Let $X_1, X_2,\dots$ be an i.i.d sequence of random variables on a probability space $(\Omega,$ F$, \mathbb{P})$ with $\mathbb{P}(X_1 = 1) = \mathbb{P}(X_1 = −1) = \frac12$.
- Show that $\phi_{X_i} (t) = \cos(t)$. ($\phi$ denotes the characteristic function)
- Use 1 to prove that for every $t\in \mathbb{R}$ $\lim_{n\to\infty}\cos^n(\frac{t}n)=1$.
- Use 2 to show that the weak law of large numbers holds for $X_i$’s, i.e., that $\frac{X_1+\dots+X_n}n\to0$ in prob.
My attempt:
We have $\phi_{X_i}(t)=\mathbb{E}[e^{iX_it}]=\frac{e^{it}+e^{-it}}2=\cos(t)$
Here I already get stuck: how should I use my prevoius result? Do I have to consider maybe $(\cos(\frac {t}n))^n=(\frac{e^{i\frac{t}n}+e^{-i\frac{t}n}}2)^n$??
- From a Corollary I know that for $S:=X_1+\dots+X_n$ we have $\phi_S(t)=\phi_{X_1}(t)\cdots\phi_{X_n}(t)=\cos^n(t)\not=\cos^n(\frac{t}n)$, so how do I employ part (2)? Then maybe we should get that $\frac{X_1+\dots+X_n}n\to\frac1n\to0$ pointwise (so why ask for convergence in probability?
Thanks for any advice
For (2) write $\cos^n(t/n)=\exp(n\ln(\cos(t/n)))$. Then \begin{align} n\ln(\cos(t/n))&=\frac{\ln(\cos(t/n))}{1-\cos(t/n)}\times \frac{1-\cos(t/n)}{1/n} \\ &\to -1\times 0 \end{align} as $n\to\infty$. (Note that $\lim_{x\to 0}\ln(1-x)/x=-1$.)
As for $(3)$, write $$ \varphi_{S_n/n}(t)=\left(\varphi_{X_1}(t/n)\right)^n=\cos^n(t/n). $$ Since the RHS converges to $1$ (the ch.f of $\delta_0$), $S_n/n\xrightarrow{d}0$, implying that $S_n/n\xrightarrow{p}0$.