コルモゴロフの二級数定理
よって任意の ε > 0 {\displaystyle \epsilon >0} に対し
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}&\mathrm {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 \mathrm {P} \left(2\max _{k\in \mathbb {N} }\left|\sum _{i=1}^{k}X_{m+i}\right|\geq \epsilon \ \right)\\&=\mathrm {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}}}