 Dudley's theorem

Dudley's theorem, named after the statistician R. M. Dudley, is a result in the field of probability theory and functional analysis, specifically concerning the behavior of sums of independent random variables. The theorem is particularly significant in the context of proving the almost sure convergence of certain types of series of random variables. In its basic form, Dudley's theorem states that if you have a series of independent, identically distributed (i.i.d.) random variables that are centered (i.e.