Weak Law of Large Numbers proof I want to know if there is a proof of the Weak Law of Large Numbers without using the Chebyshev's Inequality? please can anyone give me some references
 A: Let $\{X_n\}$ be a sequence of i.i.d. random variables with finite expectation $\mu$ and finite variance. One can prove that
$$ \frac{1}{n}\sum_{k=1}^n X_k \stackrel{P}{\rightarrow} \mu $$
using characteristic functions. 
Let $\varphi$ be the characteristic function of $X=X_n$ (it is the same for any $n$): $\varphi(t)=E[e^{itX}]$. We have $\varphi'(t)=E[iXe^{itX}]$ (because $X$ is integrable) and $\varphi''(t)=-E[X^2e^{itX}]$ (because $X^2$ is integrable). By Taylor's theorem, there exists $\xi_t$ between $0$ and $t$ such that
$$  \varphi(t)=1+i\mu t-\frac{E[X^2e^{i\xi_t X}]}{2}t^2. $$
The characteristic function of $\frac{1}{n}\sum_{k=1}^n X_k$ is
$$\psi_n(t)=\varphi\left(\frac{t}{n}\right)^n=\left(1+i\mu \frac{t}{n}-\frac{E[X^2e^{i\xi_{t/n} X}]}{2}\frac{t^2}{n^2}\right)^n=\left(1+\frac{i\mu t-E[X^2e^{i\xi_{t/n} X}]t^2/(2n)}{n}\right)^n. $$
Now use the following result: if $\{x_n\}\rightarrow x$ and $\lambda_n\rightarrow\infty$, then $(1+x_n/\lambda_n)^{\lambda_n}\rightarrow e^x$. In this case, as
$$ \left|\frac{E[X^2e^{i\xi_{t/n} X}]t^2}{2n}\right|\leq \frac{t^2 E[X^2]}{2n}\rightarrow0,$$
we have 
$$\lim_n\psi_n(t)=i\mu t, $$
which is the characteristic function of the constant random variable $\mu$. By Lévy's continuity theorem, 
$$ \frac{1}{n}\sum_{k=1}^n X_k \stackrel{L}{\rightarrow} \mu. $$
Since $\mu$ is constant, convergence in probability and in law are equivalent:
$$ \frac{1}{n}\sum_{k=1}^n X_k \stackrel{P}{\rightarrow} \mu. $$
