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On Products of Gaussian Random Variables On products of Gaussian random variables Zeljkaˇ Stojanac∗1, Daniel Suessy1, and Martin Klieschz2 1Institute for Theoretical Physics, University of Cologne, Germany 2 Institute of Theoretical Physics and Astrophysics, University of Gda´nsk,Poland May 29, 2018 Sums of independent random variables form the basis of many fundamental theorems in probability theory and statistics, and therefore, are well understood. The related problem of characterizing products of independent random variables seems to be much more challenging. In this work, we investigate such products of normal random vari- ables, products of their absolute values, and products of their squares. We compute power-log series expansions of their cumulative distribution function (CDF) based on the theory of Fox H-functions. Numerically we show that for small arguments the CDFs are well approximated by the lowest orders of this expansion. For the two non-negative random variables, we also compute the moment generating functions in terms of Mei- jer G-functions, and consequently, obtain a Chernoff bound for sums of such random variables. Keywords: Gaussian random variable, product distribution, Meijer G-function, Cher- noff bound, moment generating function AMS subject classifications: 60E99, 33C60, 62E15, 62E17 1. Introduction and motivation Compared to sums of independent random variables, our understanding of products is much less comprehensive. Nevertheless, products of independent random variables arise naturally in many applications including channel modeling [1,2], wireless relaying systems [3], quantum physics (product measurements of product states), as well as signal processing. Here, we are particularly motivated by a tensor sensing problem (see Ref. [4] for the basic idea). In this problem we consider n1×n2×···×n tensors T 2 R d and wish to recover them from measurements of the form yi := hAi;T i 1 2 d j nj with the sensing tensors also being of rank one, Ai = ai ⊗ai ⊗· · ·⊗ai with ai 2 R . Additionally, j we assumed that the entries of ai are iid Gaussian random variables. Applying such maps to arXiv:1711.10516v2 [math.PR] 27 May 2018 properly normalized rank-one tensor results in a product of d Gaussian random variables. Products of independent random variables have already been studied for more than 50 years [5] but are still subject of ongoing research [6{9]. In particular, it was shown that the probability density function of a product of certain independent and identically distributed (iid) random variables from the exponential family can be written in terms of Meijer G-functions [10]. In this work, we characterize cumulative distribution functions (CDFs) and moment generating functions arising from products of iid normal random variables, products of their squares, and products of their absolute values. We provide power-log series expansions of the CDFs of these distributions and demonstrate numerically that low-order truncations of these series provide tight ∗E-mail: [email protected] yE-mail: [email protected] zE-mail: [email protected] Comparison in Prop. 1.1 of true bound and C for N= 3 Comparison in Prop. 1.1 of true bound and C for N= 6 Comparison in Prop. 1.1 of true bound and C for N= 13 3 3 3 1.2 1.2 1 0.9 1 1 0.8 ) ) ) t t t 5 5 0.8 0.8 5 0.7 2 2 2 i i i g g g Q Q Q 0.6 ( ( ( P P 0.6 0.6 P y y y 0.5 t t t i i i l l l i i i b b b a a a 0.4 b b b o o 0.4 0.4 o r r r p p p 0.3 0.2 0.2 true bound 0.2 true bound true bound upper bound C upper bound C upper bound C 3 3 0.1 3 const=1 const=1 const=1 0 0 0 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 t t t QN 2 Figure 1: Comparison of the CDF P( i=1 gi ≤ t) (blue lines) with the bound stated in Proposi- tion1 (red lines) for different values of N. Here, we chose θ for each value t numerically from all possible values between 10−5 and 0:5 − 10−5 with step size 10−5 such that the right hand side in (1) is minimal. approximations. Moreover, we express the moment generating functions of the two latter distribu- tions in terms of Meijer-G functions. We state the corresponding Chernoff bounds, which bound the CDFs of sums of such products. Finally, we find simplified estimates of the Chernoff bounds and illustrate them numerically. 