An Inequality for the Correlation of Two Functions Operating on Symmetric Bivariate Normal Variables

TL;DR

This paper derives a simple inequality relating the correlation of two functions of symmetric bivariate normal variables, based on the Cauchy-Schwarz inequality, providing theoretical insight into their dependence structure.

Contribution

It introduces a new inequality for the correlation of functions of symmetric bivariate normal variables, expanding understanding of their dependence.

Findings

The inequality is a direct consequence of the Cauchy-Schwarz inequality.

It provides bounds on the correlation of transformed variables.

The result enhances theoretical understanding of symmetric bivariate normal functions.

Abstract

An inequality is derived for the correlation of two univariate functions operating on symmetric bivariate normal random variables. The inequality is a simple consequence of the Cauchy-Schwarz inequality.

Equations

$${\mathbb{E}}^2\left[g_1(Z_1)g_2(Z_2)\right]\le \mathbb{E}\left[g_1(Z_1)g_1(Z_2)\right]\mathbb{E}\left[g_2(Z_1)g_2(Z_2)\right],$$

$$⟨f,g⟩\triangleq \mathbb{E}\left[f(Z)g(Z)\right]$$

$$f_{Z_1,Z_2}(z_1,z_2)$$

$$\displaystyle=\frac{1}{2\pi\sigma^{2}}e^{-\frac{1}{2}\frac{z_{1}^{2}+z_{2}^{2}}{\sigma^{2}}}\sum_{n=0}^{\infty}\frac{1}{n!}H\!e_{n}\big{(}\tfrac{z_{1}}{\sigma}\big{)}H\!e_{n}\big{(}\tfrac{z_{2}}{\sigma}\big{)}\rho^{n},$$

$$H{e}_n(x)=(-1{)}^n{e}^{\frac{x^2}{2}}\frac{d^n}{d{x}^n}{e}^{-\frac{x^2}{2}},n\ge 0.$$

$$\displaystyle\langle\widetilde{H}\!e_{n},\widetilde{H}\!e_{m}\rangle=\mathbb{E}\big{[}\widetilde{H}\!e_{n}(Z)\widetilde{H}\!e_{m}(Z)\big{]}=n!\,\delta_{n,m}$$

$$g_i(x)=\sum _{n=0}^{\infty }\frac{1}{n!}{a}_{g_i,n}H{e}_n(\frac{x}{\sigma }),i=1,2.$$

$$\mathbb{E}\left[g_1(Z_1)g_2(Z_2)\right]=\sum _{n=0}^{\infty }\frac{1}{n!}{a}_{g_1,n}{a}_{g_2,n}{\rho }^n,$$

$$\mathbb{E}\left[g_i(Z_1)g_i(Z_2)\right]=\sum _{n=0}^{\infty }\frac{1}{n!}{a}_{g_i,n}^2{\rho }^n,i=1,2.$$

$${\mathbb{E}}^2\left[g_1(Z_1)g_2(Z_2)\right]$$

$$={\left(\sum _{n=0}^{\infty }\frac{a_{g_1,n}{\rho }^{\frac{n}{2}}}{\sqrt{n!}}\frac{a_{g_2,n}{\rho }^{\frac{n}{2}}}{\sqrt{n!}}\right)}^2$$

$$\le \left(\sum _{n=0}^{\infty }\frac{a_{g_1,n}^2{\rho }^n}{n!}\right)\left(\sum _{n=0}^{\infty }\frac{a_{g_2,n}^2{\rho }^n}{n!}\right)$$

$$=\mathbb{E}\left[g_1(Z_1)g_1(Z_2)\right]\mathbb{E}\left[g_2(Z_1)g_2(Z_2)\right],$$

$${\mathbb{E}}^2\left[g_1(Z_1)g_2(Z_2)\right]\le \mathbb{E}\left[g_1(Z_1)g_1(Z_2)\right]\mathbb{E}\left[g_2(Z_1)g_2(Z_2)\right],$$

$${\mathbb{E}}^2\left[g_1(Z_1)g_2(Z_2)\right]\le {\rho }^2\mathbb{E}\left[g_1^2(Z_i)\right]\mathbb{E}\left[g_2^2(Z_i)\right].$$

$${\mathbb{E}}^2\left[g_1(Z_1)g_2(Z_2)\right]\le \mathbb{E}\left[g_1(Z_1)g_1(Z_2)\right]\mathbb{E}\left[g_2(Z_1)g_2(Z_2)\right],$$

$$h_i(z)$$

$$=\mathbb{E}\left[g_i(Z_2)|Z=z\right]$$

$$\mathbb{E}\left[g_1(Z_1)g_2(Z_2)\right]$$

$$=\mathbb{E}\left[h_1(Z)h_2(Z)\right].$$

$$\mathbb{E}\left[g_i(Z_1)g_i(Z_2)\right]$$

$${\mathbb{E}}^2\left[h_1(Z)h_2(Z)\right]\le \mathbb{E}\left[h_1^2(Z)\right]\mathbb{E}\left[h_2^2(Z)\right].$$

$$y_n$$

$${\hat{z}}_n$$

$$=g(f(z_n))$$

$$=h(z_n),$$

$$K_1$$

$$K_2$$

$$\mathbb{E}[g(z_n)g(\alpha x_n)]=\mathbb{E}[g(x_n+w_n^{\prime })g(\alpha x_n)].$$

