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The Basic Distributional Theory for the Product of Zero Mean Correlated

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The Basic Distributional Theory for the Product of Zero Mean Correlated The basic distributional theory for the product of zero mean correlated normal random variables Robert E. Gaunt∗ Abstract The product of two zero mean correlated normal random variables has received much attention in the statistics literature and appears in many application areas. However, many important distributional properties are yet to tabulated. This paper fills this gap by providing the basic distributional theory for the product of two zero mean correlated normal random variables. Keywords: Product of correlated normal random variables; distributional theory AMS 2010 Subject Classification: Primary 60E05; 62E15 1 Introduction 2 2 Let (X,Y ) be a bivariate normal random vector with zero mean vector, variances (σX , σY ) and correlation coefficient ρ. The distribution of the product Z = XY arises in nu- merous applications, with recent examples including statistics of Lagrangian power in two-dimensional turbulence (Bandi and Connaughton [3]), product confidence limits for indirect effects (MacKinnon et al. [28]) and statistical mediation analysis (MacKinnon −1 [27]). The mean Zn = n (Z1 + Z2 + + Zn), where Z1,Z2,...,Zn are independent and identical copies of Z, has also found··· application in areas such as electrical engineering (Ware and Lad [42]), astrophysics (Mangilli, Plaszczynski and Tristram [29], Watts et al. [43]) and quantum cosmology (Grishchuk [18]). The exact distribution of the product Z = XY has been studied since 1936 (Craig arXiv:2106.02897v1 [math.ST] 5 Jun 2021 [5]), with subsequent contributions coming from Aroian [1], Aroian, Taneja and Cornwell [2], Bandi and Connaughton [3], Haldane [20], Meeker et al. [31]; see Nadarajah and Pog´any [32] for an overview of these and further contributions. Recently, Nadarajah and Pog´any [32] used characteristic functions to find exact formulas for the probability density function (PDF) of the product Z and more generally the mean Zn; these formulas are starting to become well-known and have found many recent applications. Independently, in the physics literature, Grishchuk [18] had obtained the formula for the PDF of Z and Watts et al. [43] then obtained the more general formula for the PDF of Zn. Much earlier, a formula for PDF of Z was given in the book of Springer [38, equation (4.8.22)], but there were an unfortunate typo. ∗Department of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, UK, [email protected] 1 Despite the interest in the distributions of Z and Zn, many other important distribu- tional properties beyond the PDF are not tabulated in the literature. A recent work of Gaunt [14] identified Z and Zn as variance-gamma (VG) random variables (leading to a simple independent derivation of their PDFs), which means that some of their distribu- tional properties can be inferred from results for the VG distribution; see, for example, Chapter 4 of the book of Kotz, Kozubowski and Podg´orski [25]. However, searching through the VG literature can difficult; it is tedious to convert results for the VG dis- tribution to results for the distributions of Z and Z; and results that exploit the special structure of the random variables Z and Zn are not available. In this paper, we aim to fill a gap in the literature by providing the basic distributional theory for the product of two zero mean correlated normal random variables. We present results for the more general mean Zn, with results for the product Z following on setting n = 1. The end result is that many of the most important distributional properties of Z and Zn are now collected in a single handy reference. The properties covered include: for- mulas for the PDF (Section 2.1), list of related distributions (Section 2.2), the cumulative distribution function (Section 2.3), generating functions and infinite divisibility (Section 2.4), representations in terms of other random variables (Section 2.5), Stein characterisa- tion (Section 2.6), moments and cumulants (Section 2.7), and mode and median (Section 2.8). Basic properties of the modified Bessel function of the second kind will be needed throughout