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Expected Logarithm and Negative Integer Moments of a Noncentral 2-Distributed Random Variable

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Expected Logarithm and Negative Integer Moments of a Noncentral 2-Distributed Random Variable entropy Article Expected Logarithm and Negative Integer Moments of a Noncentral c2-Distributed Random Variable † Stefan M. Moser 1,2 1 Signal and Information Processing Lab, ETH Zürich, 8092 Zürich, Switzerland; [email protected]; Tel.: +41-44-632-3624 2 Institute of Communications Engineering, National Chiao Tung University, Hsinchu 30010, Taiwan † Part of this work has been presented at TENCON 2007 in Taipei, Taiwan, and at ISITA 2008 in Auckland, New Zealand. Received: 21 July 2020; Accepted: 15 September 2020; Published: 19 September 2020 Abstract: Closed-form expressions for the expected logarithm and for arbitrary negative integer moments of a noncentral c2-distributed random variable are presented in the cases of both even and odd degrees of freedom. Moreover, some basic properties of these expectations are derived and tight upper and lower bounds on them are proposed. Keywords: central c2 distribution; chi-square distribution; expected logarithm; exponential distribution; negative integer moments; noncentral c2 distribution; squared Rayleigh distribution; squared Rice distribution 1. Introduction The noncentral c2 distribution is a family of probability distributions of wide interest. They appear in situations where one or several independent Gaussian random variables (RVs) of equal variance (but potentially different means) are squared and summed together. The noncentral c2 distribution contains as special cases among others the central c2 distribution, the exponential distribution (which is equivalent to a squared Rayleigh distribution), and the squared Rice distribution. In this paper, we present closed-form expressions for the expected logarithm and for arbitrary negative integer moments of a noncentral c2-distributed RV with even or odd degrees of freedom. Note that while the probability density function (PDF), the moment-generating function (MGF), and the moments of a noncentral c2-distributed RV are well-known, the expected logarithm and the negative integer moments have only been derived relatively recently for even degrees of freedom [1–6], but—to the best of our knowledge—for odd degrees of freedom they have been completely unknown so far. These expectations have many interesting applications. So, for example, in the field of information theory, there is a close relationship between the expected logarithm and entropy, and thus the expected logarithm of a noncentral c2-distributed RV plays an important role, e.g., in the description of the capacity of multiple-input, multiple-output noncoherent fading channels [1,2]. Many more examples in the field of information theory can be found in [7]. We will see that the expected logarithm and the negative integer moments can be expressed using two families of functions gm(·) and hn(·) that will be defined in Section3. Not unexpectedly, gm(·) and hn(·) are not elementary, but contain special functions like the exponential integral function ([8], Sec. 8.21), the imaginary error function [9], or a generalized hypergeometric function ([8], Sec. 9.14). While numerically this does not pose any problem as the required special functions Entropy 2020, 22, 1048; doi:10.3390/e22091048 www.mdpi.com/journal/entropy Entropy 2020, 22, 1048 2 of 45 are commonly implemented in many mathematical programming environments, working with them analytically can be cumbersome. We thus investigate gm(·) and hn(·) more in detail, present important properties, and derive tight elementary upper and lower bounds to them. The structure of this paper is as follows. After a few comments about our notation, we will formally define the noncentral c2 distribution in the following Section2 and also state some fundamental properties of the expected logarithm and the negative integer moments. In Section3 we present the two families of functions gm(·) and hn(·) that are needed for our main results in Section4. Section5 summarizes properties of gm(·) and hn(·), and Section6 presents tight upper and lower bounds on them. Many proofs are deferred to the appendices. We use upper-case letters to denote random quantities, e.g., U, and the corresponding lower-case