Full text

I Introduction

Statistical characterization of the output of non-linear systems operating on stochastic processes is in general difficult. Nonetheless, when the input process is Gaussian and the system is a memoryless non-linearity, several particularly simple and useful properties are known. Among these are Bussgang’s theorem  [1] and its generalizations (e.g., [2, 3]), results concerning the maximal correlation coefficient [4], as well as results on the output distortion-to-signal power ratio [5]. In the present note, we describe another simple result as described in the next section. A generalization to a more general class of random variables is described in Section III. An application is presented in Section IV.

II Statement of Result and Proof

Lemma 1**.**

Let $Z_{1}$ and $Z_{2}$ be zero-mean bivariate normal random variables with variance $\sigma^{2}$ and correlation coefficient $\rho>0$. Then,

[TABLE]

for any $g_{1}$ and $g_{2}$ for which the expectations exist, with equality if and only if $g_{1}$ and $g_{2}$ are equal up to a multiplicative constant.

Note that for $\rho=1$ the lemma reduces to the the standard probabilistic Cauchy-Schwarz inequality.

Proof.

Consider the inner product of real functions defined by

[TABLE]

where $Z\sim\mathcal{N}(0,\sigma^{2})$.

According to Mehler’s formula [6] (see also [7]) the joint density function of $(Z_{1},Z_{2})$ may be written as,

[TABLE]

where $H\!e_{n}(x)$ are the probabilists’ Hermite polynomials defined as,

[TABLE]

The Hermite polynomials constitute a complete orthogonal basis of polynomials with respect to the standard normal probability density function  [8], so that

[TABLE]

where $\widetilde{H}\!e_{n}(x)=H\!e_{n}(\tfrac{x}{\sigma})$, and $\delta_{n,m}$ is the Kronecker delta function.

Let $a_{g_{i},n}=\langle g_{i},\widetilde{H}\!e_{n}\rangle$. The function $g_{i}(x)$ may be represented by the series

[TABLE]

The following expectations are obtained by applying (II):

[TABLE]

Using these identities and assuming $\rho>0$ we have,

[TABLE]

where the inequality follows by the Cauchy–Schwarz inequality for sequences, which holds with equality only when $a_{g_{1},n}=c\cdot a_{g_{2},n}$ for some constant $c$, $\rho>0$, and for all $n\geq 0$. Since the case of equality holds only when the coefficients in the series (2) are equal up to a multiplicative constant, it follows that equality holds only when $g_{1}$ and $g_{2}$ are equal up to a multiplicative constant. ∎

When both functions are even or odd, we may apply the lemma to $Z_{1}$ and $-Z_{2}$ to obtain,

Corollary 1**.**

Let $Z_{1}$ and $Z_{2}$ be zero-mean bivariate normal random variables with variance $\sigma^{2}$ and correlation coefficient $\rho\neq 0$. Then, for $g_{1}$ and $g_{2}$ that are both even or odd functions,

[TABLE]

for any such $g_{1}$ and $g_{2}$ for which the expectations exist, with equality if and only if $g_{1}$ and $g_{2}$ are equal up to a multiplicative constant.

Remark 1**.**

It is interesting to contrast the lemma with the maximal correlation property of normal vectors. Specifically, consider the case where $Z_{1}$ and $Z_{2}$ are zero mean and both $g_{1}$ and $g_{2}$ are odd, so that $\mathbb{E}\left[g_{i}(Z_{j})\right]=0$, for all $i=1,2$ and $j=1,2$. Then the maximal correlation property [4] yields the inequality

[TABLE]

III Generalization

The lemma may be generalized to a broader class of random variables $(Z_{1},Z_{2})$, where $Z_{1}$ and $Z_{2}$ result from passing some random variable $Z$ through two independent realizations of the same “channel” (in information-theoretic terms). This generalization is stated in the next lemma. We note however that unlike Lemma 1, the “only if” condition for equality does not necessarily apply.

