# Convergence in probability over compact set

Let $$(\Omega,\mathcal{F},P)$$ be a probability space. Suppose $$(X_n)\subseteq (\mathbb{R}^k)^{\Omega}$$ is a sequence of random vectors converging in probability to the vector $$c\in \mathbb{R}^k$$. Let $$(\Psi_n)$$ be a sequence of functions $$\Psi_n:\mathbb{R}^k\times \Omega\to \mathbb{R}$$, such that for all $$n$$ and all $$x\in \mathbb{R}^k$$, $$\Psi_n(x,\cdot)$$ is measurable. Consider also a continuous function $$\Psi:K\to \mathbb{R}$$ where $$K$$ is a compact subset of $$\mathbb{R}^k$$ containing $$c$$. Do we have that: $$\forall \varepsilon>0,\ \lim\limits_nP(\sup\limits_{x\in K}|\Psi_n(x,\cdot)-\Psi(x)|>\varepsilon)=0\Rightarrow \forall \varepsilon>0,\ \lim\limits_nP(|\Psi_n(X_n,\cdot)-\Psi(c)|>\varepsilon)=0\ ?$$

• I think you might need a condition that $c$ is interior to $K$. It seems the statement is incorrect e.g. if $K=\{c\}$. Jun 19 at 17:07
• Indeed, that's the problem I had and it is the reason for the question mark up there xD Jun 19 at 20:28
• @Arizabalaga Does your question in anyway relate to my post (Click here)? Jun 20 at 1:32
• I could not say, I will think more about it Jun 20 at 18:02

In retrospect, as pointed in the comment section, the hypothesis should allow $$c$$ to be an interior point $$K$$.
Let $$\tilde{\Psi}$$ be any extension of $$\Psi$$ to an continuous function on $$\mathbb{R}^k$$.
Because $$c$$ is an interior point of $$K$$ hence there is a positive number $$r>0$$ such that $$B_r(c) \subset K$$.
So if $$|X_n-c|, we have \begin{align}|\Psi_n(X_n,\cdot)-\Psi(c)| &\le |\Psi_n(X_n,\cdot)-\Psi(X_n)|+|\Psi(X_n) - \Psi(c)| \\ & \le \sup_{x \in K} |\Psi_n(x,\cdot)-\Psi(x)|+|\Psi(X_n) - \Psi(c)| \\ &= \sup_{x \in K} |\Psi_n(x,\cdot)-\Psi(x)|+|\tilde{\Psi}(X_n) - \tilde{\Psi}(c)| \end{align} Thus if $$\epsilon< |\Psi_n(X_n,\cdot)-\Psi(c)|$$ and $$|X_n-c|, we have either $$\epsilon/2 < \sup_{x \in K} |\Psi_n(x,\cdot)-\Psi(x)|$$ or $$\epsilon/2 <|\tilde{\Psi}(X_n) - \tilde{\Psi}|(c)|$$ So : \begin{align} \mathbb{P}(\epsilon< |\Psi_n(X_n,\cdot)-\Psi(c)|) &\le \mathbb{P}( |X_n-c| \ge r)+ \mathbb{P} \left( \epsilon/2 < \sup_{x \in K} |\Psi_n(x,\cdot)-\Psi(x)|\right)\\ &+ \mathbb{P} \left( \epsilon/2 <|\tilde{\Psi}(X_n) - \tilde{\Psi}(c)|\right) \end{align} Hence the conclusion.