scholarly journals An urn model arising from all-optical networks

2006 ◽  
Vol 43 (04) ◽  
pp. 952-966
Author(s):  
John A. Morrison
Keyword(s):  
Positive Integer ◽  
Explicit Expression ◽  
Optical Networks ◽  
Generating Function ◽  
Random Variables ◽  
Urn Model ◽  
Factorial Moments ◽  
Occupancy Model ◽  
All Optical ◽  
Independent Identically Distributed

An occupancy model that has arisen in the investigation of randomized distributed schedules in all-optical networks is considered. The model consists of B initially empty urns, and at stage j of the process d j ≤ B balls are placed in distinct urns with uniform probability. Let M i (j) denote the number of urns containing i balls at the end of stage j. An explicit expression for the joint factorial moments of M 0(j) and M 1(j) is obtained. A multivariate generating function for the joint factorial moments of M i (j), 0 ≤ i ≤ I, is derived (where I is a positive integer). Finally, the case in which the d j , j ≥ 1, are independent, identically distributed random variables is investigated.

scholarly journals An urn model arising from all-optical networks

2006 ◽  
Vol 43 (4) ◽  
pp. 952-966
Author(s):  
John A. Morrison
Keyword(s):  
Positive Integer ◽  
Explicit Expression ◽  
Optical Networks ◽  
Generating Function ◽  
Random Variables ◽  
Urn Model ◽  
Factorial Moments ◽  
Occupancy Model ◽  
All Optical ◽  
Independent Identically Distributed

An occupancy model that has arisen in the investigation of randomized distributed schedules in all-optical networks is considered. The model consists of B initially empty urns, and at stage j of the process dj ≤ B balls are placed in distinct urns with uniform probability. Let Mi(j) denote the number of urns containing i balls at the end of stage j. An explicit expression for the joint factorial moments of M0(j) and M1(j) is obtained. A multivariate generating function for the joint factorial moments of Mi(j), 0 ≤ i ≤ I, is derived (where I is a positive integer). Finally, the case in which the dj, j ≥ 1, are independent, identically distributed random variables is investigated.

A clustering law for some discrete order statistics

2003 ◽  
Vol 40 (01) ◽  
pp. 226-241 ◽  
Author(s):  
Sunder Sethuraman
Keyword(s):  
Distribution Function ◽  
Positive Integer ◽  
Order Statistics ◽  
Law Of Large Numbers ◽  
Asymptotic Properties ◽  
Random Variables ◽  
Large Numbers ◽  
The Law ◽  
Sample Maximum ◽  
Independent Identically Distributed

Let X 1, X 2, …, X n be a sequence of independent, identically distributed positive integer random variables with distribution function F. Anderson (1970) proved a variant of the law of large numbers by showing that the sample maximum moves asymptotically on two values if and only if F satisfies a ‘clustering’ condition, In this article, we generalize Anderson's result and show that it is robust by proving that, for any r ≥ 0, the sample maximum and other extremes asymptotically cluster on r + 2 values if and only if Together with previous work which considered other asymptotic properties of these sample extremes, a more detailed asymptotic clustering structure for discrete order statistics is presented.

An Enumeration Problem Related to the Number of Labelled Bi-Coloured Graphs

1961 ◽  
Vol 13 ◽  
pp. 217-220 ◽  
Author(s):  
C. Y. Lee
Keyword(s):  
Positive Integer ◽  
Explicit Expression ◽  
Generating Function ◽  
Finite Sets ◽  
Enumeration Problem ◽  
Ordered Pairs

We will consider the following enumeration problem. Let A and B be finite sets with α and β elements in each set respectively. Let n be some positive integer such that n ≦ αβ. A subset S of the product set A × B of exactly n distinct ordered pairs (ai, bj) is said to be admissible if given any a ∈ A and b ∈ B, there exist elements (ai, bj) and (ak, bl) (they may be the same) in S such that ai = a and bl = b. We shall find here a generating function for the number N(α, β n) of distinct admissible subsets of A × B and from this generating function, an explicit expression for N(α, β n). In obtaining this result, the idea of a cut probability is used. This approach in a problem of enumeration may be of interest.

A clustering law for some discrete order statistics

2003 ◽  
Vol 40 (1) ◽  
pp. 226-241 ◽  
Author(s):  
Sunder Sethuraman
Keyword(s):  
Distribution Function ◽  
Positive Integer ◽  
Order Statistics ◽  
Law Of Large Numbers ◽  
Asymptotic Properties ◽  
Random Variables ◽  
Large Numbers ◽  
The Law ◽  
Sample Maximum ◽  
Independent Identically Distributed

Let X1, X2, …, Xn be a sequence of independent, identically distributed positive integer random variables with distribution function F. Anderson (1970) proved a variant of the law of large numbers by showing that the sample maximum moves asymptotically on two values if and only if F satisfies a ‘clustering’ condition, In this article, we generalize Anderson's result and show that it is robust by proving that, for any r ≥ 0, the sample maximum and other extremes asymptotically cluster on r + 2 values if and only if Together with previous work which considered other asymptotic properties of these sample extremes, a more detailed asymptotic clustering structure for discrete order statistics is presented.