1.1. Simple bound We first state a simple but quite loose bound to the CDF of the product of iid squared standard Gaussian random variables. N Proposition 1. Let fgigi=1 be a set of iid standard Gaussian random variables (i.e., gi ∼ N (0; 1)). Then " N # Y 2 N θ P gi ≤ t ≤ inf Cθ t ; (1) θ2(0;1=2) i=1 where Γ(−θ + 1 ) C = p 2 > 0: θ π2θ 1 2 1 Proof. For θ < 2 the moment generating function of log(gi ) is given by Z 2 −θ log g2 −2θ 1 −2θ − x Ee i = E jgij = p jxj e 2 dx 2π R (2) 2−θ Z 1 1 2−θ Γ(−θ + 1 ) = p z−θ− 2 e−zdz = p 2 ; π 0 π x2 1 where we have used the substitution z = 2 . Note that for θ ≥ 2 the integral diverges. Now, the proposition is a simple consequence of Chernoff bound: N ! N ! Y 2 X 2 P gi ≤ t = P log(gi ) ≤ log t i=1 i=1 N θ log t −θ log g2 ≤ inf e Ee i (3) θ>0 θ N t = π− 2 inf Γ(−θ + 1 )N : 1 2N 2 θ2(0; 2 ) 1Throughout the article, log(x) denotes the natural logarithm of x. 2 Figure1 shows that this bound is indeed quite loose. This holds in particular for values of t that are very small. However, since Proposition1 is based on Chernoff inequality { a tail bound on sums of random variables { one cannot expect good results for those intermediate values. As a matter of fact, the upper bound in (1) becomes slightly larger than the trivial bound P(Y ≤ t) ≤ 1 already for quite small values of t. Deriving a good approximations for such values t and any N is in the focus of the next section. 2. Power-log series expansion for cumulative distribution functions N Throughout the article, we use the following notation. Let N 2 N and denote by fgigi=1 a set of iid standard Gaussian random variables (i.e., gi ∼ N (0; 1), for all i 2 [N]). We denote the random variables considered in this work by N N N Y Y 2 Y X := gi;Y := gi ;Z := jgij: (4) i=1 i=1 i=1 The probability density function of X can be written in terms of Meijer G-functions [10]. We provide a similar representation for the CDF of X, Y , and Z as well as the corresponding power-log series expansion using the theory developed for the more general Fox H-functions [11]. It is worth noting that series of Meijer-G functions have already been extensively studied. For example, in [12] the expansions in series of Meijer G-functions as well as in series of Jacobi and Chebyshev's polynomials have been provided. In this work, we investigate the power-log series expansion of special instances of Meijer-G functions. An advantage of this expansion is that it does not contain any special functions (excpet derivatives of the Gamma function at 1). For the sake of completeness, we give a brief introduction on Meijer G-functions together with some of their properties in AppendixA; see Refs. [6, 10] for more details. The following proposition is a special case of a result by Cook [13, p. 103] which relies on the Mellin transform. However, for this particular case we provide an elementary proof. N Proposition 2 (CDFs in terms of Meijer G-functions). Let fgigi=1 be a set of independent iden- QN tically distributed standard Gaussian random variables (i.e., gi ∼ N (0; 1)) and X := i=1 gi, QN 2 QN Y := i=1 gi , Z := i=1 jgij with N 2 N. Define the function Gα by 1 1 1; 1=2;:::; 1=2 G (z) := 1 − · G0;N+1 z : (5) α α N N+1;1 2 π 2 0 Then, for any t > 0, 2N (X ≤ t) = (X ≥ −t) = G (6) P P 1 t2 2N (Y ≤ t) = G (7) P 0 t 2N (Z ≤ t) = G : (8) P 0 t2 In Ref. [4] we have provided a proof of the above proposition by deriving first a result for the random variable X. The results for random variables Y and Z then follow trivially. For completeness, we derive a proof for a random variable Y in the AppendixB (see Lemma 16). The statements for X and Z are simple consequences of this result, since for t > 0, 2 2 2 P(Z ≤ t) = P(Z ≤ t ) = P(Y ≤ t ) ; (9) 1 1 (X ≤ t) = (2 (X ≤ t)) = ( (X ≤ t) + 1 − (X ≤ −t)) P 2 P 2 P P 1 1 1 = ( (−t ≤ X ≤ t) + 1) = (X2 ≤ t2) + 1 = (Y ≤ t2) + 1 : (10) 2 P 2 P 2 P 3 Now we are ready to present the main result of this section.
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