Convergence of Random Variables

Abstract

In elementary mathematics courses (such as Calculus) one speaks of the convergence of functions: f_n:R → R, then lim_n→∞ f_n = f if lim_n→∞ f_n(x) = f( _x ) for all x in R. This is called pointwise convergence of functions. A random variable is of course a function (X: Ω → R for an abstract space Ω), and thus we have the same notion: a sequence X _n : Ω → R converges pointwise to X if lim_n→∞X _n (ω) = X(ω), for all ω ∈ Ω. This natural definition is surprisingly useless in probability. The next example gives an indication why.