Suppose that $V_i$, $i \in \mathbb{N}$ are i.i.d. standard normal random variables and let $Y_i = \sum_{k=1}^i V_i$ for $i\geq 1$ and $Y_0 = 0$. Consider the following ratio random variable
$$R = \frac{Y_n \sum_{i=1}^n X_i Y_{i-1}}{\sum_{i=1}^{n}Y_{i-1}^2}.$$
How can I show that $E[R^2]$ is upperbounded by $O(\frac{1}{n})$?
So far I did the following:
(1) Let $X = Y_n^2 (\sum_{i=1}^n X_i Y_{i-1})^2$ and $Z = \sum_{i=1}^{n}Y_{i-1}^2$. I used Taylor expansion to estimate the expectation $E[R^2]$:
$$ E\Big[\frac{X}{Z}\Big] \approx \frac{E[X]}{E[Z]} - \frac{\text{cov}(X,Z)}{E[Z]^2} + \frac{E[X]}{E[Y]^3}\text{var}(Z).$$
By computing all these terms, I showed that the above approximation is $\Theta(1/n)$. I also evaluated third-order Taylor expansion and got a similar result.
(2) I tried computing the Taylor expansion error using the mean-value theorem. However, I was not able to bound the error because $Z$, which appears in the denominator, can be close to zero.
(3) I ran some simulations and $E[R^2]$ indeed is bounded by $O(1/n)$.