# Types of Convergence

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

**Uniform Convergence**

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

**Pointwise Convergence**

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or, equivalently,

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

Convergence With Probability 1

Convergence in Probability

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

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