When does $f_n(\beta_n)\rightarrow^p f(\beta)$ and Let's suppose that 


*

*$\beta_n\rightarrow^p \beta$

*$f_n(x)\rightarrow f(x)$. 


When can I be sure that $f_n(\beta_n)\rightarrow^p f(\beta)$ ?
I'm also looking for assumptions/theorems valid for a multivariate setting.
Any help would be appreciated.
 A: One should first address a simpler question: suppose that $x_n\to x$, $n\to\infty$. Under what conditions $f_n(x_n)\to f(x)$, $n\to\infty$? 
This is something like joint continuity of $f(n,y) = f_n(y)$ at the point $(\infty,x)$. The necessary and sufficient condition is that for any $\varepsilon>0$ there are $\delta>0$ and $N>1$ such that $|f_n(y)-f(x)|<\varepsilon$ provided that $|y-x|<\delta$ and $n\ge N$. 
One can formulate simpler sufficient conditions like: 1) $f_n\to f$ locally uniformly; 2) $f_n\to f$ pointwise, and the family $\{f_n\}$ is locally equicontinuous.
Now let us get back to the original problem. We know that a sequence is convergent in probability to $\xi$ iff any subsequence contains a subsubsequence converging to $\xi$ almost surely. 
With this in mind, let $\{\beta_{n_k}\}$ be any subsequence of $\{\beta_n\}$. It has an almost surely convergent subsequence; for simplicity (and without the loss of generality) let it be $\{\beta_{n_k}\}$ itself. Let $\Omega_0$ be the set where $\beta_{n_k}(\omega) \to \beta(\omega)$, $k\to\infty$. If we had that for $\omega\in \Omega_0$, $f_{n_k}(\beta_{n_k}(\omega))\to f(\beta(\omega))$, $k\to\infty$, as well, then we would have the convergence in probability $f_n(\beta_{n}(\omega))\overset{P}\longrightarrow f(\beta(\omega))$, $n\to\infty$. Indeed, in that case any subsequence of $\{f_n(\beta_{n})\}$ would have a subsubsequence converging to $f(\beta)$ almost surely.
Thus, we come to the above deterministic problem: for fixed $\omega \in \Omega_0$ we want to deduce the convergence $f_{n_k}(x_k)\to f(x)$, $k\to \infty$, from the convergence $x_k = \beta_{n_k}(\omega)\to x = \beta(\omega)$, $k\to\infty$. Consequently, one can formulate some sufficient conditions for the convergence: 


*

*$f(n,x)=f_n(x)$ is jointly continuous at $(\infty,x)$ in the above sense for each $x\in \operatorname{supp} \xi$. This is the weakest condition, but hardest to check.

*$f_n \to f$ locally uniformly. 

*$f_n\to f$ pointwise on $\operatorname{supp} \xi$, and the family $\{f_n, n\ge 1\}$ is locally equicontinuous.
Note that these sets of conditions are universal: neither of them appeals to the speed at which $\beta_n\to \beta$ in reality. If one has some information, then it might be possible to formulate weaker conditions about $f_n$.