Two characterizations of the geometric distribution

1980 ◽  
Vol 17 (02) ◽  
pp. 570-573 ◽  
Author(s):  
Barry C. Arnold
Keyword(s):  
Positive Integer ◽  
Order Statistics ◽  
Conditional Distribution ◽  
Geometric Distribution ◽  
Random Variables ◽  
Random Variable ◽  
Mild Conditions ◽  
Independent Identically Distributed

Let X 1, X 2, …, Xn be independent identically distributed positive integer-valued random variables with order statistics X 1:n , X 2:n , …, Xn :n. If the Xi 's have a geometric distribution then the conditional distribution of Xk +1:n – Xk :n given Xk+ 1:n – Xk :n > 0 is the same as the distribution of X 1:n–k . Also the random variable X 2:n – X 1:n is independent of the event [X 1:n = 1]. Under mild conditions each of these two properties characterizes the geometric distribution.

A characterization of the geometric distribution

1983 ◽  
Vol 20 (01) ◽  
pp. 209-212 ◽  
Author(s):  
M. Sreehari
Keyword(s):  
Positive Integer ◽  
Order Statistics ◽  
Geometric Distribution ◽  
Random Variables ◽  
Random Variable ◽  
Characterization Problem ◽  
Independent Identically Distributed

Let X 1, X 2, …, Xn be independent identically distributed positive integer-valued random variables with order statistics X 1:n , X 2:n , …, X n:n . We prove that if the random variable X2:n – X 1:n is independent of the events [X1:n = m] and [X1:n = k], for fixed k > m > 1, then the Xi 's are geometric. This is related to a characterization problem raised by Arnold (1980).

A characterization of the geometric distribution

1983 ◽  
Vol 20 (1) ◽  
pp. 209-212 ◽  
Author(s):  
M. Sreehari
Keyword(s):  
Positive Integer ◽  
Order Statistics ◽  
Geometric Distribution ◽  
Random Variables ◽  
Random Variable ◽  
Characterization Problem ◽  
Independent Identically Distributed

Let X1, X2, …, Xn be independent identically distributed positive integer-valued random variables with order statistics X1:n, X2:n, …, Xn:n. We prove that if the random variable X2:n – X1:n is independent of the events [X1:n = m] and [X1:n = k], for fixed k > m > 1, then the Xi's are geometric. This is related to a characterization problem raised by Arnold (1980).

scholarly journals Selecting the last consecutive record in a record process

2010 ◽  
Vol 42 (03) ◽  
pp. 739-760
Author(s):  
Shoou-Ren Hsiau
Keyword(s):  
Asymptotic Behavior ◽  
Positive Integer ◽  
Explicit Expression ◽  
Optimal Stopping ◽  
Continuous Distribution ◽  
Stopping Time ◽  
Random Variables ◽  
Maximum Probability ◽  
Optimal Stopping Time ◽  
Bernoulli Random Variables

Suppose that I 1, I 2,… is a sequence of independent Bernoulli random variables with E(I n ) = λ/(λ + n − 1), n = 1, 2,…. If λ is a positive integer k, {I n } n≥1 can be interpreted as a k-record process of a sequence of independent and identically distributed random variables with a common continuous distribution. When I n−1 I n = 1, we say that a consecutive k-record occurs at time n. It is known that the total number of consecutive k-records is Poisson distributed with mean k. In fact, for general is Poisson distributed with mean λ. In this paper, we want to find an optimal stopping time τλ which maximizes the probability of stopping at the last n such that I n−1 I n = 1. We prove that τλ is of threshold type, i.e. there exists a t λ ∈ ℕ such that τλ = min{n | n ≥ t λ, I n−1 I n = 1}. We show that t λ is increasing in λ and derive an explicit expression for t λ. We also compute the maximum probability Q λ of stopping at the last consecutive record and study the asymptotic behavior of Q λ as λ → ∞.

scholarly journals The average position of the first maximum in a sample of geometric random variables

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Margaret Archibald ◽  
Arnold Knopfmacher
Keyword(s):  
Generating Function ◽  
Probability Generating Function ◽  
Random Variables ◽  
Expected Value ◽  
Factorial Moments ◽  
Maximum Value ◽  
Fourier Expansions ◽  
First Occurrence ◽  
International Audience ◽  
Average Position

International audience We consider samples of n geometric random variables $(Γ _1, Γ _2, \dots Γ _n)$ where $\mathbb{P}\{Γ _j=i\}=pq^{i-1}$, for $1≤j ≤n$, with $p+q=1$. The parameter we study is the position of the first occurrence of the maximum value in a such a sample. We derive a probability generating function for this position with which we compute the first two (factorial) moments. The asymptotic technique known as Rice's method then yields the main terms as well as the Fourier expansions of the fluctuating functions arising in the expected value and the variance.

scholarly journals Strong consistencies of the bootstrap moments

1991 ◽  
Vol 14 (4) ◽  
pp. 797-802 ◽  
Author(s):  
Tien-Chung Hu
Keyword(s):  
Positive Integer ◽  
Sample Size ◽  
Real Number ◽  
Random Variables ◽  
Random Variable ◽  
Bootstrap Sample ◽  
Data Set ◽  
Sample Moment ◽  
Almost All ◽  
Independent Identically Distributed

LetXbe a real valued random variable withE|X|r+δ<∞for some positive integerrand real number,δ,0<δ≤r, and let{X,X1,X2,…}be a sequence of independent, identically distributed random variables. In this note, we prove that, for almost allw∈Ω,μr;n*(w)→μrwith probability1. iflimn→∞infm(n)n−β>0for someβ>r−δr+δ, whereμr;n*is the bootstraprthsample moment of the bootstrap sample some with sample sizem(n)from the data set{X,X1,…,Xn}andμris therthmoment ofX. The results obtained here not only improve on those of Athreya [3] but also the proof is more elementary.

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