# 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:

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