Beta-hypergeometric distributions and random continued fractions

Asci, Claudio; Letac, Gérard; Piccioni, Mauro

doi:10.1016/j.spl.2008.01.047

Cited by 4 publications

(18 citation statements)

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“…Then the distribution of X and Y are called the beta distributions of the first and of the second kind with parameters (p, q) and are denoted by β (1) p,q and β (2) p,q respectively.…”

Section: Introductionmentioning

confidence: 99%

“…The beta distributions of the first and of second kind on IR have many remarkable properties. For instance, it is well known (see [1]) that if W ′ ∼ β (2) a+a ′ ,a ′ is independent of X ∼ β (1) a,a ′ , then…”

Section: Introductionmentioning

confidence: 99%

“…And it is shown in [3] that if W ∼ β (2) a+a ′ ,a , X ∼ β (1) a,a ′ , W ′ ∼ β (2) a+a ′ ,a ′ are independent, then…”

Section: Introductionmentioning

confidence: 99%

“…In these two properties, the random variables W and W ′ are beta distributed with first parameter equal to the sum of the parameters of the distribution of the variable X. Asci, Letac and Piccioni [1] have used the so-called real beta-hypergeometric distribution to extend these results to the general case where W ∼ β (2) b,a , W ′ ∼ β (2) b,a ′ with b > 0 not necessarily equal to a + a ′ . Recall that the hypergeometric function p F q is defined for positive numbers a 1 , ..., a p ; b 1 , ..., b q , by p F q (a 1 , ..., a p ; b 1 , ..., b q ; x) = ∞ n=0 (a 1 ) n ...(a p ) n n!…”

Section: Introductionmentioning

confidence: 99%

“…Note that the distribution µ a,a ′ ,b , reduces to a β (1) a,a ′ when b = a + a ′ . Asci, Letac and Piccioni have shown that if X ∼ µ a,a ′ ,b and W ′ ∼ β (2) b,a ′ are independent, then…”

Section: Introductionmentioning

confidence: 99%

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Beta-hypergeometric probability distribution on symmetric matrices

Hassairi

Masmoudi

Regaig

2017

Advances in Applied Mathematics

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: Some remarkable properties of the beta distribution are based on relations involving independence between beta random variables such that a parameter of one among them is the sum of the parameters of an other (see (1.1) et (1.2) below). Asci, Letac and Piccioni [1] have used the real beta-hypergeometric distribution on IR to give a general version of these properties without the condition on the parameters. In the present paper, we extend the properties of the real beta to the beta distribution on symmetric matrices, we use on the positive definite matrices the division algorithm defined by the Cholesky decomposition to define a matrix-variate beta-hypergeometric distribution, and we extend to this distribution the proprieties established in the real case by Asci, Letac and Piccioni.

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“…Then the distribution of X and Y are called the beta distributions of the first and of the second kind with parameters (p, q) and are denoted by β (1) p,q and β (2) p,q respectively.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…And it is shown in [3] that if W ∼ β (2) a+a ′ ,a , X ∼ β (1) a,a ′ , W ′ ∼ β (2) a+a ′ ,a ′ are independent, then…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Beta-hypergeometric probability distribution on symmetric matrices

Hassairi

Masmoudi

Regaig

2017

Advances in Applied Mathematics

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Angular Processes Related to Cauchy Random Walks

Cammarota

Orsingher²

2011

Theory Probab. Appl.

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В статье изучается угловой процесс, связанный со случайным блужданиями в евклидовом и неевклидовом пространстве, шаги которых имеют распределение Коши.Это приводит к различного рода нелинейным преобразованиям случайных величин с распределением Коши, сохраняющим плотность Коши. Мы приводим явный вид этих распределений для всевозможных комбинаций параметров масштаба и сдвига.Анализируются непрерывные дроби, содержащие случайные величины с распределением Коши. Показано, что на n-м этапе случайные величины все еще имеют распределение Коши с параметрами, связанными с числами Фибоначчи. Это позволяет нам доказать, что последовательность сходится по распределению к золотому сечению.Ключевые слова и фразы: гиперболическая тригонометрия, закон арксинуса, непрерывные дроби, числа Фибоначчи, нелинейные преобразования случайных величин.

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Random continued fractions with beta-hypergeometric distribution

Letac¹,

Piccioni²

2012

Ann. Probab.

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In a recent paper (Asci et al., 2008) it has been shown that certain random continued fractions have a density which is a product of a beta density and a hypergeometric function 2F1. In the present paper we fully exploit a formula due to Thomae (1879) in order to generalize substantially the class of random continuous fractions with a density of the above form. This involves the design of seven particular graphs. Infinite paths on them lead to random continued fractions with an explicit distribution. A careful study about the set of five real parameters leading to a beta-hypergeometric distribution is required, relying on almost forgotten results mainly due to Felix Klein. Theorem 2.2. Suppose that r / ∈ −N. Then (p, q, r) ∈ P if and only if 1. either r ≥ 1, r − p − q ≤ 0 and p, q ≥ 0; 2. or r ≥ 1, r − p − q ≥ 0 and r − p, r − q ≥ 0; 3. or r < 1, r − p − q ≤ 0, r − p, r − q ≤ 1 and (p, q, r) is in S; 4. or r < 1, r − p − q ≥ 0, p, q ≤ 1 and (r − p, r − q, r) is in S.Remark. The most obvious case of positivity is p, q, r > 0. It can be checked that it fulfills the sufficient conditions above to be in P. The same holds for r − p, r − q, r > 0.Proof. From Klein's Theorem 2.1 a necessary condition for (p, q, r) ∈ P is X(p, q, r) = 0. We now consider three cases.1. If r − p − q ≥ 0 and r − p or r − q is −n with n ∈ N then

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Beta-hypergeometric distributions and random continued fractions

Cited by 4 publications

References 4 publications

Beta-hypergeometric probability distribution on symmetric matrices

Beta-hypergeometric probability distribution on symmetric matrices

Angular Processes Related to Cauchy Random Walks

Random continued fractions with beta-hypergeometric distribution

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