Martingales and Integrals question I'm stuck with an martingales exercise here:
$$\lim_{n\to\infty}\int_0^1\int_0^1\cdots\int_0^1\sin\left(\frac{x_1+x_2+\cdots+x_n}{n}\right)dx_1dx_2\cdots dx_n$$
I tried to do it without martingales and it work fine for me.
But I can't see how to apply the definition of Martingales here.
Can someone please explain me what I exactly have to do?
Thank's!
 A: Let $X_1,X_2,X_3,...$ be a sequence of i.i.d. random variables which are distributed uniformly over $[0,1]$.
Then we know that $$\int_0^1\int_0^1\cdots\int_0^1\sin\left(\frac{x_1+x_2+\cdots+x_n}{n}\right)dx_1dx_2\cdots dx_n \;\;= \;\;E \bigg[\sin \bigg( \frac{X_1+...+X_n}{n} \bigg) \bigg]$$
By the law of large numbers, we know that $\frac{X_1+...+X_n}{n}$ converges almost surely to $1/2$. Thus, by continuity of the sine function, we know that $\sin \big( \frac{X_1+...+X_n}{n} \big)$ converges almost surely to $\sin(1/2)$.
Hence by using the Dominated convergence theorem as well as the fact that sine is bounded, we see that $$ \lim_{n \to \infty} \int_0^1\int_0^1\cdots\int_0^1\sin\left(\frac{x_1+x_2+\cdots+x_n}{n}\right)dx_1dx_2\cdots dx_n$$ $$=\lim_{n \to \infty} E \bigg[\sin \bigg( \frac{X_1+...+X_n}{n} \bigg) \bigg] = E \bigg[ \lim_{n \to \infty} \;\sin \bigg( \frac{X_1+...+X_n}{n} \bigg) \bigg] $$$$ = E\big[\sin(1/2)\big]=\sin(1/2)$$
I apologize in advance if there is a mistake somewhere in my calculations.
