A note on the distribution of the product of zero mean correlated normal random variables
aa r X i v : . [ m a t h . S T ] J u l A note on the distribution of the product of zero meancorrelated normal random variables
Robert E. Gaunt ∗ Abstract
The problem of finding an explicit formula for the probability density functionof two zero mean correlated normal random variables dates back to 1936. Perhapssurprisingly, this problem was not resolved until 2016. This is all the more surprisinggiven that a very simple proof is available, which is the subject of this note; weidentify the product of two zero mean correlated normal random variables as avariance-gamma random variable, from which an explicit formula for probabilitydensity function is immediate.
Keywords:
Product of correlated normal random variables; probability density function;variance-gamma distribution
AMS 2010 Subject Classification:
Primary 60E05; 62E15
Let (
X, Y ) be a bivariate normal random vector with zero mean vector, variances ( σ X , σ Y )and correlation coefficient ρ . The exact distribution of the product Z = XY has beenstudied since 1936 [3]; see [14] for an overview of some of the contributions. The distribu-tion of Z has been used in numerous applications since 1936, with some recent examplesbeing: product confidence limits for indirect effects [13]; statistics of Lagrangian powerin two-dimensional turbulence [1]; statistical mediation analysis [12]. However, despitethis interest, the problem of finding an exact formula for the probability density function(PDF) of Z remained open for many years.Recently in 2016, some 80 years after the problem was first studied, an approachbased on characteristic functions was used by [14] to obtain an explicit formula for thePDF of Z . As a by-product, the exact distribution was obtained for the mean Z = n ( Z + Z + · · · + Z n ), where Z , Z , . . . , Z n are independent and identical copies of Z .This distribution is of interest itself; see, for example [15] for an application from electricalengineering. Since the work of [14], an exact formula for the PDF of a product of correlatednormal random variables with non-zero means was obtained by [4]. This formula takes acomplicated form, involving a double sum of modified Bessel functions of the second kind.Shortly before the work of [14], it was shown that Z had a variance-gamma distribution(see Section 2 for further details regarding this distribution) in the thesis [7] and the paper ∗ School of Mathematics, The University of Manchester, Manchester M13 9PL, UK
18] (see part (iii) of Proposition 1.2). A formula for the PDF is then immediate, and whilstnot noted in those works, a formula for the PDF of Z can then be obtained from standardproperties of the variance-gamma distribution. In this note, we fill in this gap to provide asimple alternative proof of the main results of [14]. Given the simplicity of our approach,it is surprising that such a natural problem in probability and statistics remained openfor so many years. Moreover, an advantage of our approach is that the distributions of Z and Z are identified as being from the variance-gamma class, for which a well establisheddistributional theory exists; see Chapter 4 of the book [10].The rest of this note is organised as follows. In Section 2, we introduce the variance-gamma distribution and record some basic properties that will be useful in the sequel. InSection 3, we provide an alternative proof of the main results of [14] by noting that Z and Z are variance-gamma distributed. The variance-gamma distribution with parameters r > θ ∈ R , σ > µ ∈ R has PDF f ( x ) = 1 σ √ π Γ( r ) e θσ ( x − µ ) (cid:18) | x − µ | √ θ + σ (cid:19) r − K r − (cid:18) √ θ + σ σ | x − µ | (cid:19) , x ∈ R , (2.1)where the modified Bessel function of the second kind is given, for x >
0, by K ν ( x ) = R ∞ e − x cosh( t ) cosh( νt ) d t . If a random variable W has density (2.1) then we write W ∼ VG( r, θ, σ, µ ). This parametrisation was given in [8]. It is similar to the parametrisationgiven by [6] and alternative parametrisations are given by [5], and the book [10] in whichthe name generalized Laplace distribution is used. The distribution has semi-heavy tails,which are useful for modelling financial data [11], and an overview of this and otherapplications are given in [10].We now review some basic properties of the variance-gamma distribution that will beneeded in Section 3. We stress that, with a standard handbook on definite integrals athand (such as [9]), all that is required to establish these properties is a working knowl-edge of a first course in undergraduate probability. Throughout, we shall set the locationparameter µ equal to 0. Firstly, we note a fundamental representation in terms of inde-pendent normal and gamma random variables ([10], Proposition 4.1.2). Let S ∼ Γ( r , )(with PDF r/ Γ( r/ x r/ − e − x/ , x >
0) and T ∼ N (0 ,
1) be independent. Then θS + σ √ ST ∼ VG( r, θ, σ, . (2.2)We will need the following special case of (2.2). Let U and V be independent N (0 , θU + σU V ∼ VG(1 , θ, σ, . (2.3)This follows from (2.2) due to the standard facts that U ∼ Γ( , ) and | U | V D = U V .Finally, we note that the class of variance-gamma distributions is closed under convolution(provided the random variables have common values of θ and σ ) [2] and scaling by aconstant. Let W ∼ VG( r , θ, σ,
0) and W ∼ VG( r , θ, σ,
0) be independent. Then W + W ∼ VG( r + r , θ, σ, . (2.4)2t is also clear from (2.2) that aW ∼ VG( r , aθ, aσ, . (2.5) Here, we provide an alternative proof of the main results of [14] by noting that Z and Z are variance-gamma distributed. Theorem 3.1.
Let ( X, Y ) denote a bivariate normal random vector with zero means,variances ( σ X , σ Y ) and correlation coefficient ρ .(i) Let Z = XY . Then Z ∼ VG(1 , ρσ X σ Y , σ X σ Y p − ρ , .(ii) Let Z , Z , . . . , Z n be independent random variables with the same distribution as Z . Let Z denote their sample mean. Then Z ∼ VG( n, n ρσ X σ Y , n σ X σ Y p − ρ , .(iii) Consequently, the PDFs of Z and Z are given by f Z ( x ) = 1 πσ X σ Y p − ρ exp (cid:18) ρxσ X σ Y (1 − ρ ) (cid:19) K (cid:18) | x | σ X σ Y (1 − ρ ) (cid:19) , x ∈ R , and, for n ≥ , f Z ( x ) = n ( n +1) / (1 − n ) / | x | ( n − / ( σ X σ Y ) ( n +1) / p π (1 − ρ )Γ (cid:0) n (cid:1) exp (cid:18) ρnxσ X σ Y (1 − ρ ) (cid:19) K n − (cid:18) n | x | σ X σ Y (1 − ρ ) (cid:19) ,x ∈ R , where K ν ( · ) is the modified Bessel function of the second kind of order ν .Proof. We consider the case σ X = σ Y = 1; the general case follows from (2.5).(i) Define the random variable W by W = √ − ρ ( Y − ρX ). It is straightforward toshow that W and X are jointly standard normally distributed with correlation 0. Thus, Z can be expressed in terms of independent N (0 ,
1) random variables X and W as follows: Z = XY = X ( p − ρ W + ρX ) = p − ρ XW + ρX . Hence, from (2.3), we have that Z ∼ VG(1 , ρ, p − ρ , ) = √ π . Acknowledgements
The author is supported by a Dame Kathleen Ollerenshaw Research Fellowship.
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