Convergence of Functions
The concept of convergence of a real sequence in general is that, given some measure of distance or size in a space, a sequence converges to a limit if the distance between the sequence and the limit becomes arbitrarily small eventually.
When considering convergence of sequences of functions we need to be more careful. There are several different kinds of functional convergence with differing strengths. In order from strongest to weakest the following are all types of functional convergence:
- [[uniform-convergence]]
- pointwise convergence
- Convergence in Norm
- [[convergence-in-measure]]
1 Pointwise Convergence
The sequence of functions \(f_n:[0,1]\to\mathbb{R}\) converges pointwise to \(f\), written \(f_n\stackrel{p.w.}{\to}f\) \((n\to\infty)\), if for every \(x\in[0,1]\) we have \(\lim_{n\to\infty}f_n(x)=f(x)\).
2 Vague Convergence
The sequence of distribution functions \(F_{n}\) converges vaguely to \(F\) (\(F_{n}\stackrel{v}{\to}F\)), if for every finite interval of continuity \(I=(a,b]\) of \(F\) (i.e. \(a,b \in \mathcal{C}(F)\)), we have \(F_{n}(I)\to F(I)\).