Example 4.5.2 (Behavior of exponential random variables) We assume that $\left\{E_{n}, n \geq 1\right\}$ are iid unit exponential random variables; that is, \[ P\left[E_{n}>x\right]=e^{-x}, \quad x>0 . \] Then \[ P\left[\limsup _{n \rightarrow \infty} E_{n} / \log _{n}=1\right]=1 . \] This result is sometimes considered surprising. There is a (mistaken) tendency to think of iid sequences as somehow roughly constant, and therefore the division by $\log n$ should send the ratio to 0 . However, every so often, the sequence $\left\{E_{n}\right\}$ spits out a large value and the growth of these large values approximately matches that of $\{\log n, n \geq 1\}$.

(a) Suppose $\left\{X_{n}, n \geq 1\right\}$ are iid random variables and suppose $\left\{a_{n}\right\}$ is a sequence of constants. Show \[ P\left\{\left[X_{n}>a_{n}\right] \text { i.o. }\right\}=\left\{\begin{array}{ll} 0, & \text { iff } \sum_{n} P\left[X_{1}>a_{n}\right]<\infty \\ 1, & \text { iff } \sum_{n} P\left[X_{1}>a_{n}\right]=\infty \end{array}\right. \]

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