Random continued fractions with beta-hypergeometric distribution

Abstract

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

Convolution of beta prime distribution

“…which, as will be observed in Remark 11 (b) below, is a particular case of a general identity in law derived in [22]. Let us however give our own argument for the sake of completeness.…”

“…It is worth mentioning that all size-biases of 1 − B b,a B d,c are betahypergeometric random variables, but that the converse is not true -see Theorems 2.1 and 2.2 in [22]. Observe finally that Thomae's relations play a crucial role in the proof of (5) in [22].…”

“…(b) The identity ( 27) is an instance of a general result of [22] for beta-hypergeometric distributions.…”

Random continued fractions: Lévy constant and Chernoff-type estimate

“…We say that X is a random continued fractions generated by the stochastic process {A n , n ≥ 1}. Different models of random continued fractions have been considered in literature for instance [17], [19], [20], [32] and the references therein.…”