On Maxima and Minima of Partial Sums of Strongly Interchangeable Random Variables

1990 ◽  
Vol 4 (3) ◽  
pp. 319-332 ◽  
Author(s):  
Teunis J. Ott ◽  
J. George Shanthikumar
Keyword(s):  
Random Vector ◽  
Queueing Networks ◽  
Joint Distribution ◽  
Random Variables ◽  
Markov Property ◽  
Partial Sums ◽  
Random Vectors ◽  
Symmetric Distributions ◽  
Voice Communication ◽  
Special Instance

We introduce the concept of “strong interchangeability” of random vectors. Strongly interchangeable random vectors arise naturally in packetized voice channels, M/G/1 queues, symmetric queueing networks, and other standard symmetric distributions. We study some properties of strongly interchangeable random vectors. We show that if (X1, …, XN) is a strongly interchangeable random vector, then even though there is no Markov property, taboo probabilities can be used to compute the joint distribution of ŽN = min1≤n≤N σnk=IXk and ZN = max1≤n≤N σnk=1Xk. For a special instance of this problem that arises in packetized voice communication, it is shown that the resulting algorithm essentially has a complexity of order N4. When ( σnk=1Xk, n = 1,… N) is an associated random vector bound for the joint distribution of ŽN and ZN are obtained and applied to the packetized voice communication problem.

scholarly journals Stochastic Comparisons of Symmetric Supermodular Functions of Heterogeneous Random Vectors

2013 ◽  
Vol 50 (02) ◽  
pp. 464-474
Author(s):  
Antonio Di Crescenzo ◽  
Esther Frostig ◽  
Franco Pellerey
Keyword(s):  
Queueing Theory ◽  
Queueing Networks ◽  
Random Variables ◽  
Random Vectors ◽  
Convex Order ◽  
Stochastic Comparisons ◽  
Increasing Convex Order ◽  
Supermodular Functions ◽  
Stop Loss ◽  
Series Systems

Consider random vectors formed by a finite number of independent groups of independent and identically distributed random variables, where those of the last group are stochastically smaller than those of the other groups. Conditions are given such that certain functions, defined as suitable means of supermodular functions of the random variables of the vectors, are supermodular or increasing directionally convex. Comparisons based on the increasing convex order of supermodular functions of such random vectors are also investigated. Applications of the above results are then provided in risk theory, queueing theory, and reliability theory, with reference to (i) net stop-loss reinsurance premiums of portfolios from different groups of insureds, (ii) closed cyclic multiclass Gordon-Newell queueing networks, and (iii) reliability of series systems formed by units selected from different batches.

scholarly journals Joint Distributions of Counts of Strings in Finite Bernoulli Sequences

2012 ◽  
Vol 49 (3) ◽  
pp. 758-772 ◽  
Author(s):  
Fred W. Huffer ◽  
Jayaram Sethuraman
Keyword(s):  
Poisson Process ◽  
Random Vector ◽  
Joint Distribution ◽  
Infinite Sequence ◽  
Random Variables ◽  
Finite Sequence ◽  
Factorial Moments ◽  
Bernoulli Random Variables ◽  
Joint Distributions ◽  
Bernoulli Sequence

An infinite sequence (Y1, Y2,…) of independent Bernoulli random variables with P(Yi = 1) = a / (a + b + i - 1), i = 1, 2,…, where a > 0 and b ≥ 0, will be called a Bern(a, b) sequence. Consider the counts Z1, Z2, Z3,… of occurrences of patterns or strings of the form {11}, {101}, {1001},…, respectively, in this sequence. The joint distribution of the counts Z1, Z2,… in the infinite Bern(a, b) sequence has been studied extensively. The counts from the initial finite sequence (Y1, Y2,…, Yn) have been studied by Holst (2007), (2008b), who obtained the joint factorial moments for Bern(a, 0) and the factorial moments of Z1, the count of the string {1, 1}, for a general Bern(a, b). We consider stopping the Bernoulli sequence at a random time and describe the joint distribution of counts, which extends Holst's results. We show that the joint distribution of counts from a class of randomly stopped Bernoulli sequences possesses the mixture of independent Poissons property: there is a random vector conditioned on which the counts are independent Poissons. To obtain these results, we extend the conditional marked Poisson process technique introduced in Huffer, Sethuraman and Sethuraman (2009). Our results avoid previous combinatorial and induction methods which generally only yield factorial moments.

