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Hi I need some help with the next exercise: Let $Y_1,Y_2...$ be independent random variables with mean zero and finite variances and let $X_n=Y_1+Y_2+....+Y_n$

Use Doob´s inequality to show that:

$$P(\max_{1\leq k\leq n}|X_k|>\epsilon)\leq\frac{\sum_{k=1}^nVar Y_k}{\epsilon^2}\ \ \ for\ \ \epsilon>0$$

Here, I did the next:

Using the Doob's inequality, I have:

$P(\max_{1\leq k\leq n}|X_k|>\epsilon)\leq\frac{E[X_n^2]}{\epsilon^2}\ \ \ for\ \ \epsilon>0$, and since the expectation of every random variable is $0$. I have: $$Var(X_n)=E[X_n^2]-E[X_n]E[X_n]$$, note that $E[X_n]=E[Y_1]+E[Y_2]+...=0$ and also $Var(X_n)=Var[Y_1+...+Y_n]=\sum_{k=1}^nVar[Y_k]$. using these facts I get the claim. Is ok my propose?

2)Asuume further that $$\sum_{k=1}^{\infty}\frac{1}{k^2}Var(Y_k)<\infty$$

Use the previous inequality to show that $$Z_n=\sum_{k=1}^{\infty}\frac{Yk}{k}\xrightarrow{a.s}Z$$ as $n\rightarrow\infty$ for some finite random variable $Z$.

3) Deduce that $$\frac{1}{n}\sum_{k=1}^{n}Y_k\xrightarrow{a.s}0\ \ as\ \ n\rightarrow\infty$$

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I think Part(i) is correct.

Part (ii) is the Kolmogorov's two series theorem, whose proof uses the Kolmogorov inequality in Part(i).

Part (iii): Kronecker’s Lemma: if $\sum_{k=1}^n \frac{Y_k}{k}$ converges, then $\frac{\sum_{k=1}^n Y_k}{n} \rightarrow 0.$

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