scholarly journals Samples of geometric random variables with multiplicity constraints

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Margaret Archibald ◽  
Arnold Knopfmacher
Keyword(s):  
Geometric Distribution ◽  
Random Variables ◽  
Fixed Integer ◽  
Forbidden Set ◽  
International Audience ◽  
Independent Identically Distributed

International audience We investigate the probability that a sample $\Gamma=(\Gamma_1,\Gamma_2,\ldots,\Gamma_n)$ of independent, identically distributed random variables with a geometric distribution has no elements occurring exactly $j$ times, where $j$ belongs to a specified finite $\textit{'forbidden set'}$ $A$ of multiplicities. Specific choices of the set $A$ enable one to determine the asymptotic probabilities that such a sample has no variable occuring with multiplicity $b$, or which has all multiplicities greater than $b$, for any fixed integer $b \geq 1$.

scholarly journals Compositions and samples of geometric random variables with constrained multiplicities

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Margaret Archibald ◽  
Arnold Knopfmacher ◽  
Toufik Mansour
Keyword(s):  
Positive Integer ◽  
Geometric Distribution ◽  
Random Variables ◽  
Related Case ◽  
Forbidden Set ◽  
International Audience ◽  
Nous Examinons ◽  
Independent Identically Distributed

International audience We investigate the probability that a random composition (ordered partition) of the positive integer $n$ has no parts occurring exactly $j$ times, where $j$ belongs to a specified finite $\textit{`forbidden set'}$ $A$ of multiplicities. This probability is also studied in the related case of samples $\Gamma =(\Gamma_1,\Gamma_2,\ldots, \Gamma_n)$ of independent, identically distributed random variables with a geometric distribution. Nous examinons la probabilité qu'une composition faite au hasard (une partition ordonnée) du nombre entier positif $n$ n'a pas de parties qui arrivent exactement $j$ fois, où $j$ appartient à une série interdite, finie et spécifiée $A$ de multiplicités. Cette probabilité est aussi étudiée dans le cas des suites $\Gamma =(\Gamma_1,\Gamma_2,\ldots,\Gamma_n)$ de variables aléatoires identiquement distribuées et indépendantes avec une distribution géométrique.

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 A combinatorial and probabilistic study of initial and end heights of descents in samples of geometrically distributed random variables and in permutations

2007 ◽  
Vol Vol. 9 no. 1 (Analysis of Algorithms) ◽  
Author(s):  
Guy Louchard ◽  
Helmut Prodinger
Keyword(s):  
Geometric Distribution ◽  
Analysis Of Algorithms ◽  
Markov Chain Model ◽  
Random Variables ◽  
Discrete Distributions ◽  
Chain Model ◽  
Initial Height ◽  
Probabilistic Study ◽  
International Audience ◽  
Expected Values

Analysis of Algorithms International audience In words, generated by independent geometrically distributed random variables, we study the lth descent, which is, roughly speaking, the lth occurrence of a neighbouring pair ab with a>b. The value a is called the initial height, and b the end height. We study these two random variables (and some similar ones) by combinatorial and probabilistic tools. We find in all instances a generating function Ψ(v,u), where the coefficient of vjui refers to the jth descent (ascent), and i to the initial (end) height. From this, various conclusions can be drawn, in particular expected values. In the probabilistic part, a Markov chain model is used, which allows to get explicit expressions for the heights of the second descent. In principle, one could go further, but the complexity of the results forbids it. This is extended to permutations of a large number of elements. Methods from q-analysis are used to simplify the expressions. This is the reason that we confine ourselves to the geometric distribution only. For general discrete distributions, no such tools are available.

scholarly journals d-records in geometrically distributed random variables

2006 ◽  
Vol Vol. 8 ◽  
Author(s):  
Helmut Prodinger
Keyword(s):  
Geometric Distribution ◽  
Random Variables ◽  
Random Permutations ◽  
Weak Model ◽  
Asymptotic Formulae ◽  
International Audience

International audience We study d-records in sequences generated by independent geometric random variables and derive explicit and asymptotic formulæ for expectation and variance. Informally speaking, a d-record occurs, when one computes the d-largest values, and the variable maintaining it changes its value while the sequence is scanned from left to right. This is done for the "strict model," but a "weak model" is also briefly investigated. We also discuss the limit q → 1 (q the parameter of the geometric distribution), which leads to the model of random permutations.

Weak Records of Geometric Distribution

2005 ◽  
Vol 56 (1-4) ◽  
pp. 295-304
Author(s):  
M. Ahsanullah
Keyword(s):  
Geometric Distribution ◽  
Random Variables ◽  
Record Values ◽  
Distributional Properties ◽  
Weak Records ◽  
Independent Identically Distributed

Summary Upper weak record values from a sequence of independent identically distributed random variables are considered. Several distributional properties of the upper weak records from geometric distribution are presented. Based on these distributional properties some characterizations of the geometric distribution are given.

scholarly journals Characterization of the Geometric Distribution Via Linear Combinations of Observations and of Records

Sankhya A ◽  
2021 ◽  
Author(s):  
Barry C. Arnold ◽  
Jose A. Villasenor
Keyword(s):  
Linear Combination ◽  
Geometric Distribution ◽  
Random Variables ◽  
Record Values ◽  
Linear Combinations ◽  
Distributional Property ◽  
Weak Records ◽  
Geometric Variables ◽  
Independent Identically Distributed

AbstractIn a sequence of independent identically distributed geometric random variables, the sum of the first two record values is distributed as a simple linear combination of geometric variables. It is verified that this distributional property characterizes the geometric distribution. A related characterization conjecture is also discussed. Related discussion in the context of weak records is also provided.

Two characterizations of the geometric distribution

1980 ◽  
Vol 17 (2) ◽  
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 X1, X2, …, Xn be independent identically distributed positive integer-valued random variables with order statistics X1:n, X2: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 X1:n–k. Also the random variable X2:n – X1:n is independent of the event [X1:n = 1]. Under mild conditions each of these two properties characterizes the geometric distribution.

scholarly journals On subcritical multi-type branching process in random environment

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Elena Dyakonova
Keyword(s):  
Random Walk ◽  
Random Environment ◽  
Survival Probability ◽  
Branching Process ◽  
Random Variables ◽  
International Audience ◽  
Watson Process ◽  
Independent Identically Distributed

International audience We investigate a multi-type Galton-Watson process in a random environment generated by a sequence of independent identically distributed random variables. Suppose that the associated random walk constructed by the logarithms of the Perron roots of the reproduction mean matrices has negative mean and assuming some additional conditions, we find the asymptotics of the survival probability at time $n$ as $n \to \infty$.

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