E · (·) letter for their realization, e.g., u. The expectation operator is denoted by [ ]; ln is the naturalp logarithm; G(·) denotes the Gamma function ([8], Sec. 8.31–8.33); and i is the imaginary number i , −1. We use Neven to denote the set of all even natural numbers: even N , f2, 4, 6, 8, . .g. (1) Accordingly, Nodd , N n Neven is the set of all odd natural numbers. For a function x 7! f (x), f (`)(x) denotes its `th derivative: ` (`) d f (x) f (x) , . (2) dx` 2 2 The real Gaussian distribution of mean m 2 R and variance s > 0 is denoted by NR m, s , 2 2 while NC h, s describes the complex Gaussian distribution of mean h 2 C and variance s > 0. Thus, if X1 and X2 are independent standard Gaussian RVs, X1, X2 ∼ NR(0, 1), X1 ?? X2, then 1 1 Z , p X1 + i p X2 (3) 2 2 is circularly symmetric complex Gaussian, Z ∼ NC(0, 1). 2. The Noncentral c2 Distribution n Definition 1. For some n 2 N, let fXkgk=1 be independent and identically distributed (IID) real, n standard Gaussian RVs, Xk ∼ NR(0, 1), let fmkgk=1 2 R be real constants, and define n 2 t , ∑ mk. (4) k=1 Then the nonnegative RV n 2 U , ∑ (Xk + mk) (5) k=1 is said to have a noncentral c2 distribution with n degrees of freedom and noncentrality parameter t. Note that the distribution of U depends on the constants fmkg only via the sum of their squares (4). The corresponding PDF is ([10], Ch. 29) n−2 1 u 4 − t+u p p (u) = e 2 I − tu , u ≥ 0, (6) U 2 t n/2 1 Entropy 2020, 22, 1048 3 of 45 where In(·) denotes the modified Bessel function of the first kind of order n 2 R ([8], Eq. 8.445): ¥ 1 x n+2k I (x) , x ≥ 0. (7) n , ∑ ( + + ) k=0 k! G n k 1 2 For t = 0 we obtain the central c2 distribution, for which the PDF (6) simplifies to 1 n u 2 −1 − 2 pU(u) = n u e , u ≥ 0. (8) 2 n 2 G 2 Note that in this work any RV U will always be defined as given in (5). Sometimes we will write U[n,t] to clarify the degrees of freedom n and the noncentrality parameter t of U. If the number of degrees of freedom n is even (i.e., if n = 2m for some natural number m), there exists a second, slightly different definition of the noncentral c2 distribution that is based on complex Gaussian random variables. m m Definition 2. For some m 2 N, let fZjgj=1 be IID ∼ NC(0, 1), let fhjgj=1 2 C be complex constants, and define m 2 l , ∑ hj . (9) j=1 Then the nonnegative RV m 2 V , ∑ Zj + hj (10) j=1 is said to have a noncentral c2 distribution with 2m degrees of freedom and noncentrality parameter l. It has a PDF m−1 p v 2 −v−l p (v) = e I − 2 lv , v ≥ 0, (11) V l m 1 which in the central case of l = 0 simplifies to 1 p (v) = vm−1 e−v, v ≥ 0. (12) V G(m) Note that in this work any RV V will always be defined as given in (10). Sometimes we will write V[m,l] to clarify the degrees of freedom 2m and the noncentrality parameter l of V. Lemma 1. Let n 2 Neven be an even natural number and t ≥ 0 a nonnegative constant. Then U[n,t] L= 2V[n/2,t/2], (13) where “=L ” denotes equality in probability law. n n Proof. Let fXkgk=1 be IID ∼ NR(0, 1) and fmkgk=1 2 R as given in Definition1. Define m , n/2 2 N and for all j 2 f1, . , mg, 1 1 Zj , p X2j−1 + i p X2j, (14) 2 2 Entropy 2020, 22, 1048 4 of 45 1 1 hj , p m2j−1 + i p m2j. (15) 2 2 Then m 2 l , ∑ hj (16) j=1 m 1 2 1 2 = ∑ m2j−1 + m2j (17) j=1 2 2 n 1 2 1 = ∑ mk = t (18) 2 k=1 2 and m [m,l] 2 V = ∑ Zj + hj (19) j=1 m 2 1 1 1 1 = ∑ p X2j−1 + i p X2j + p m2j−1 + i p m2j (20) j=1 2 2 2 2 m 1 2 2 = ∑ X2j−1 + m2j−1 + X2j + m2j (21) 2 j=1 n 1 2 = ∑ (Xk + mk) (22) 2 k=1 1 = U[n,t]. (23) 2 Proposition 1 (Existence of Negative Integer Moments). For ` 2 N, the negative `th moment of a noncentral c2-distributed RV of n 2 N degrees of freedom and noncentrality parameter t ≥ 0, h −`i E U[n,t] , (24) is finite if, and only if, n ` < . (25) 2 Proof. See Appendix A.1. Proposition 2 (Monotonicity in Degrees of Freedom). The expected logarithm of a noncentral c2-distributed RV of n 2 N degrees of freedom and noncentrality parameter t ≥ 0, h i E lnU[n,t] , (26) is monotonically strictly increasing in n (for fixed t). n th [n,t] Similarly, for any ` 2 N \ 1, 2 − 1 , the negative ` moment of U , h −`i E U[n,t] , (27) Entropy 2020, 22, 1048 5 of 45 is monotonically strictly decreasing in n (for fixed t). Proof. See Appendix A.2. Proposition 3 (Continuity in Noncentrality Parameter). For a fixed n, the expected logarithm (26) is continuous in t for every finite t ≥ 0.
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