Questions tagged [weak-convergence]

Let $X_n$ and $X$ be r.v.s taking values in $\mathbb{Z}$. Suppose that $\lim \inf_{n \to \infty} \mathbb{P}(X_n=k) \geq \mathbb{P}(X=k)$ for $k \in \mathbb{Z}$. Show that $X_n$ converges to $X$ in distribution. We need to show that $F_{X_n}(a) \to…

I have a problem that I am not sure so I need your help. If we have ${X_n} \Rightarrow X$ and $P(X=0)=0$ then we can show $X_n^{-1} \Rightarrow X^{-1}$ ($0^{-1}$ is defined as $0$) but can we prove it without the condition $P(X=0)=0$? Any comments…

Let $ (H, (·, ·)) $ be a (separable) Hilbert space and let $ K \subset H$ be a nonempty, closed, convex set. Let $x \in H$ and denote by $x_{k} \in K$ the unique element such that $ d(x,K) = \parallel x − x_{k} \parallel $. (i) Prove that…

Im trying to prove that the series \begin{eqnarray} f_n(x) & = & 1\quad \mbox{for } n < x \leq n+1 \\ {} & = & 0 \quad \mbox{otherwise} \end{eqnarray} converges weakly in $L_2[0,\inf]$ I did the following : $ = |\int_{n}^{n+1} g(t)dt| \leq…