Asymptotic behaviour of a stopping time related to cumulative sum procedures and single-server queues

1987 ◽  
Vol 24 (1) ◽  
pp. 200-214 ◽  
Author(s):  
Wolfgang Stadje
Keyword(s):  
Asymptotic Behaviour ◽  
Queueing Theory ◽  
Asymptotic Properties ◽  
Stopping Time ◽  
Random Variables ◽  
Stopping Times ◽  
Single Server ◽  
Cumulative Sum ◽  
Single Server Queues

For a sequence ξ1, ξ2, · ·· of i.i.d. random variables let X0 = 0 and Xk = max(Xk–1 + ξ k, 0) for k = 1, 2, ···. Let . These stopping times are used in Page's (1954) one-sided cusum procedures and are also important in queueing theory. Various asymptotic properties of Nx are derived.

Asymptotic behaviour of a stopping time related to cumulative sum procedures and single-server queues

1987 ◽  
Vol 24 (01) ◽  
pp. 200-214 ◽  
Author(s):  
Wolfgang Stadje
Keyword(s):  
Asymptotic Behaviour ◽  
Queueing Theory ◽  
Asymptotic Properties ◽  
Stopping Time ◽  
Random Variables ◽  
Stopping Times ◽  
Single Server ◽  
Cumulative Sum ◽  
Single Server Queues

For a sequence ξ 1 , ξ 2 , · ·· of i.i.d. random variables let X 0 = 0 and X k = max(X k –1 + ξ k , 0) for k = 1, 2, ···. Let . These stopping times are used in Page's (1954) one-sided cusum procedures and are also important in queueing theory. Various asymptotic properties of N x are derived.

Correction

1987 ◽  
Vol 24 (4) ◽  
pp. 1014-1014
Keyword(s):  
Asymptotic Behaviour ◽  
Stopping Time ◽  
Single Server ◽  
Cumulative Sum ◽  
Single Server Queues

In his paper ‘Asymptotic behaviour of a stopping time related to cumulative sum procedures and single-server queues' (J. Appl. Prob. 24 (1987), 200–214), Wolfgang Stadje proved the following theorem:

Correction

1987 ◽  
Vol 24 (04) ◽  
pp. 1014
Keyword(s):  
Asymptotic Behaviour ◽  
Stopping Time ◽  
Single Server ◽  
Cumulative Sum ◽  
Single Server Queues

In his paper ‘Asymptotic behaviour of a stopping time related to cumulative sum procedures and single-server queues' (J. Appl. Prob. 24 (1987), 200–214), Wolfgang Stadje proved the following theorem:

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.

Optimal Stopping Under General Dependence Conditions

1983 ◽  
Vol 26 (3) ◽  
pp. 260-266
Author(s):  
M. Longnecker
Keyword(s):  
Time Series ◽  
Stochastic Processes ◽  
Optimal Stopping ◽  
Stopping Time ◽  
Random Variables ◽  
Stopping Times ◽  
Dependence Conditions ◽  
Series Type

AbstractLet {Xn} be a sequence of random variables, not necessarily independent or identically distributed, put and Mn =max0≤k≤n|Sk|. Effective bounds on in terms of assumed bounds on , are used to identify conditions under which an extended-valued stopping time τ exists. That is these inequalities are used to guarantee the existence of the stopping time τ such that E(ST/aτ) = supt ∈ T∞ E(|Sτ|/at), where T∞ denotes the class of randomized extended-valued stopping times based on S1, S2, … and {an} is a sequence of constants. Specific applications to stochastic processes of the time series type are considered.

scholarly journals On Asymptotics of Optimal Stopping Times

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 194
Author(s):  
Hugh N. Entwistle ◽  
Christopher J. Lustri ◽  
Georgy Yu. Sofronov
Keyword(s):  
Asymptotic Behaviour ◽  
Optimal Stopping ◽  
Upper Bound ◽  
Stopping Time ◽  
Stopping Times ◽  
Independent Random Variables ◽  
Algebraic Form ◽  
Asymptotic Expressions ◽  
Optimal Stopping Problems ◽  
Optimal Stopping Times

We consider optimal stopping problems, in which a sequence of independent random variables is drawn from a known continuous density. The objective of such problems is to find a procedure which maximizes the expected reward. In this analysis, we obtained asymptotic expressions for the expectation and variance of the optimal stopping time as the number of drawn variables became large. In the case of distributions with infinite upper bound, the asymptotic behaviour of these statistics depends solely on the algebraic power of the probability distribution decay rate in the upper limit. In the case of densities with finite upper bound, the asymptotic behaviour of these statistics depends on the algebraic form of the distribution near the finite upper bound. Explicit calculations are provided for several common probability density functions.

Optimal stopping problems with generalized objective functions

1990 ◽  
Vol 27 (04) ◽  
pp. 828-838
Author(s):  
T. P. Hill ◽  
D. P. Kennedy
Keyword(s):  
Optimal Stopping ◽  
Stopping Time ◽  
Random Variables ◽  
Stopping Times ◽  
Objective Functions ◽  
Monotone Functions ◽  
Optimal Stopping Problems ◽  
Universal Bounds ◽  
Optimal Stopping Times ◽  
Entire Vector

Optimal stopping of a sequence of random variables is studied, with emphasis on generalized objectives which may be non-monotone functions ofEXt, wheretis a stopping time, or may even depend on the entire vector (E[X1I{t=l}], · ··,E[XnI{t=n}]),such as the minimax objective to maximize minj{E[XjI{t=j}]}.Convexity is used to establish a prophet inequality and universal bounds for the optimal return, and a method for constructing optimal stopping times for such objectives is given.

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.

On the validity of Wald's equation

1994 ◽  
Vol 31 (04) ◽  
pp. 949-957 ◽  
Author(s):  
Markus Roters
Keyword(s):  
Random Walk ◽  
Stopping Time ◽  
Random Variables ◽  
Stopping Times ◽  
Stieltjes Transform ◽  
Positive Part ◽  
Negative Drift ◽  
The Given ◽  
Independent Identically Distributed

In this paper we review conditions under which Wald's equation holds, mainly if the expectation of the given stopping time is infinite. As a main result we obtain what is probably the weakest possible version of Wald's equation for the case of independent, identically distributed (i.i.d.) random variables. Moreover, we improve a result of Samuel (1967) concerning the existence of stopping times for which the expectation of the stopped sum of the underlying i.i.d. sequence of random variables does not exist. Finally, we show by counterexamples that it is impossible to generalize a theorem of Kiefer and Wolfowitz (1956) relating the moments of the supremum of a random walk with negative drift to moments of the positive part of X 1 to the case where the expectation of X 1 is —∞. Here, the Laplace–Stieltjes transform of the supremum of the considered random walk plays an important role.

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