Expectation of the ratio of dependent random variables where the expectation of the numerator is known to be zero

Question

Let $\mathbf{x} = \begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix}$ be a complex normal random vector with $\mathbf{x} \sim \mathcal{CN}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$, where $\boldsymbol{\mu} \neq \mathbf{0}$ and $\boldsymbol{\Sigma}$ is not diagonal. Furthermore, I have that $\mathbb{E}_{x,y}[x_{1}^{*} x_{2}] = 0$, with $(\cdot)^{*}$ denoting complex conjugate.

I need to show that the following expectation is also equal to zero:

$$\mathbb{E}_{x_{1},x_{2}} \bigg[ \frac{x_{1}^{*} x_{2}}{|x_{1}|^{2} + |x_{2}|^{2} + c} \bigg]$$

where $c \in \mathbb{R}_{+}$.

Answer 1

Based on a classic counter-example for another question:

Let $X$ have a real standard normal distribution $N(0,1)$, and let $Y=X$ when $|X| \lt k$ and $Y=-X$ when $|X| \ge k$, where $k$ is square root of the median of a $\chi^2$ random variable with $3$ degrees of freedom, about $1.538172$. Clearly $Y$ also has a real standard normal distribution $N(0,1)$. Then

• $\mathbb{E}_{X,Y}[X^*Y] =0$ even though though there is a bijection between $X$ and $Y$
• $\mathbb{E}_{X,Y} \left[ \frac{X^{*} Y}{|X|^{2} + |Y|^{2} + c} \right] \gt 0 $ for $c \gt 0$ since the numerator is positive for small magnitude $X,Y$ and the denominator is less than $2k^2+c$, and the numerator is negative for large magnitude $X,Y$ and the denominator is greater than or equal to $2k^2+c$. For example with $c=1$ this expectation is about $0.118$