Lemma 2**.**

Let $Z_{1}$ and $Z_{2}$ be random variables such that for some random variable $Z$, $Z_{1}$ and $Z_{2}$ are independent and identically distributed given $Z$. Then,

[TABLE]

for any $g_{1}$ and $g_{2}$ for which the expectations exist, with equality if (but not necessarily only if) $g_{1}$ and $g_{2}$ are equal up to a multiplicative constant.

Proof.

Denote for $i=1,2$,

[TABLE]

since $Z_{1}$ and $Z_{2}$ are i.i.d. given $Z$. Then,

[TABLE]

Following the same steps, we also have,

[TABLE]

The claim now follows by applying the Cauchy-Schwarz inequality to obtain

[TABLE]

∎

Example 1**.**

Let $Z$ be a vector of length $N$ whose entries are i.i.d. Bernoulli$(p)$ and similarly, let $W_{1}$ and $W_{2}$ be two independent random vectors (of length $N$) whose entries are i.i.d. Bernoulli$(q)$. Finally, let $Z_{i}=Z\oplus W_{i}$ for $i=1,2$, where $\oplus$ denotes the binary exclusive or operation. Then the lemma holds for any two functions $g_{i}:\{0,1\}^{N}\rightarrow\mathbb{R}$.

Remark 2**.**

We note the method that is used in [4] to prove the maximal correlation property of bivariate normal random variables utilizes series expansions involving (probabilists’) Hermite polynomials via Mehler’s formula, similar to the approach taken in Lemma 1. In contrast, the proof in Lemma 2 follows the approach taken in [9], where an alternative proof to the maximal correlation property is derived.

IV Application: A Criterion for Identification of A Memoryless Non-Linearity

We now consider an application of Lemma 1. Consider a memoryless non-linearity $f$ operating on a discrete-time signal corrupted by additive white Gaussian noise (AWGN), as depicted in Figure 1.

Thus, the input $z_{n}$ consists of the sum of the signal $x_{n}$ and AWGN $w_{n}$ having variance $\sigma_{w}^{2}$. The output is thus,

[TABLE]

We assume that we observe both the input sequence $x_{n}$ as well as the output $y_{n}$. The function $f$ on the other hand is unknown and we wish to estimate it.

Let us consider first the case where it is known that $f$ is invertible. A possible means to identify $f$ is as follows. Apply another function $g$ to the output to obtain

[TABLE]

where $h=g\circ f$ is the composition of the functions $f$ and $g$. Assume now that $x_{n}$ is an AWGN process as well (i.e., a training sequence drawn according to such statistics) with variance $\sigma_{x}^{2}$. We may further assume for simplicity that $\mathbb{E}[h(z_{n})]=0$. Define

[TABLE]

Then, since for bivariate normal random variables nonlinear functions cannot increase (the absolute value of) correlation [4], it follows that $K_{1}$ is maximized (only) when $g=c\cdot f^{-1}$ for some constant $c$, so that $h$ is a linear function. As $K_{1}$ may be estimated by replacing expectations with time averages, we have obtained a simple criterion for identification of the non-linearity $f$ (up to to a scale factor that may easily be subsequently estimated).

A limitation of the identification criterion described above, is that it does not apply to non-linearities that are not invertible.111Nonetheless, it’s a suitable criterion for memoryless nonlinear compensation, where the inverse of the non-linearity (if exists) is desired (see, e.g., [10]). We now outline how the inequality derived in this note may serve to overcome this limitation. We note, however, that a drawback of the system described next is that we need to assume that the signal-to-noise ratio $\sigma_{x}^{2}/\sigma_{w}^{2}$ at the input of the non-linearity is known, unlike for the system described above.

Let $\alpha=\sqrt{\frac{\sigma_{x}^{2}+\sigma_{w}^{2}}{\sigma_{x}^{2}}}$ so that $\alpha x_{n}$ has the same variance as $z_{n}$. Consider now passing $\alpha x_{n}$ through a non-linearity $g$ as depicted in Figure 2,

to obtain $u_{n}=g(\alpha x_{n})$. Define

[TABLE]

It follows from Lemma 1 that $K_{2}$ is maximized only when $g=c\cdot f$ for some constant $c$.222Note that the “only when” property, which is crucial for the identification problem, follows by the condition for equality in Lemma 1. Again, $K_{2}$ may be computed by replacing expectations with time averages. This is the case, as although we do not observe $w_{n}$, we can replace it with AWGN noise $w^{\prime}_{n}$ generated with the same variance, to compute

[TABLE]

We note that for a practical implementation of the scheme, one would need some explicit (parametric) representation for the function $g(\cdot)$. For instance, one could employ a series expansion in orthogonal polynomials (see, e.g., [10]).

Acknowledgment

The authors are grateful to Or Ordentlich for helpful discussions concerning the contents of Section IV.