For non-independent random vectors $X, Y$, I have an upper bound on the expectations $E[\|X\|_2] \leq a, E[\|Y\|^2_2] \leq b$. How can I compute an upper bound for $E[\|X^\intercal Y\|_2]$ or $E[\|X^\intercal Y\|^2_2]$?
I've tried the following:
$$E[\|X^\intercal Y\|_2] \leq E[\|X\|_2 \|Y\|_2] \leq E[\|X\|_2^2]^{1/2} E[\|Y\|_2^2]^{1/2}.$$
But I don't have an upper bound on $E[\|X\|_2^2]$ and $Y$ is not uniformly bounded (it's a Gaussian random variable). I also thought of using Holder's inequality for $p = 1$, but I don't know how to compute $\lim_{q \rightarrow \infty }E[\|Y\|_2^q]^{1/q}$.
Best Answer
There’s no bound for either expectation. This is true already in the univariate case. Consider $X=0$ with probability $\frac{n-1}n$ and $X=n$ with probability $\frac1n$, and $Y=\sqrt X$. Then $E[|X|]=E[Y^2]=1$, yet $E[|XY|]\to\infty$ and $E[(XY)^2]\to\infty$ for $n\to\infty$.