# Show that $S_n = X_1+ \cdots X_n$ the probability density function $f_{s{_n}}$

We consider $$(\Omega, \mathcal{F}, \mathcal{P})$$ a probability space and $$(X_{n})_{n \in \mathbb{N}}$$ a sequence of independent and identically distributed random variables such that $$X_1 \sim \operatorname{Unif}(-1,1)$$. Show that $$S_n = X_1+ \cdots X_n$$ for $$n \ge 2$$ has the probability density function $$f_{s{_n}} = \frac{1}{n} \int_{0}^{\infty} \Bigg(\frac{\sin(t)}{t} \bigg)^n \cos(xt)dt$$

I have found out that I could use that $$|\frac{\sin(t)}{t}|\le1$$ for $$\forall t \in \mathbb{R}$$ and I have also calculated that $$\int_{0}^{\infty} \Bigg(\frac{\sin(t)}{t} \bigg)^2 dt = \frac{\pi}{2}$$ At this point I have no clue how should I continue it I would really appreciate any kind of help since I'm learning the topics on my own.

• If $X_1,\dots,X_n$ are any independent r.v.s with probability distributions $\mu_1,\dots,\mu_n$, then the probability distribution of their sum is the convolution $\mu_1*\dots*\mu_n$. For the rest, just use the Fourier transform. Commented May 29, 2022 at 17:04
• math.stackexchange.com/q/3615897/321264 Commented May 29, 2022 at 19:29

The characteristic function of $$S_n$$ is $$\phi_{S_n}(t) = E[e^{itS_n}] = \prod_{k=1}^n E[e^{itX_k}] = \left(\frac{e^{it} - e^{-it}}{2it}\right)^n = \left(\frac{\sin t}{t}\right)^n.$$
The inversion theorem shows how you can compute the density $$f_{S_n}$$ if you know the characteristic function $$\phi_{S_n}$$.
\begin{align} f_{S_n}(x) &= \frac{1}{2\pi} \int_{-\infty}^\infty e^{-itx} \phi_{S_n}(t) \, dt \\ &= \frac{1}{2\pi} \int_0^\infty e^{-itx} \phi_{S_n}(t) \, dt + \frac{1}{2\pi} \int_0^\infty e^{iux} \phi_{S_n}(-u) \, du & u = -t \\ &= \frac{1}{2\pi} \int_0^\infty (e^{itx} + e^{-itx}) \phi_{S_n}(t) \, dt & \phi_{S_n}(-t) = \phi_{S_n}(t) \\ &= \frac{1}{\pi} \int_0^\infty \cos(tx) \phi_{S_n}(t) \, dt. \end{align}
• $\frac{1}{\pi}$ makes more sense than $\frac{1}{n}$ Commented May 29, 2022 at 17:24
Characteristic function will work. Ley $$f(x)$$ be density function for $$X_k$$. $$\phi(t)=\int_R e^{itx}f(x)dx$$. Then density function for $$S$$ is given by $$f_S(x)=\frac{1}{2\pi}\int_R e^{-itx}\phi^n(t)dt$$.