I have to show that $\frac{1}{n} \sum_{i=1}^n Z_i^2 $ converges almost surely.

Define $Z_n =X_n + W_n Y_n$

Where $X_n$, $Y_n$, $W_n$ are independent.

$X_n$ are IID with $E \ X_n = 0$ and $E \ X_n^2 = 1$.

$Y_n$ have $E \ Y_n =0$ and $E \ Y_n^2 =n^2$.

$P(W_n = 0) = 1 - \frac{1}{n^2}$ and $P(W_n = 1 ) = \frac{1}{n^2}$.

So far I've done the following:

$$\frac{1}{n} \sum_{i=1}^n Z_i^2 = \frac{1}{n} \left( \sum_{i=1}^n X_i^2 + \sum_{i=1}^n (W_iY_i)^2 + 2 \sum_{i=1}^n X_i W_i Y_i \right)$$

I can apply SLLN to the first term and I've shown that $\frac{1}{n} \sum_{i=1}^n W_iY_i \to 0$ a.s.

  • 1
    $\begingroup$ Are you familiar with the three series theorem? That's usually the best way to show that a sum converges almost surely. $\endgroup$ – Marcus M Sep 28 '17 at 14:27

Recall that a sequence of random variables $\xi _n$ converges to a random variable $\xi$ if and only if for every $\varepsilon>0$ $$ P\left (\sup _{k\geqslant n}|\xi _k-\xi |>\varepsilon\right )\to 0\quad (n\to \infty ) $$ and observe that $$ P\left (\sup _{k\geqslant n}\left |\dfrac{1}{k}\sum _{i=1}^kX_iW_iY_i \right |>\varepsilon \right )\leqslant P\left (\exists k\geqslant n \ W_k=1\right ) \leqslant \sum _{k\geqslant n}P(W_k=1)=\sum _{k\geqslant n}\dfrac{1}{n^2}\to 0 $$ as $n\to \infty $ (notice that $\sum (1/n^2)<\infty $). In a similar way, we can prove that $n^{-1}\sum _{i=1}^n(W_iY_i)^2\to 0$ a.s.


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