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IID sequence

The acronym IID stands for "Independent and Identically Distributed". A sequence of random variables (or random vectors) is IID if the terms of the sequence are mutually independent and they all have the same probability distribution.

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A formal definition follows.

Definition Let [eq1] be a sequence of random vectors. Let [eq2] be the joint distribution function of a generic term of the sequence [eq3]. We say that [eq4] is an IID sequence if and only if[eq5]and any subset of terms of the sequence is a set of mutually independent random vectors.


The requirement that a sequence of random variables be IID is often found in asymptotic theory, for example to derive some versions of the Law of Large Numbers and of the Central Limit Theorem.

The requirement that a sequence be IID is very strong and is often replaced by milder conditions such as covariance stationarity or mixing (see the lecture entitled Sequences of random variables).

More details

More details about IID sequences are presented in the lecture entitled Sequences of random variables.

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