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.

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