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Let $X_n$ be an iid sequence of random variable with uniform distribution on $(-1,1)$. Using characteristic functions prove that $X_1+X_2+...+X_n$ has density

$$f(x)= \frac{1}{\pi} \int_{0}^{\infty}{(\frac{\sin(t)}{t})^n \cos(tx)dt}$$

I proved that the characteristic function of $X_n$ is $\phi(t)= (\frac{\sin(t)}{t}) $ Thus the characteristic function of $S_n$ is $(\frac{\sin(t)}{t})^n $

I don't know how this could work. Please help me!

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  • $\begingroup$ Shouldn't you now apply the inverse Fourier transform to get the pdf from the characteristic function? $\endgroup$
    – gt6989b
    Commented Nov 21, 2013 at 17:36
  • $\begingroup$ Try to find the characteristic function of $X$, where $X$ has the density $f$. You get your $\phi$ then you have finished. $\endgroup$
    – Lucien
    Commented Nov 21, 2013 at 18:19

1 Answer 1

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Hint: the characteristic function of $X_1+\dots+X_n$ is indeed $\left(\frac{\sin t}t\right)^n$. The inversion formula for an integrable characteristic function ($n\geqslant 2$) will give the result (cut the integral $\int_{-\infty}^0+\int_0^\infty$ and use the substitution $s=-t$ in the first integral).

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