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This is exercise 2.3.17 in Probabilty:Theory and Example p67.

$X_1,X_2,X_3\cdots$ are i.i.d random variables. The reader is reqeusted to find the equivalent condition for each of the four statement.

  1. $\frac{X_n}{n}\rightarrow 0$ almost surely
  2. $(\max_{m\leq n}X_m)/n \rightarrow0$ almost surely
  3. $\frac{X_n}{n}\rightarrow 0$ in probability
  4. $(\max_{m\leq n}X_m)/n \rightarrow0$ in probability

I think the equivalent conditions should be some statement about the first moments,which is better working definition. May someone some kind give me the right route to tackle these problems? Especially about how the i.i.d. condition should be used in these problem.

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