Topics from Sequences of Independent Random Variables

In reference to Exercise 6 in Chapter 11, explain why $\frac{S_{n}}{n} \underset {n \rightarrow \infty }{\stackrel{P}{\nrightarrow }} 0$.

For a r.v. $X$, show that $\mathbb {E}\left[\left|X\right|\right] < \infty$ if and only if $\sum_{n = 1}^{\infty } \mathbb {P}\left(\left|X\right| \geq n c\right) < \infty$ for some fixed constant $c > 0$.

Let $X_{1}, X_{2}, \ldots$ be independent r.v.s, and let the r.v. $X$ be $\mathcal{A}_{n} = \sigma \left(X_{n}, X_{n+1}, \ldots \right)$ measurable for every $n \geq 1$. Then show that $\mathbb {P}\left(X = c\right) = 1$ for some (finite) constant $c$.

For $n = 1,2, \ldots$, suppose that the r.v.s $X_{n}$ are independent and that $\sum_{n = 1}^{\infty } \mathbb {E}\left[\left|X_{n}\right|\right] < \infty$. Then show that $\sum_{n = 1}^{\infty }\left|X_{n}\right|$ converges a.s.

For $n \geq 1$, let $X_{n}$ and $X$ be r.v.s defined on the measure space $(\Omega , \mathcal{A}, \mu )$. Then, by Theorem 4 in Chapter 3, $X_{n} \xrightarrow [n \rightarrow \infty ]{ } X$ a.e. if and only if $\mu \left(\cap_{n = 1}^{\infty } \cup_{\nu = 0}^{\infty }\left(\left|X_{n+v}-X\right| \geq \frac{1}{k}\right)\right) = 0$ for each arbitrary but fixed $k = 1,2, \ldots$ Replace $\mu$ by a probability measure $P$, and show that $X_{n} \xrightarrow [n \rightarrow \infty ]{ } 0$ a.s. if and only if $\mathbb {P}\left(\lim \sup_{n \rightarrow \infty } A_{n}\right) = 0$, where $A_{n} = \left(\left|X_{n}\right| \geq \frac{1}{k}\right)$ for each arbitrary but fixed $k = 1,2, \ldots$

Consider any events (independent or not) $A_{n}, n \geq 1$, and suppose that $\sum_{n = 1}^{\infty } \mathbb {P}\left(A_{n}\right) < \infty$. Then $\mathbb {P}\left(\lim \sup_{n \rightarrow \infty } A_{n}\right) = 0$. This is so by Theorem 3 in this chapter (see also Theorem 4(i)). Regarding the converse of this statement, we have that, if $\mathbb {P}\left(\limsup_{n \rightarrow \infty } A_{n}\right) = 0$ and the events $A_{n}, n \geq 1$, are independent, then $\sum_{n = 1}^{\infty } \mathbb {P}\left(A_{n}\right) < \infty$. Justify this assertion.

Regarding the converse stated in Exercise 6, if the events $A_{n}, n \geq 1$, are not independent, then $\mathbb {P}\left(\lim \sup_{n \rightarrow \infty } A_{n}\right) = 0$ need not imply that $\sum_{n = 1}^{\infty } \mathbb {P}\left(A_{n}\right) < \infty$. Give one or two concrete examples to demonstrate this assertion.