Let $X$ be a random variable with $E[X] = \mu < \infty$.

For $n=1,2,\dots$, construct a triangular array of random variables as

\begin{equation} Y_{n,i} = X_i \frac{\sqrt{\mu}}{\sqrt{\sum_{j=1}^n X_j/n}}. \end{equation}

Then, does the following hold? \begin{equation} \frac{1}{n} \sum_{i=1}^n Y_{n,i} \overset{p}{\to} \mu, \end{equation} where $\overset{p}{\to}$ denotes the convergence in probability.

Or more generally, for any bounded measurable function $f$, does the following hold? \begin{equation} \frac{1}{n} \sum_{i=1}^n f(Y_{n,i}) \overset{p}{\to} E[f(X)]. \end{equation}

If not, is there any necessary condition?

Thanks in advance.