This question is related to my previous post : The law of large numbers with dependent random variable
Let $X\in\mathbb{R}^{n\times n}$ be a random matrix with i.i.d. entries, zero mean and variance $1/n$. Let $y\in\mathbb{R}^{n\times 1}$ be a random vector which is given by $y = Xz$ where $z\in\mathbb{R}^{n\times 1}$ is another random vector with i.i.d. entries (zero mean), finite variance, and independent on $X$.
Let $f(\cdot)$ be some deterministic and continuous function. It is known that $|f(x)|\leq 1$ for all $x\in\mathbb{R}$.
Can we say something regard the convergence of the following series $$ \frac{1}{n}\sum_{i=1}^nf\left(X_i^Ty\right) $$ as $n\to\infty$, and $X_i$ is the $i$th column of the matrix $X$.
EDIT: Ignore.