Does the limit in probability equal almost surely to the liminf?

Question

Let $(X_n)_n$ be a sequence of random variables which converges in probability $\mathbb{P}$ to $X$. Is the following true? $$ \mathrm{liminf}\, X_n = X \,(\mathbb{P}-a.e) $$ In other words, does the liminf coincide with the limit in probability almost everywhere?

Answer 1

You can modify a classical example people use to show that convergence in probability does not imply almost sure convergence.

For any $0\leq k <2^m$, define $I_{m,k}= \Big[ \frac{k}{2^m},\frac{k}{2^{m+1}} \Big]$. Taking the random variables, for $n=2^m+k$ by $X_n= -1 \cdot\mathbf{1}_{I_{m,k}}$ on the probability space $([0,1],Leb)$ gives you that $X_n\to X\equiv 0$ in probability. However, $\liminf X_n \equiv -1$, since for all $t\in [0,1]$ there are infinitely many $n$'s such that $X_n(t)=-1$.