Convergence of random variables

In the stochastics are different concepts of a threshold concept for random variables. Unlike in the case of real number sequences there is no natural definition for the limiting behavior of random variables with increasing sample size, because the asymptotic behavior of the experiments always depends on the individual realizations and we have therefore formally dealing with the convergence of functions. Therefore, over time, different strong concepts have emerged, the most important of these convergence types are briefly described below.

Requirements

We will formulate the classical concepts of convergence always in the following model: Given a sequence of random variables defined on a probability space. A realization of this sequence is usually denoted by.

Almost sure convergence

The notion of almost sure convergence is most comparable with the formulation of a Sequence. Used especially in the formulation of high laws of large numbers.

We say that the sequence converges almost surely to a random variable if

Applies. Translated, this means that the classical notion of convergence is true for almost all realizations of the sequence of real analysis.

Notation: .

The almost sure convergence corresponds to the pointwise convergence almost everywhere of measure theory.

Convergence in pth mean

An integration of theoretical approach is taken with the notion of convergence in the designated funds. There are not considered individual realizations, but expected values ​​of random variables.

Formal converges in designated funds for a random variable, if

Applies. It is assumed here. This means that the difference in Lp- space converges to. We call this convergence therefore also called convergence.

Because of the inequality of the generalized mean values ​​followed by the convergence in designated funds for in -th from the convergence means.

Convergence in probability

A slightly weaker notion of convergence is the stochastic convergence or convergence in probability. As the name suggests, are not specific realizations of the random variables considered, but probabilities for specific events. A classic application of stochastic convergence are weak laws of large numbers.

The mathematical formulation is as follows: The sequence converges stochastically to a random variable if

Usually following notations are used for the convergence in probability used: or or.

It can be shown that a sequence converges if and only stochastically to, if

That is, the convergence in probability corresponds to the convergence with respect to the metric. The space of all random variables equipped with this metric is a topological vector space, which is not locally convex in general.

The stochastic convergence corresponds to the convergence in measure from the measure theory.

Weak convergence

The fourth prominent notion of convergence is the weak convergence of measures or convergence in distribution ( for random variables ).

Definition of dimensions

Be a metric space and the corresponding Borel σ - algebra. A sequence of finite measures on the measurement space converges weakly to a measure if for all bounded and continuous functions with respect to

Applies.

Application to random variables

A sequence of random variables converges in distribution to the random variable, if the sequence of induced image measure converges weakly to the image size. That is, applies to any continuous function bounded

For real random variable, the following characterization is equivalent to: For the distribution functions of and

At all points at which it is continuous. The most well-known applications of the convergence in distribution are central limit theorems.

Since the convergence are defined in the distribution only by the image dimensions or by the distribution function of the random variables, it is not necessary that the random variables are defined in the same probability space.

As notation is used in the rule or. The letter "W " and " D" stand for the corresponding terms in English, so weak convergence or convergence in distribution.

Associated with characteristic functions

Following relationship exists between the pointwise convergence of the characteristic function of a sequence of random variables and the weak convergence of:

Is still for the opposite direction, the continuity of the limit function to detect zero:

Often the detection of the convergence point by point of the characteristic function is simpler than the direct detection of the convergence of the distribution functions. With the uniqueness of the characteristic function can be determined as well as the limit of the random variable.

Correlation between the individual types of convergence

In the series of the most important concepts of convergence in the stochastic, the two first terms presented the strongest convergence dar. Both types of almost sure convergence and convergence in the pth mean can always be the stochastic convergence of a sequence of random variables are derived. It also follows automatically the convergence in distribution is the weakest one of the types of convergence of stochastic convergence.

In exceptional cases, other implications hold: If a sequence of random variables converges in distribution to a random variable X and X converges almost surely constant, then this sequence converges stochastically.

From the convergence in pth mean it does not follow the almost sure convergence in general. It is regarded by convergence in pth mean but always that there exists a subsequence that converges almost surely.

Conversely, it can be concluded from almost sure convergence in general no convergence in pth mean. However, this conclusion is allowed if there is a common majorant in ( see Theorem from the dominated convergence ). A sequence of random variables converges in, if it converges stochastically and is uniformly integrable.

Example

On the probability space with the Borel sets and the Lebesgue measure, consider the random variable and the sequence of random variables with for ( each has a natural unique decomposition of this type ) is defined as follows:

Because of

Converges in the pth mean against. It follows from the above relation between the individual types of convergence that also stochastically converges to, as also from

Reveals.

The functions are as it were increasingly thin spikes that run over the interval. But For every fixed for infinitely many, as is for infinitely many, so so there is no almost sure convergence of.

For each subsequence of can, however, find a partial subsequence that converges to. If there were a topology of the almost sure convergence, it would follow from this property that almost surely converges to. Thus, this example also shows that there can not be almost certain topology convergence.

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