Upper bound for mean of squared variable

Question

Assume we have two random variables $x \sim (\mu_x, \sigma_x^2)$ and $y \sim (\mu_y, \sigma_y^2)$ and covariance $\sigma_{xy}$. I was wondering if there is a way to find an upper bound to $\mathbb{E}(x^2y^2)$? I know that I have $\mathbb{E}(xy) = \sigma_{xy} + \mu_x\mu_y$, but I don't know how to find the upper bound for the mean of the squared product of the two variables.

Answer 1

Let's denote $X = \frac{\sigma_{XY}}{\sigma_Y^2}\cdot Y + Z$ where $Z$ is a random variable indepenent to $Y$ that will be determined. It's easy to verify that $Cov(X,Y)= \frac{\sigma_{XY}}{\sigma_Y^2}\cdot Cov(Y,Y)= \sigma_{XY}$.

We will determine the expectation and variance of $Z$:

$$\mu_X = \frac{\sigma_{XY}}{\sigma_Y^2}\cdot \mu_Y + \mu_Z \implies \mu_Z =\mu_X - \frac{\sigma_{XY}}{\sigma_Y^2}\cdot \mu_Y \tag{1}$$ $$\sigma_X^2 = \frac{\sigma_{XY}^2}{\sigma_Y^4}\cdot \sigma_Y^2 + \sigma_Z^2 \implies \sigma_Z^2 = \sigma_X^2 - \frac{\sigma_{XY}^2}{\sigma_Y^2} \tag{2}$$

Now, we compute $\mathbb{E}(X^2Y^2)$:

$$\begin{align} \mathbb{E}(X^2Y^2) &= \mathbb{E}\left(\left( \frac{\sigma_{XY}}{\sigma_Y^2}\cdot Y + Z \right)^2 Y^2 \right) \\ & = \mathbb{E}\left( \frac{\sigma_{XY}^2}{\sigma_Y^4}\cdot Y^4 + 2\frac{\sigma_{XY}}{\sigma_Y^2} Y^3Z + Z^2Y^2 \right) \\ & = \frac{\sigma_{XY}^2}{\sigma_Y^4}\mathbb{E}\left( Y^4\right) + 2\frac{\sigma_{XY}}{\sigma_Y^2} \mathbb{E}(Y^3)\mu_Z + \sigma_Z^2 \sigma_Y^2 \\ \mathbb{E}(X^2Y^2)&= \frac{\sigma_{XY}^2}{\sigma_Y^4}\color{red}{\mathbb{E}\left( Y^4\right)} + 2\frac{\sigma_{XY}}{\sigma_Y^2} \left(\mu_X - \frac{\sigma_{XY}}{\sigma_Y^2}\mu_Y \right)\color{red}{\mathbb{E}(Y^3)} + \sigma_Y^2 \left(\sigma_X^2 - \frac{\sigma_{XY}^2}{\sigma_Y^2} \right) \end{align}$$

It is impossible to determine the range of $\mathbb{E}\left( Y^4\right)$ and $\mathbb{E}\left( Y^3\right)$ (the two red terms) from the expectations, variances and covariance of $X,Y$ only. In particular, with the given information, it is impossible to determine the maximum (or upper bound) of $\mathbb{E}(X^2Y^2)$.

However, if you know the range of $\mathbb{E}\left( Y^4\right)$ and $\mathbb{E}\left( Y^3\right)$ (or the range of $\mathbb{E}\left( X^4\right)$ and $\mathbb{E}\left( X^3\right)$ ), you can determine easily the bounds of $\mathbb{E}(X^2Y^2)$.