Convergence properties of Kemp’s q-binomial distribution

Abstract

Abstract. We consider Kemp's q-analogue of the binomial distribution. Several convergence results involving the classical binomial, the Heine, the discrete normal, and the Poisson distribution are established. Some of them are qanalogues of classical convergence properties. Besides elementary estimates, we apply Mellin transform asymptotics.

On a family of q-binomial distributions

“…α = 1 2 ) the limit distributions are the Heine distribution and the discrete normal distribution. This was done by Gerhold and Zeiner [6]. We will show that these results can be generalized to the case α > 0.…”

“…For properties and applications of this distribution see [6,7,10,12]. In the limit q → 1 Kemp's distribution converges to a binomial distribution:…”

“…The case n < f (n) α can be treated similarly. In [6] Gerhold and Zeiner studied the behaviour of the means of Kemp's q-binomial distribution in the limit q → 0 and c(a, α, q) in the limit q → 1. We will do the same analysis here.…”

A q-binomial extension of the CRR asset pricing model

“…As noted in Charalambides (2016), the Central Limit Theorem does not hold for the q-binomial distribution. The convergence of the q-binomial distribution has been studied for fixed q ∈ (0, 1), and convergence to a discrete Heine distribution has been shown in Gerhold and Zeiner (2010), see also Kyriakoussis and Vamvakari (2013).…”

Convergence properties of the q-deformed binomial distribution

“…This paper is devoted to the study of sequences of q-deformed binomially distributed random variables X n ∼ QD(n, τ n , q) with parameter sequence (τ n ) depending on n (a similar analysis for Kemp's qbinomial distribution has been done by Gerhold and Zeiner [7]). …”