On the maximum and its uniqueness for geometric random samples

1990 ◽  
Vol 27 (03) ◽  
pp. 598-610 ◽  
Author(s):  
F. Thomas Bruss ◽  
Colm Art O'cinneide
Keyword(s):  
Asymptotic Behavior ◽  
Random Variables ◽  
Expected Value ◽  
Asymptotic Result ◽  
Random Samples ◽  
Original Motivation ◽  
Independent Identically Distributed

Given n independent, identically distributed random variables, let ρ n denote the probability that the maximum is unique. This probability is clearly unity if the distribution of the random variables is continuous. We explore the asymptotic behavior of the ρ n 's in the case of geometric random variables. We find a function Φsuch that (ρ n – Φ(n)) → 0 as n →∞. In particular, we show that ρ n does not converge as n →∞. We derive a related asymptotic result for the expected value of the maximum of the sample. These results arose out of a random depletion model due to Bajaj, which was the original motivation for this paper and which is included.

On the maximum and its uniqueness for geometric random samples

1990 ◽  
Vol 27 (3) ◽  
pp. 598-610 ◽  
Author(s):  
F. Thomas Bruss ◽  
Colm Art O'cinneide
Keyword(s):  
Asymptotic Behavior ◽  
Random Variables ◽  
Expected Value ◽  
Asymptotic Result ◽  
Random Samples ◽  
Original Motivation ◽  
Independent Identically Distributed

Given n independent, identically distributed random variables, let ρ n denote the probability that the maximum is unique. This probability is clearly unity if the distribution of the random variables is continuous. We explore the asymptotic behavior of the ρ n's in the case of geometric random variables. We find a function Φsuch that (ρ n – Φ(n)) → 0 as n →∞. In particular, we show that ρ n does not converge as n →∞. We derive a related asymptotic result for the expected value of the maximum of the sample. These results arose out of a random depletion model due to Bajaj, which was the original motivation for this paper and which is included.

A note on the waiting times between record observations

1969 ◽  
Vol 6 (03) ◽  
pp. 711-714 ◽  
Author(s):  
Paul T. Holmes ◽  
William E. Strawderman
Keyword(s):  
Distribution Function ◽  
Infinite Sequence ◽  
Waiting Times ◽  
Continuous Distribution ◽  
Random Variables ◽  
Expected Value ◽  
Underlying Distribution ◽  
Upper Record ◽  
Image Position ◽  
Independent Identically Distributed

Let X 1, X 2, X 3,··· be independent, identically distributed random variables with a continuous distribution function and let the sequence of indices {Vr } be defined as follows: and for r ≧ 1, V r is the trial on which the rth (upper) record observation occurs. {V r} will be an infinite sequence of random variables since the underlying distribution function of the X's is continuous. It is well known that the expected value of V r. is infinite for every r (see, for example, Feller [1], page 15). Also define and for r > 1 δr is the number of trials between the (r - l)th and the rth record. The distributions of the random variables Vr and δ r do not depend on the distribution of the original random variables. It can be shown (see Neuts [2], page 206 or Tata 1[4], page 26) that The following theorem is due to Neuts [2].

scholarly journals Online Selection of Alternating Subsequences from a Random Sample

2011 ◽  
Vol 48 (4) ◽  
pp. 1114-1132 ◽  
Author(s):  
Alessandro Arlotto ◽  
Robert W. Chen ◽  
Lawrence A. Shepp ◽  
J. Michael Steele
Keyword(s):  
Asymptotic Behavior ◽  
Random Sample ◽  
Random Sequence ◽  
Random Variables ◽  
Sequential Selection ◽  
Exact Asymptotic Behavior ◽  
Full Knowledge ◽  
Independent Identically Distributed ◽  
Selection Of

We consider sequential selection of an alternating subsequence from a sequence of independent, identically distributed, continuous random variables, and we determine the exact asymptotic behavior of an optimal sequentially selected subsequence. Moreover, we find (in a sense we make precise) that a person who is constrained to make sequential selections does only about 12 percent worse than a person who can make selections with full knowledge of the random sequence.

