Mean Square Convergence with Square Root n Term

Question

If $\sqrt{n}(X_n-X)$ converges in mean squre ($L^2$), do we have $X_n$ converges to $X$ in mean square? How can we show this?

Moreover, Is there a Slutsky Theorem for $L^p$ convergence?

Thank you!

Answer 1

In general if $W_{n}\to W$ in $L^p$ and $Z_{n}\to Z$ in ${L^p}$, then $W_nZ_n\to WZ$ in $L^p$. Indeed, let $\lVert\cdot\rVert$ denote the $L^p$ norm and note the inequality $$ \lVert W_{n}Z_{n}-WZ\rVert\leq \lVert W_n\rVert\lVert Z_n-Z\rVert+ \lVert Z\rVert \lVert W_n-W\rVert $$ from which the result follows.

Returning to the original question, suppose $ \sqrt n(X_n-X)\to U $ in $L^2$ for some $U$. Because $1/\sqrt{n}\to 0$ in $L^2$, it follows that the product $X_n-X\to 0$ in $L^2$ as desired.