 StatLect

# Sequences of random vectors and their convergence

In this lecture, we generalize the concepts introduced in the lecture entitled Sequences of random variables and their convergence We no longer consider sequences whose elements are random variables, but we now consider sequences whose generic element is a random vector. The generalization is straightforward, as the terminology and the basic concepts are almost the same used for sequences of random variables. ## Terminology

### Realization of a sequence

Let be a sequence of real vectors and a sequence of random vectors. If the real vector is a realization of the random vector for every , then we say that the sequence of real vectors is a realization of the sequence of random vectors and we write ### Sequences on a sample space

Let be a sample space. Let be a sequence of random vectors. We say that is a sequence of random vectors defined on the sample space if all the random vectors belonging to the sequence are functions from to .

### Independent sequences

Let be a sequence of random vectors defined on a sample space . We say that is an independent sequence of random vectors (or a sequence of independent random vectors) if every finite subset of (i.e. every finite subset of random vectors belonging to the sequence) is a set of mutually independent random vectors.

### Identically distributed sequences

Let be a sequence of random vectors. Denote by the joint distribution function of a generic element of the sequence . We say that is a sequence of identically distributed random vectors if any two elements of the sequence have the same joint distribution function: ### IID sequences

Let be a sequence of random vectors defined on a sample space . We say that is a sequence of independent and identically distributed random vectors (or an IID sequence of random vectors), if is both a sequence of independent random vectors and a sequence of identically distributed random vectors.

### Stationary sequences

Let be a sequence of random vectors defined on a sample space . Take a first group of successive terms of the sequence , ..., . Now take a second group of successive terms of the sequence , ..., . The second group is located positions after the first group. Denote the joint distribution function of the first group of terms by and the joint distribution function of the second group of terms by The sequence is said to be stationary (or strictly stationary) if and only if for any and for any vector .

In other words, a sequence is strictly stationary if and only if the two random vectors and have the same distribution (for any , and ). Requiring strict stationarity is weaker than requiring that a sequence be IID (see IID sequences above): if is an IID sequence, then it is also strictly stationary, while the converse is not necessarily true.

### Weakly stationary sequences

Let be a sequence of random vectors defined on a sample space . We say that is a covariance stationary sequence (or weakly stationary sequence) if where and are, of course, integers. Property (1) means that all the random vectors belonging to the sequence have the same mean. Property (2) means that the cross-covariance between a term of the sequence and the term that is located positions before it ( ) is always the same, irrespective of how has been chosen. In other words, depends only on and not on . Note also that property (2) implies that all the random vectors in the sequence have the same covariance matrix (because ): ### Mixing sequences

The definition of mixing sequence of random vectors is a straightforward generalization of the definition of mixing sequence of random variables, which has been discussed in the lecture entitled Sequences of random variables and their convergence. Therefore, we report here the definition of mixing sequence of random vectors without further comments and we refer the reader to the aforementioned lecture for an explanation of the concept of mixing sequence.

Definition We say that a sequence of random vectors is mixing (or strongly mixing) if and only if for any two functions and and for any and .

### Ergodic sequences

As in the previous section, we report here a definition of ergodic sequence of random vectors, which is a straightforward generalization of the definition of ergodic sequence of random variables, and we refer the reader to the lecture entitled Sequences of random variables and their convergence for explanations of the concept of ergodicity.

Denote by the set of all possible sequences of real vectors. When is a sequence of real vectors, denote by the subsequence obtained by dropping the first term of , that is, We say that a subset is a shift invariant set if and only if belongs to whenever belongs to .

Definition A set is shift invariant if and only if Shift invariance is used to define ergodicity.

Definition A sequence of random vectors is said to be an ergodic sequence if an olny if whenever is a shift invariant set.

## Convergence

Similarly to what happens for sequences of random variables, there are several different notions of convergence also for sequences of random vectors. In particular, all the modes of convergence found for random variables can be generalized to random vectors:

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