scholarly journals Joint Distributions of Counts of Strings in Finite Bernoulli Sequences

2012 ◽  
Vol 49 (03) ◽  
pp. 758-772 ◽  
Author(s):  
Fred W. Huffer ◽  
Jayaram Sethuraman
Keyword(s):  
Poisson Process ◽  
Random Vector ◽  
Joint Distribution ◽  
Infinite Sequence ◽  
Random Variables ◽  
Finite Sequence ◽  
Factorial Moments ◽  
Bernoulli Random Variables ◽  
Joint Distributions ◽  
Bernoulli Sequence

An infinite sequence (Y 1, Y 2,…) of independent Bernoulli random variables with P(Y i = 1) = a / (a + b + i - 1), i = 1, 2,…, where a > 0 and b ≥ 0, will be called a Bern(a, b) sequence. Consider the counts Z 1, Z 2, Z 3,… of occurrences of patterns or strings of the form {11}, {101}, {1001},…, respectively, in this sequence. The joint distribution of the counts Z 1, Z 2,… in the infinite Bern(a, b) sequence has been studied extensively. The counts from the initial finite sequence (Y 1, Y 2,…, Y n ) have been studied by Holst (2007), (2008b), who obtained the joint factorial moments for Bern(a, 0) and the factorial moments of Z 1, the count of the string {1, 1}, for a general Bern(a, b). We consider stopping the Bernoulli sequence at a random time and describe the joint distribution of counts, which extends Holst's results. We show that the joint distribution of counts from a class of randomly stopped Bernoulli sequences possesses the mixture of independent Poissons property: there is a random vector conditioned on which the counts are independent Poissons. To obtain these results, we extend the conditional marked Poisson process technique introduced in Huffer, Sethuraman and Sethuraman (2009). Our results avoid previous combinatorial and induction methods which generally only yield factorial moments.

ALMOST SURE CENTRAL LIMIT THEOREMS OF THE PARTIAL SUMS AND MAXIMA FROM COMPLETE AND INCOMPLETE SAMPLES OF STATIONARY SEQUENCES

2012 ◽  
Vol 12 (03) ◽  
pp. 1150026 ◽  
Author(s):  
ZUOXIANG PENG ◽  
BIN TONG ◽  
SARALEES NADARAJAH
Keyword(s):  
Central Limit ◽  
Random Vector ◽  
Limit Theorems ◽  
Random Sequence ◽  
Random Variables ◽  
Central Limit Theorems ◽  
Partial Sums ◽  
Stationary Sequences ◽  
Vector Formula ◽  
Weakly Dependent

Let (Xn) denote an independent and identically distributed random sequence. Let [Formula: see text] and Mn = max {X1, …, Xn} be its partial sum and maximum. Suppose that some of the random variables of X1, X2,… can be observed and denote by [Formula: see text] the maximum of observed random variables from the set {X1, …, Xn}. In this paper, we consider the joint limiting distribution of [Formula: see text] and the almost sure central limit theorems related to the random vector [Formula: see text]. Furthermore, we extend related results to weakly dependent stationary Gaussian sequences.

scholarly journals Stochastic Comparisons of Symmetric Supermodular Functions of Heterogeneous Random Vectors

2013 ◽  
Vol 50 (2) ◽  
pp. 464-474 ◽  
Author(s):  
Antonio Di Crescenzo ◽  
Esther Frostig ◽  
Franco Pellerey
Keyword(s):  
Queueing Theory ◽  
Queueing Networks ◽  
Random Variables ◽  
Random Vectors ◽  
Convex Order ◽  
Stochastic Comparisons ◽  
Increasing Convex Order ◽  
Supermodular Functions ◽  
Stop Loss ◽  
Series Systems

