Liminf of a sequence of random variables

Question

let the $X_{i}$'s be an independent sequence of random variables. How can I write the event [Liminf $X_{n}$ = 0] as a liminf of limsup of sets? Thanks

Answer 1

Try $$ [\,\liminf_{n\to\infty}X_n=0\,]=\bigcap_{k\geqslant1}\left(A_{1/k}\cap B_{-1/k}\right), $$ with $$ A_x=\limsup_{n\to\infty}\,[\,X_n\leqslant x\,]\quad\text{and}\quad B_x=\liminf_{n\to\infty}\,[\,X_n\geqslant x\,]. $$ In the formula above, $\bigcap\limits_{k\geqslant1}$ can be replaced by $\lim\limits_{k\to\infty}$ or $\limsup\limits_{k\to\infty}$ or $\liminf\limits_{k\to\infty}$.