the paper, and are collected in Appendix A. 2 Distributional theory for the product of zero mean correlated normal random variables 2.1 Probability density function For independent derivations of the following formulas for the PDFs of Z and Zn we refer the reader to Watts et al. [43], Nadarajah and Pog´any [32] and Gaunt [14, 15]. The formula for the PDF of Z was also earlier derived by Grishchuk [18] and Springer [38], with the latter reference having a typo in the formula. For x R, ∈ 1 ρx x fZ(x)= exp 2 K0 | | 2 , (2.1) πσ σ 1 ρ2 σX σY (1 ρ ) σX σY (1 ρ ) X Y − − − and, for n 1, p ≥ n(n+1)/22(1−n)/2 x (n−1)/2 ρnx n x f (x)= | | exp K n−1 | | . Zn 2 2 2 (σ σ )(n+1)/2 π(1 ρ2)Γ(n/2) σX σY (1 ρ ) σX σY (1 ρ ) X Y − − − (2.2) p Here Kν(x) is a modified Bessel function of the second kind; see Appendix A for a defi- nition and some standard properties. The modified Bessel function in the PDFs (2.1) and (2.2) make them difficult to parse on first inspection. We can gain some understanding from the following limiting forms. 2 Using the limiting form (A.30), we have that 1 log x , x 0, n =1, 2 −πσX σY 1 ρ | | → fZn (x) 2 n/2−−1 (2.3) ∼ n(1 ρ ) Γ((n 1)/2) − p − , x 0, n 2. 2√πσX σY Γ(n/2) → ≥ We see that the density has a singularity at the origin if n = 1. In fact, for all parameter values (n 1, 1 <ρ< 1, σ , σ > 0) the distribution of Z is unimodal; see Section ≥ − X Y n 2.8 for further details. For n 2, the density is bounded. As observed by Watt et al. [43], the tail behaviour of the density≥ is obtained by applying the limiting form (A.31): (n/2)n/2 nx f (x) xn/2−1 exp , x , (2.4) Zn ∼ (σ σ )n/2Γ(n/2) − σ σ (1 + ρ) →∞ X Y X Y and (n/2)n/2 nx f (x) ( x)n/2−1 exp , x . (2.5) Zn ∼ (σ σ )n/2Γ(n/2) − σ σ (1 ρ) → −∞ X Y X Y − We observe that the distribution of Zn has semi-heavy tails. In the case that n is even, we can take advantage of a standard simplification of the modified Bessel function of the second kind (see (A.28)) to obtain a representation of the PDF in terms of elementary functions: (n/2)n/2 n(ρx x ) f (x)= x n/2−1 exp −| | Zn (σ σ )n/2Γ(n/2)| | σ σ (1 ρ2) × X Y X Y − n/2−1 (n/2 1+ j)! σ σ (1 ρ2) j − X Y − , x R, n 2Z+. (2.6) × (n/2 1 j)!j! 2n x ∈ ∈ j=0 X − − | | 2.2 Related distributions 1. Setting n = 2 in (2.6) yields 1 2(ρx x ) f (x)= exp −| | , (2.7) Z2 σ σ σ σ (1 ρ2) X Y X Y − which is the PDF of the asymmetric Laplace distribution (see Chapter 3 of Kotz, Kozubowski and Podg´orski [25] for a comprehensive account of its distributional theory). 2. As ρ 1, Zn converges in distribution to a scaled chi-square random variable: Zn d (σ σ /n↑)V , where V χ2 , the chi-square distribution with n degrees of freedom.→ By X Y ∼ (n) symmetry, Z (σ σ /n)V , as ρ 1. This is hardly surprising given the definition n →d − X Y ↓ − of Z , and can be confirmed rigorously by letting ρ 1 in the formula (2.11) below for n → ± the characteristic function of Zn and comparing to the chi-square characteristic function using L´evy’s continuity theorem. 3 2 2 2 3. By the central limit theorem, √n(Zn ρσX σY ) d N(0, σX σY (1 + ρ )), as n . E 2 2 − 2 → → ∞ Here, [Z]= ρσX σY and Var(Z)= σX σY (1 + ρ ) (see (2.20) and (2.21)). 4. The variance-gamma (VG) distribution with parameters r > 0, θ R, σ > 0, µ R has PDF ∈ ∈ 2 r−1 eθ(x−µ)/σ x µ 2 √θ2 + σ2 fVG(x)= | − | K r−1 x µ , x R, (2.8) σ√πΓ(r/2) 2√θ2 + σ2 2 σ2 | − | ∈ If a random variable V has PDF (2.8), we write V VG(r,θ,σ,µ). This parametrisation was given in Gaunt [11]. It is similar to the parametrisation∼ given by Finlay and Seneta [9] and other parametrisations are given by Eberlein and Hammerstein [8] and Kotz, Kozubowski and Podg´orski [25], with Chapter 4 of the latter reference giving the most comprehensive account of the distributional theory of the VG distribution in the literature. The VG distribution is also known as the Bessel function distribution or the McKay Type II distribution (McKay [30]), as well as the generalized Laplace distribution (Kotz, Kozubowski and Podg´orski [25]). 2 It was noted in the thesis of Gaunt [10] that Z VG(1, ρσX σY , σX σY 1 ρ , 0) and later by Gaunt [14] that also ∼ − p Z VG(n, ρσ σ /n, σ σ 1 ρ2/n, 0). (2.9) n ∼ X Y X Y − p 5. An exact formula for the PDF of the product of correlated normal random variables with non-zero means was derived by Cui et al.
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