# Convergence of random variable

In this post, we will pursue a approach to the strong law of large numbers based on the convergence of random variables.

Definition (Tail $\sigma$-algebra) Let  $(X_n)_{n\geq 1}$ be a sequence of random variables. Let $\mathcal{F}_n=\sigma(X_n,X_{n+1},\dots)$ be the smallest $\sigma$-algebra with respect to which all $X_n:m\geq n$ are measure. The tail $\sigma$-algebra $\mathcal{T}:=\cap_n \mathcal{F}_n$ is the intersection of all $\mathcal{F}_n$.

Theorem (Kolmogorov’s 0-1 law) If $X_1,X_2,\dots$ are independent and $A\in\mathcal{T}$, then $P(A)=0$ or $1$.

Theorem (Kolmogorov’s maximal inequality) Suppose that $X_1,\dots,X_n$ are independent with $\mathbf{E}X_i=0$ and $\mathbf{Var}X_i<\infty$. If $S_n=X_1+\cdots+X_n$ then

$\displaystyle \mathbf{P}\left(\max_{1\leq k\leq n}|S_k|\geq x\right)\leq \frac{\mathbf{Var}S_n}{x^2}.$

Theorem Suppose that $X_1,X_2,X_3,\dots$ are independent and have $\mathbf{E}X_n=0$. If

$\displaystyle \sum_{n=1}^\infty \mathbf{Var} X_n<\infty,$

then with probability one $\sum_{n=1}^\infty X(\omega)$ converges.