Consider random vectors formed by a finite number of independent groups of independent and identically distributed random variables, where those of the last group are stochastically smaller than those of the other groups. Conditions are given such that certain functions, defined as suitable means of supermodular functions of the random variables of the vectors, are supermodular or increasing directionally convex. Comparisons based on the increasing convex order of supermodular functions of such random vectors are also investigated. Applications of the above results are then provided in risk theory, queueing theory, and reliability theory, with reference to (i) net stop-loss reinsurance premiums of portfolios from different groups of insureds, (ii) closed cyclic multiclass Gordon-Newell queueing networks, and (iii) reliability of series systems formed by units selected from different batches.

A note on Spitzer's lemma and its application to the maximum of partial sums of dependent random variables

1976 ◽  
Vol 13 (2) ◽  
pp. 361-364 ◽  
Author(s):  
M. E. Solari ◽  
J. E. A. Dunnage
Keyword(s):  
Infinite Sequence ◽  
Random Variables ◽  
Finite Sequence ◽  
Partial Sums ◽  
Dependent Random Variables ◽  
Finite Segment ◽  
Exchangeable Random Variables ◽  
Partial Sum

We give an expression for the expectation of max (0, S1, …, Sn) where Sk is the kth partial sum of a finite sequence of exchangeable random variables X1, …, Xn. When the Xk are also independent, the formula we give has already been obtained by Spitzer; and when the sequence is a finite segment of an infinite sequence of exchangeable random variables, it is a consequence of a theorem of Hewitt.

Central Limit Theorems for Interchangeable Processes

1958 ◽  
Vol 10 ◽  
pp. 222-229 ◽  
Author(s):  
J. R. Blum ◽  
H. Chernoff ◽  
M. Rosenblatt ◽  
H. Teicher
Keyword(s):  
Stochastic Process ◽  
Probability Measure ◽  
Central Limit ◽  
Joint Distribution ◽  
Limit Theorems ◽  
Random Variables ◽  
Central Limit Theorems ◽  
Probability Measures ◽  
Image Position ◽  
Positive Integers

Let {Xn} (n = 1, 2 , …) be a stochastic process. The random variables comprising it or the process itself will be said to be interchangeable if, for any choice of distinct positive integers i 1, i 2, H 3 … , ik, the joint distribution of depends merely on k and is independent of the integers i 1, i 2, … , i k. It was shown by De Finetti (3) that the probability measure for any interchangeable process is a mixture of probability measures of processes each consisting of independent and identically distributed random variables.

On the ordered partial sums of real random variables

1977 ◽  
Vol 14 (1) ◽  
pp. 75-88 ◽  
Author(s):  
Lajos Takács
Keyword(s):  
Random Variables ◽  
Partial Sums ◽  
Markov Sequence ◽  
Stieltjes Transforms ◽  
Independent And Identically Distributed

In 1952 Pollaczek discovered a remarkable formula for the Laplace-Stieltjes transforms of the distributions of the ordered partial sums for a sequence of independent and identically distributed real random variables. In this paper Pollaczek's result is proved in a simple way and is extended for a semi-Markov sequence of real random variables.

scholarly journals Distribution of Minimal Path Lengths when Edge Lengths are Independent Heterogeneous Exponential Random Variables

2012 ◽  
Vol 49 (3) ◽  
pp. 895-900
Author(s):  
Sheldon M. Ross
Keyword(s):  
Joint Distribution ◽  
Shortest Paths ◽  
Random Variables ◽  
Simulated Data ◽  
Minimal Path ◽  
Exponential Random Variables ◽  
Path Lengths

We find the joint distribution of the lengths of the shortest paths from a specified node to all other nodes in a network in which the edge lengths are assumed to be independent heterogeneous exponential random variables. We also give an efficient way to simulate these lengths that requires only one generated exponential per node, as well as efficient procedures to use the simulated data to estimate quantities of the joint distribution.

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