In probability theory, Kolmogorov's two-series theorem is a result about the convergence of random series. It follows from Kolmogorov's inequality and is used in one proof of the strong law of large numbers.
Statement of the theorem Let ( X n ) n = 1 ∞ {\displaystyle \left(X_{n}\right)_{n=1}^{\infty }} be independent random variables with expected values E [ X n ] = μ n {\displaystyle \mathbf {E} \left[X_{n}\right]=\mu _{n}} and variances V a r ( X n ) = σ n 2 {\displaystyle \mathbf {Var} \left(X_{n}\right)=\sigma _{n}^{2}} , such that ∑ n = 1 ∞ μ n {\displaystyle \sum _{n=1}^{\infty }\mu _{n}} converges in R {\displaystyle \mathbb {R} } and ∑ n = 1 ∞ σ n 2 {\displaystyle \sum _{n=1}^{\infty }\sigma _{n}^{2}} converges in R {\displaystyle \mathbb {R} } . Then ∑ n = 1 ∞ X n {\displaystyle \sum _{n=1}^{\infty }X_{n}} converges in R {\displaystyle \mathbb {R} } almost surely.
Proof Assume WLOG μ n = 0 {\displaystyle \mu _{n}=0} . Set S N = ∑ n = 1 N X n {\displaystyle S_{N}=\sum _{n=1}^{N}X_{n}} , and we will see that lim sup N S N − lim inf N S N = 0 {\displaystyle \limsup _{N}S_{N}-\liminf _{N}S_{N}=0} with probability 1. For every m ∈ N {\displaystyle m\in \mathbb {N} } ,
lim sup N → ∞ S N − lim inf N → ∞ S N = lim sup N → ∞ ( S N − S m ) − lim inf N → ∞ ( S N − S m ) ≤ 2 max k ∈ N | ∑ i = 1 k X m + i | {\displaystyle \limsup _{N\to \infty }S_{N}-\liminf _{N\to \infty }S_{N}=\limsup _{N\to \infty }\left(S_{N}-S_{m}\right)-\liminf _{N\to \infty }\left(S_{N}-S_{m}\right)\leq 2\max _{k\in \mathbb {N} }\left|\sum _{i=1}^{k}X_{m+i}\right|}
Thus, for every m ∈ N {\displaystyle m\in \mathbb {N} } and ϵ > 0 {\displaystyle \epsilon >0} ,
P ( lim sup N → ∞ ( S N − S m ) − lim inf N → ∞ ( S N − S m ) ≥ ϵ ) ≤ P ( 2 max k ∈ N | ∑ i = 1 k X m + i | ≥ ϵ ) = P ( max k ∈ N | ∑ i = 1 k X m + i | ≥ ϵ 2 ) ≤ lim sup N → ∞ 4 ϵ − 2 ∑ i = m + 1 m + N σ i 2 = 4 ϵ − 2 lim N → ∞ ∑ i = m + 1 m + N σ i 2 {\displaystyle {\begin{aligned}\mathbb {P} \left(\limsup _{N\to \infty }\left(S_{N}-S_{m}\right)-\liminf _{N\to \infty }\left(S_{N}-S_{m}\right)\geq \epsilon \right)&\leq \mathbb {P} \left(2\max _{k\in \mathbb {N} }\left|\sum _{i=1}^{k}X_{m+i}\right|\geq \epsilon \ \right)\\&=\mathbb {P} \left(\max _{k\in \mathbb {N} }\left|\sum _{i=1}^{k}X_{m+i}\right|\geq {\frac {\epsilon }{2}}\ \right)\\&\leq \limsup _{N\to \infty }4\epsilon ^{-2}\sum _{i=m+1}^{m+N}\sigma _{i}^{2}\\&=4\epsilon ^{-2}\lim _{N\to \infty }\sum _{i=m+1}^{m+N}\sigma _{i}^{2}\end{aligned}}}
While the second inequality is due to Kolmogorov's inequality. By the assumption that ∑ n = 1 ∞ σ n 2 {\displaystyle \sum _{n=1}^{\infty }\sigma _{n}^{2}} converges, it follows that the last term tends to 0 when m → ∞ {\displaystyle m\to \infty } , for every arbitrary ϵ > 0 {\displaystyle \epsilon >0} .
References
Durrett, Rick. Probability: Theory and Examples. Duxbury advanced series, Third Edition, Thomson Brooks/Cole, 2005, Section 1.8, pp. 60–69. M. Loève, Probability theory, Princeton Univ. Press (1963) pp. Sect. 16.3 W. Feller, An introduction to probability theory and its applications, 2, Wiley (1971) pp. Sect. IX.9