scholarly journals Online Selection of Alternating Subsequences from a Random Sample

2011 ◽  
Vol 48 (04) ◽  
pp. 1114-1132 ◽  
Author(s):  
Alessandro Arlotto ◽  
Robert W. Chen ◽  
Lawrence A. Shepp ◽  
J. Michael Steele
Keyword(s):  
Asymptotic Behavior ◽  
Random Sample ◽  
Random Sequence ◽  
Random Variables ◽  
Sequential Selection ◽  
Exact Asymptotic Behavior ◽  
Full Knowledge ◽  
Independent Identically Distributed ◽  
Selection Of

We consider sequential selection of an alternating subsequence from a sequence of independent, identically distributed, continuous random variables, and we determine the exact asymptotic behavior of an optimal sequentially selected subsequence. Moreover, we find (in a sense we make precise) that a person who is constrained to make sequential selections does only about 12 percent worse than a person who can make selections with full knowledge of the random sequence.

A note on the waiting times between record observations

1969 ◽  
Vol 6 (3) ◽  
pp. 711-714 ◽  
Author(s):  
Paul T. Holmes ◽  
William E. Strawderman
Keyword(s):  
Distribution Function ◽  
Infinite Sequence ◽  
Waiting Times ◽  
Continuous Distribution ◽  
Random Variables ◽  
Expected Value ◽  
Underlying Distribution ◽  
Upper Record ◽  
Image Position ◽  
Independent Identically Distributed

Let X1,X2,X3,··· be independent, identically distributed random variables with a continuous distribution function and let the sequence of indices {Vr} be defined as follows: and for r ≧ 1, Vr is the trial on which the rth (upper) record observation occurs. {Vr} will be an infinite sequence of random variables since the underlying distribution function of the X's is continuous. It is well known that the expected value of Vr. is infinite for every r (see, for example, Feller [1], page 15). Also define and for r > 1 δr is the number of trials between the (r - l)th and the rth record. The distributions of the random variables Vr and δr do not depend on the distribution of the original random variables. It can be shown (see Neuts [2], page 206 or Tata 1[4], page 26) that The following theorem is due to Neuts [2].

A Note on the Accuracy of Normal Approximation of Random Quantities

2021 ◽  
Vol 73 (1) ◽  
pp. 62-67
Author(s):  
Ibrahim A. Ahmad ◽  
A. R. Mugdadi
Keyword(s):  
Sample Size ◽  
Order Statistic ◽  
Normal Approximation ◽  
Order Approximation ◽  
Random Variables ◽  
Random Variable ◽  
Limiting Distribution ◽  
Exact Order ◽  
Fixed Sample ◽  
Independent Identically Distributed

For a sequence of independent, identically distributed random variable (iid rv's) [Formula: see text] and a sequence of integer-valued random variables [Formula: see text], define the random quantiles as [Formula: see text], where [Formula: see text] denote the largest integer less than or equal to [Formula: see text], and [Formula: see text] the [Formula: see text]th order statistic in a sample [Formula: see text] and [Formula: see text]. In this note, the limiting distribution and its exact order approximation are obtained for [Formula: see text]. The limiting distribution result we obtain extends the work of several including Wretman[Formula: see text]. The exact order of normal approximation generalizes the fixed sample size results of Reiss[Formula: see text]. AMS 2000 subject classification: 60F12; 60F05; 62G30.

scholarly journals A counterexample to the existence of a general central limit theorem for pairwise independent identically distributed random variables

2021 ◽  
Vol 499 (1) ◽  
pp. 124982
Author(s):  
Benjamin Avanzi ◽  
Guillaume Boglioni Beaulieu ◽  
Pierre Lafaye de Micheaux ◽  
Frédéric Ouimet ◽  
Bernard Wong
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Variables ◽  
Independent Identically Distributed
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