Kolmogorov criterion

The sequence of variates X_i with corresponding means μ_i obeys the strong law of large numbers if, to every pair ϵ, δ>0, there corresponds an N such that there is probability 1 - δ or better that for every r>0, all r + 1 inequalities ( left bracketing bar S_n - m_n right bracketing bar )/n<ϵ for n = N, N + 1, ..., N + r will be satisfied, where S_n | congruent | sum_(i = 1)^n X_n m_n | congruent | 〈S_n 〉 = μ_1 + ... + μ_n (Feller 1968).

frivolous theorem of arithmetic | law of large numbers | law of truly large numbers | strong law of small numbers