Types of Convergence

We are interested in if a sequence of functions f_n(x) converges to f(x).  These are easier to contrast with one another if stated in a compact, purely notational form.

Uniform Convergence

\forall \epsilon, \exists N \text{ s.t. } \forall n\geq N, \forall x, |f_n(x)-f(x)|<\epsilon.

Informally, if you pick an \epsilon, I can find an N, such that for all functions after N, f_n and f never disagree by more than \epsilon.  This is a strong form of convergence, in contrast to the weak

Pointwise Convergence

\forall x, \forall \epsilon, \exists N \text{ s.t. } \forall n\geq N, |f_n(x)-f(x)|<\epsilon.

or, equivalently,

\forall x, \lim_{n\rightarrow\infty} f_n(x) = f(x).

Informally, if you pick an \epsilon, and a particular x I can find an N, such that for all functions after N, f_n and f never disagree by more than \epsilon on x. Though uniform convergence implies pointwise, the converse is not true. To see why, it might be that certain x require arbitrarily large N. That is, if we think of N as a function N(x,\epsilon), we might have that \sup_x N(x,\epsilon)=\infty.

Convergence With Probability 1

Convergence in Probability

2 Responses to Types of Convergence

  1. Hong Liu says:

    Thanks for the nice post! Waiting for the 3rd and 4th definitions🙂

  2. Anonymous says:

    thanks a lot it help me for my homework

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