# pr.probability – LLN for variable that is a function of n

Let $$x sim P$$ be a random variable and $$f_k(x)$$ be a random function of $$x$$ with parameter $$k$$. Further, let $$k$$ be a function of n such that $$lim_{n to infty}k=infty$$ and $$lim_{n to infty} frac kn=0$$. $$f_k(x)$$ is such that by dominated convergence, it holds that
begin{align*} lim_{k to infty} E(f_k(x))=E(lim_{n to infty}f_k(x))=E(f(x)) end{align*}
Now I want to show that
begin{align*} lim_{n to infty} frac 1n sum_{i=1}^nf_k(x_i)=E(f(x)) end{align*}
What I would like to do is something like this:
begin{align*} lim_{k to infty}lim_{n to infty}frac 1n sum_{i=1}^nf_k(x_i)=lim_{k to infty}E(f_k(x))=E(f(x)) end{align*}
Additionally it holds that $$f_k(x_i), f_k(x_j)$$ might have a bit of a dependency structure. Sorry for not being clearer on this issue. However, $$f(x_i), f(x_j)$$ are independent.