On the q -moment Determinacy of Probability Distributions
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On the q-moment Determinacy of Probability Distributions Sofiya Ostrovska1
· Mehmet Turan1
Received: 3 April 2019 / Revised: 27 November 2019 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract Given 0 < q < 1, every absolutely continuous distribution can be described in two different ways: in terms of a probability density function and also in terms of a q-density. Correspondingly, it has a sequence of moments and a sequence of qmoments, if they exist. In this article, new conditions on the q-moment determinacy of probability distributions are derived. In addition, results related to the comparison of the properties of probability distributions with respect to the moment- and q-moment determinacy are presented. Keywords q-density · q-moment · Moment problem · q-moment (in)determinacy · Analytic function Mathematics Subject Classification 60E05 · 30E05 · 05A30 · 62E10
1 Introduction Due to the popularity of the q-calculus, numerous q-analogues of classical probability distributions have emerged, both for discrete and absolutely continuous cases. For example, there are q-binomial, q-Poisson, q-exponential, q-Erlang, and other qdistributions. These distributions play a significant role not only in the q-calculus itself, but also in various applications, primarily in theoretical physics. See, for example, [1,6,10,14]. Comprehensive information concerning q-distributions is presented in [6] and, in this article, we follow the terminology and exposition of this monograph. Throughout the paper, q ∈ (0, 1) is taken to be fixed. The q-integral defined
Communicated by Anton Abdulbasah Kamil.
B
Sofiya Ostrovska [email protected] Mehmet Turan [email protected]
1
Department of Mathematics, Atilim University, 06836 Ankara, Incek, Turkey
123
S. Ostrovska, M. Turan
by Jackson for 0 < a < b as
a
f (t)dq t = a(1 − q)
0
∞
b
f (aq j )q j ,
f (t)dq t =
a
j=0
b
f (t)dq t −
0
a
f (t)dq t
0
will be used along with the improper q-integral on [0, +∞) defined as
∞
∞
f (t)dq t = (1 − q)
0
f (q j )q j .
j=−∞
See [11, Sec. 19]. Definition 1.1 [6] Let P be a probability distribution with a distribution function F satisfying F(0) = 0. A function f (t), t > 0, is a q-density of P if
x
F(x) =
f (t)dq t, x > 0.
(1.1)
0
Correspondingly, the nth-order q-moment of P is m q (n; P) := m q (n; f ) :=
∞
t n f (t)dq t, n ∈ N0 .
(1.2)
0
Clearly, m q (n; f ) = (1 − q)
f (q − j )q − j(n+1) n ∈ N0 .
(1.3)
j∈Z
It has to be mentioned here that if P has a q-density f , then f is the q-derivative of the distribution function F, that is, f (t) = Dq F(t) :=
F(t) − F(qt) , t > 0. t(1 − q)
It is known ([11, Theorem 20.1]) that if F(0) = 0, and it is continuous at 0, and then F can be represented in the form (1.1) and, therefore, possesses a q-density. In this paper, only probability distributions satisfying these conditions and possessing finite q-moments of all orders will be considered. The set of such distributions will be denot
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