
Limit Distributions ◽  
Joint Distributions ◽  
Poisson Limit ◽  
Image Position ◽  
Independent Identically Distributed

Let Y 1, Y2 , · ·· be a sequence of independent, identically distributed random variables, g some symmetric 0–1 function of m variables and set Silverman and Brown (1978) have shown that under certain conditions the statistic is asymptotically distributed as a Poisson random variable. They then use this result to derive limit distributions for various statistics, useful in the analysis of spatial data. In this paper, it is shown that Silverman and Brown's theorem holds under much weaker assumptions; assumptions which involve only the symmetry of the joint distributions of the X il…i m .

On the optimal stopping values induced by general dependence structures

2001 ◽  
Vol 38 (3) ◽  
pp. 672-684 ◽  
Author(s):  
Alfred Müller ◽  
Ludger Rüschendorf
Keyword(s):  
Optimal Stopping ◽  
Random Variables ◽  
Dependence Structure ◽  
Optimal Stopping Problem ◽  
Convex Order ◽  
Joint Distribution Function ◽  
Positive Dependence ◽  
Joint Distributions ◽  
Dependence Structures

The optimal stopping value of random variables X1,…,Xn depends on the joint distribution function of the random variables and hence on their marginals as well as on their dependence structure. The maximal and minimal values of the optimal stopping problem is determined within the class of all joint distributions with fixed marginals F1,…,Fn. They correspond to some sort of strong negative or positive dependence of the random variables. Any value inbetween these two extremes is attained for some dependence structures. The determination of the minimal value is based on some new ordering results for probability measures, in particular on lattice properties of stochastic orderings. We also identify properties of dependence structures leading to the minimal optimal stopping value. In the proofs we need an extension of Strassen's theorem on representation of the convex order which reveals that convex ordered distributions can be coupled by a two-step martingale (X,Y) with the additional property that Y is stochastically increasing in X.

On the optimal stopping values induced by general dependence structures

2001 ◽  
Vol 38 (03) ◽  
pp. 672-684 ◽  
Author(s):  
Alfred Müller ◽  
Ludger Rüschendorf
Keyword(s):  
Optimal Stopping ◽  
Random Variables ◽  
Dependence Structure ◽  
Optimal Stopping Problem ◽  
Convex Order ◽  
Joint Distribution Function ◽  
Positive Dependence ◽  
Joint Distributions ◽  
Dependence Structures

The optimal stopping value of random variables X 1,…,X n depends on the joint distribution function of the random variables and hence on their marginals as well as on their dependence structure. The maximal and minimal values of the optimal stopping problem is determined within the class of all joint distributions with fixed marginals F 1,…,F n . They correspond to some sort of strong negative or positive dependence of the random variables. Any value inbetween these two extremes is attained for some dependence structures. The determination of the minimal value is based on some new ordering results for probability measures, in particular on lattice properties of stochastic orderings. We also identify properties of dependence structures leading to the minimal optimal stopping value. In the proofs we need an extension of Strassen's theorem on representation of the convex order which reveals that convex ordered distributions can be coupled by a two-step martingale (X,Y) with the additional property that Y is stochastically increasing in X.

scholarly journals Limit Distributions for the Extreme Order Statistics

1978 ◽  
Vol 21 (4) ◽  
pp. 447-459 ◽  
Author(s):  
D. Mejzler
Keyword(s):  
Distribution Function ◽  
Asymptotic Behaviour ◽  
Order Statistics ◽  
Random Variables ◽  
Independent Random Variables ◽  
Maximal Term ◽  
Limit Distributions ◽  
Extreme Order Statistics

Let Xl, …, Xn be independent random variables with the same distribution function (df) F(x) and let Xln≤X2n≤…≤nr be the corresponding order statistics. The (df) of Xkn will be denoted always by Fkn(x). Many authors have investigated the asymptotic behaviour of the maximal term Xnn as n → ∞. Gnedenko [3] proved the following

scholarly journals Two-choice optimal stopping

2004 ◽  
Vol 36 (4) ◽  
pp. 1116-1147 ◽  
Author(s):  
David Assaf ◽  
Larry Goldstein ◽  
Ester Samuel-Cahn
Keyword(s):  
Distribution Function ◽  
Asymptotic Behaviour ◽  
Large Class ◽  
Optimal Stopping ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Indicator Function ◽  
Stopping Rules ◽  
A Value ◽  
Independent Identically Distributed

Let Xn,…,X1 be independent, identically distributed (i.i.d.) random variables with distribution function F. A statistician, knowing F, observes the X values sequentially and is given two chances to choose Xs using stopping rules. The statistician's goal is to stop at a value of X as small as possible. Let equal the expectation of the smaller of the two values chosen by the statistician when proceeding optimally. We obtain the asymptotic behaviour of the sequence for a large class of Fs belonging to the domain of attraction (for the minimum) 𝒟(Gα), where Gα(x) = [1 - exp(-xα)]1(x ≥ 0) (with 1(·) the indicator function). The results are compared with those for the asymptotic behaviour of the classical one-choice value sequence , as well as with the ‘prophet value’ sequence

A Poisson limit theorem for weakly exchangeable events

1979 ◽  
Vol 16 (4) ◽  
pp. 794-802 ◽  
Author(s):  
G. K. Eagleson
Keyword(s):  
Limit Theorem ◽  
Spatial Data ◽  
Random Variables ◽  
Random Variable ◽  
Poisson Random Variable ◽  
Limit Distributions ◽  
Joint Distributions ◽  
Poisson Limit ◽  
Image Position ◽  
Independent Identically Distributed

Let Y1, Y2, · ·· be a sequence of independent, identically distributed random variables, g some symmetric 0–1 function of m variables and set Silverman and Brown (1978) have shown that under certain conditions the statistic is asymptotically distributed as a Poisson random variable. They then use this result to derive limit distributions for various statistics, useful in the analysis of spatial data. In this paper, it is shown that Silverman and Brown's theorem holds under much weaker assumptions; assumptions which involve only the symmetry of the joint distributions of the Xil…im.

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