91Ó°ÊÓ

Suppose that the sequences \(\left\\{X_{n}: n \geq 1 \mid\right.\) and \(\left[Y_{n}: n \geq 1\right]\) are tail equivalent, which is to say that \(\sum_{n=1}^{\infty} P\left(X_{n} \not Y_{n}\right)<\infty\). Show that: (a) \(\sum_{n=1}^{\infty} X_{n}\) and \(\sum_{n=1}^{\infty} Y_{n}\) converge or diverge together, (b) \(\sum_{n=1}^{\infty}\left(X_{n}-Y_{n}\right)\) converges almost surely, (c) if there exist a random variable \(X\) and a sequence \(a_{n}\) such that \(a_{n} \uparrow \infty\) and \(a_{n}^{-1} \sum_{r=1}^{n} X_{r} \stackrel{\text { a.s }}{\longrightarrow} X\), then $$ \frac{1}{a_{n}} \sum_{r=1}^{n} Y_{r} \stackrel{a s}{\longrightarrow} X $$.

Let \(\left\\{X_{n}: n \geq 1\right\\}\) be independent random variables. Show that \(\sum_{n=1}^{\infty} X_{n}\) comerges a.s. if, for some \(a>0\), the following three series all converge: (a) \(\sum_{n} \mathrm{P}\left(\left|X_{n}\right|>a\right)\) (b) \(\sum_{n} \operatorname{var}\left(X_{n} I_{\left|I X_{n}\right| \leq a \mid}\right)\). (c) \(\sum_{m} \mathbb{E}\left(X_{n} I_{\left.\| X_{n} \mid \leq a\right)}\right)\), [The converse holds also, but is harder to prove.]

Let \(\left(X_{n}: n \geq 1\right)\) be independent random variables with continuous common distribution function \(F\). We call \(X_{k}\) a record value for the sequence if \(X_{k}>X_{r}\) for \(1 \leq r

Let \(\left\\{X_{n}: n \geq 1\right)\) be a sequence of independent random variables with \(\mathrm{P}\left(X_{n}=1\right)=\mathrm{P}\left(X_{n}=-1\right)=\frac{1}{2}\). Does the series \(\sum_{r=1}^{n} X_{r} / r\) converge a.s. as \(n \rightarrow \infty\) ?