If the Covariance between $(X,Y)$ is finite, then the second moment of $X$ is finite?

Question

If the Covariance between $(X,Y)$ is finite, then the second moment of $X$ is finite.

I think that this statement is false, but I can´t find a counterexample....

Answer 1

An easy counterexample is to take $Y$ to be a constant random variable. Then the covariance between $X$ and $Y$ is zero: $$ \operatorname{Cov}(X,Y)=E(XY)-E(X)E(Y)=E(cX)-E(X)E(c)=cE(X)-cE(X)=0, $$ even if $\operatorname{Var}(X)=\infty$.

Answer 2

Let $X, Y$ be random variables such that

$$X = \begin{cases} 1.9^i, & i = 1, 2, 3, ... \\ 0, & \text{otherwise}\end{cases} \\ P(X = i) = \begin{cases} (\frac 12)^i, & i = 1, 2, 3, ... \\ 0, & \text{otherwise}\end{cases} \\ Y = 0 \ \text{ with probability }\ 1.$$

$$\mathsf{E}[X^2] = \sum\limits_{i=1}^\infty 1.805^i = \infty, \mathsf{E}[XY] = \mathsf{E}[0] = 0$$

Therefore the statement is untrue.