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Can you give an example of sequence, that almost surely converge but,but doesn't converge in quadratic mean?

And how can we prove a.s convergenc of sequence of random variables?

For example does sequence of normal random variables with mean: $\lim E(X_n) = \mu$ and variance = $\lim Var(X_n) = 1$$ almost surely converge? If yes how to prove it?

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For your first question: Consider the probability space $(\Omega,\Bbb P) = ([0,1],\lambda)$ and the sequence of random variables defined by$$X_n = n1_{\left[0,\frac{1}{n}\right]}$$

Then $$X_n \to 0$$ $\Bbb P$-a.s but $$E[X_n^2] = n \to \infty$$.

For your sequence of normal r.v. consider for $\mu=0$ a standard normal r.v. $Y \sim \mathcal{N}(0,1)$ and define $$Y_n = \begin{cases} +Y & \mbox{for n even} \\ -Y & \mbox{for n odd} \end{cases}$$

Then $E[Y_n] = 0 = \mu, \text{Var}(Y_n) = 1$ for all $n\in \Bbb N$ but $$\Bbb P\left(\lim_{n\to\infty} Y_n \text{ exists }\right) = P( Y = 0) = 0$$

But you always get the almost sure convergence of a subsequence if you have convergence in any moment.

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  • $\begingroup$ Thank you for your help! But in first question can you explain how did you calculate $E[X_n^2]$? Because I didn't get it. $\endgroup$ – vitsuk Nov 16 '17 at 8:27
  • $\begingroup$ $$\begin{align*} E\left[X_n^2\right] &= E\left[\left(n1_{\left[0,\frac{1}{n}\right]}\right)^2\right] \\ &=E\left[n^21_{\left[0,\frac{1}{n}\right]}\right] \\ &= n^2 E\left[1_{\left[0,\frac{1}{n}\right]}\right]\\ &= n^2 \Bbb P\left(\left[0,\frac{1}{n}\right]\right)\\ &= n^2\cdot\frac{1}{n} = n\end{align*}$$ $\endgroup$ – Gono Nov 16 '17 at 8:53

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