Pointwise convergence of $X_n$ vs $X_nI_{\{|X_n|\leq c_n\}}$ and of $\sum X_n$ vs $\sum X_nI_{\{|X_n|\leq c_n\}}$

Question

Let $\{X_n,n\geq 1\}$ be a sequence of random variables, and $\{c_n,n\geq1\}$ a positive sequence. Let also $\sum_n P(|X_n|> c_n)<\infty$. Prove:

1. If $Y_n=X_nI_{\{|X_n|\leq c_n\}}$ and $P\left( {\mathop {\lim }\limits_n {Y_n} = X} \right) = 1$, then $$P\left( {\mathop {\lim }\limits_n {X_n}} =X\right) = 1$$
2. $$P\left( {\sum\limits_n {{X_n}} } \,\text{converges}\,\right) = P\left( {\sum\limits_n {{X_n}{I_{\left\{ {\left| {{X_n}} \right| \le {c_n}} \right\}}}} } \,\text{converges}\,\right)$$

I have tried to prove it by the following method.

First, by Borel-Cantelli Theorem, we have $$P\left( {\mathop {\lim \sup }\limits_n {\mkern 1mu} \left| {{X_n}} \right| > {c_n}} \right) = 0.$$ And$$0 \le P\left( {\mathop {\lim \inf }\limits_n {\mkern 1mu} \left| {{X_n}} \right| > {c_n}} \right) \le P\left( {\mathop {\lim \sup }\limits_n {\mkern 1mu} \left| {{X_n}} \right| > {c_n}} \right) = 0,$$then $$P\left( {\mathop {\lim \inf }\limits_n {\mkern 1mu} \left| {{X_n}} \right| > {c_n}} \right) = 0.$$ Since $P\left( {\mathop {\lim }\limits_n {Y_n} = X} \right) = 1$,we obtain $$P\left( {\mathop {\lim }\limits_n \left| {{X_n}} \right| \le {c_n}} \right) = 1.$$ Hence $$P\left( {\mathop {\lim \inf }\limits_n {\mkern 1mu} {{X_n}} } \right) = P\left( {\mathop {\lim \inf }\limits_n {\mkern 1mu} \left| {{X_n}} \right| > {c_n}} \right) + P\left( {\mathop {\lim \inf }\limits_n {\mkern 1mu} \left| {{X_n}} \right| \le {c_n}} \right) = 1.$$ Then $$1 = P\left( {\mathop {\lim \inf }\limits_n {\mkern 1mu} {X_n}} \right) \le P\left( {\mathop {\lim }\limits_n {\mkern 1mu} {X_n} = X} \right) \le 1,$$ which means $$P\left( {\mathop {\lim }\limits_n {X_n}} =X\right) = 1.$$

It seems something wrong! Thank everyone for good ideas!

Answer 1

By Hypothesis, $\sum_n P(\{X_n\neq Y_n\})<\infty$.

By Borel-Cantelli Lemma, $P\left( {\mathop {\lim \inf }\limits_n {\mkern 1mu} {{\{X_n=Y_n\}}}} \right) = 1.$

Put $A={\mathop {\lim \inf }\limits_n {\mkern 1mu} {{\{X_n=Y_n\}}}}$

For every $\omega \in A, X_n(\omega)=Y_n(\omega)$ for all large n.

This shows, for every $\omega \in A,$ $$\mathop {\lim}\limits_n X_n(\omega)=\mathop {\lim}\limits_n Y_n(\omega)$$

$$\sum_n X_n(\omega)\;converges \Leftrightarrow \sum_n Y_n(\omega)\;converges$$

Now, the required results follow easily.