# Almost certain convergence

The almost certain convergence , also P-almost certain convergence or almost certain point-by-point convergence is a term from probability theory , a branch of mathematics. The almost certain convergence is next to the convergence in the p-th mean , the stochastic convergence and the convergence in distribution one of the four most important convergence terms for sequences of random variables and is the probabilistic counterpart to the convergence almost everywhere in the measure theory . The almost certain convergence is used, for example, in the formulation of the strong law of large numbers .

## definition

### General case

Let a probability space and a separable, metric space (such as the one ) be provided with Borel's σ-algebra and random variables from to . The sequence of random variables converges almost surely or P-almost surely against when a lot there with and ${\ displaystyle (\ Omega, {\ mathcal {A}}, P)}$${\ displaystyle (M, d)}$${\ displaystyle \ mathbb {R} ^ {n}}$ ${\ displaystyle {\ mathcal {B}} (M)}$${\ displaystyle X, X_ {n}}$${\ displaystyle (\ Omega, {\ mathcal {A}}, P)}$${\ displaystyle (M, {\ mathcal {B}} (M))}$${\ displaystyle (X_ {n}) _ {n \ in \ mathbb {N}}}$${\ displaystyle X}$${\ displaystyle N \ in {\ mathcal {A}}}$${\ displaystyle P (N) = 0}$

${\ displaystyle \ lim _ {n \ to \ infty} d (X (\ omega), X_ {n} (\ omega)) = 0}$

for everyone . One then also writes , or -fs ${\ displaystyle \ omega \ in \ Omega \ setminus N}$${\ displaystyle X_ {n} {\ xrightarrow [{}] {fs}} X}$${\ displaystyle X_ {n} {\ xrightarrow [{}] {P {\ text {-fs}}}} X}$${\ displaystyle X_ {n} \ rightarrow X \; P}$

### For real random variables

Alternatively, for real random variables there is also the formulation that the random variables converge almost certainly if and only if

${\ displaystyle P \ left (\ {\ omega \ in \ Omega \ colon \ lim _ {n \ to \ infty} X_ {n} (\ omega) = X (\ omega) \} \ right) = 1}$

is.

## Examples

As an example, consider the basic set of real numbers in the interval from 0 to 1, i.e. provided with Borel's σ-algebra . The probability measure is the Dirac measure on the 1, that is ${\ displaystyle \ Omega = [0,1]}$${\ displaystyle {\ mathcal {B}} ([0,1])}$${\ displaystyle P}$

${\ displaystyle \ delta _ {1} (A): = {\ begin {cases} 1 \, & {\ text {if}} 1 \ in A \, \\ 0 \, & \ mathrm {otherwise} \. \ end {cases}}}$

for . Given are two random variables from to defined by ${\ displaystyle A \ in {\ mathcal {B}} ([0,1])}$${\ displaystyle ([0,1], {\ mathcal {B}} ([0,1]), \ delta _ {1})}$${\ displaystyle ([0,1], {\ mathcal {B}} ([0,1]))}$

${\ displaystyle X (\ omega): = {\ begin {cases} 1 & {\ text {if}} \ omega = 1 \\ 0 & {\ text {otherwise}} \ end {cases}}}$.

and

${\ displaystyle Y (\ omega): = 0 {\ text {for all}} \ omega \ in [0,1]}$

A sequence of random variables is defined by

${\ displaystyle X_ {n} (\ omega): = \ left (1 - {\ tfrac {1} {n}} \ right) \ chi _ {[0,1)} (\ omega)}$.

It denotes the characteristic function . The sequence of random variables converges for towards infinity for each towards 1 and for towards 0. Hence, ${\ displaystyle \ chi}$${\ displaystyle (X_ {n}) _ {n \ in N}}$${\ displaystyle n}$${\ displaystyle \ omega \ in [0,1)}$${\ displaystyle \ omega = 1}$

${\ displaystyle \ {\ omega \ in \ Omega | \ lim _ {n \ to \ infty} X_ {n} (\ omega) = X (\ omega) \} = \ emptyset}$,

therefore they do not almost certainly converge against , since for every probability measure holds. But it is ${\ displaystyle X_ {n}}$${\ displaystyle X (\ omega)}$${\ displaystyle P (\ emptyset) = 0}$

${\ displaystyle \ {\ omega \ in \ Omega \ colon \ lim _ {n \ to \ infty} X_ {n} (\ omega) = Y (\ omega) \} = \ {1 \}}$

But there is, they almost certainly converge to , although the point-wise convergence only takes place in a single point. However, this is maximally weighted by the Dirac measure. ${\ displaystyle \ delta _ {1} (\ {1 \}) = 1}$${\ displaystyle X_ {n}}$${\ displaystyle Y}$

## properties

The almost certain convergence of the sequence is equivalent to that ${\ displaystyle (X_ {n}) _ {n \ in \ mathbb {N}}}$

${\ displaystyle P \ left (\ bigcup _ {m = n} ^ {\ infty} \ left \ {\ omega \ in \ Omega \, | \, \ vert X_ {m} -X \ vert \ geq \ epsilon \ right \} \ right) {\ xrightarrow [{}] {n \ to \ infty}} 0}$

applies. With the Bonferroni inequality one then obtains the following sufficient criterion for the almost certain convergence:

${\ displaystyle \ sum _ {m = 1} ^ {\ infty} P (| X-X_ {m} | \ geq \ epsilon) \ quad <\ infty}$

for everyone . The terms of the shape can then be estimated using the Markov inequality, for example . ${\ displaystyle \ epsilon> 0}$${\ displaystyle P (| X-X_ {m} | \ geq \ epsilon)}$

## Relationship to other types of convergence in stochastics

In general, the implications apply to the concepts of convergence in probability theory

${\ displaystyle {\ begin {matrix} {\ text {almost certain}} \\ {\ text {convergence}} \ end {matrix}} \ implies {\ begin {matrix} {\ text {convergence in}} \\ {\ text {probability}} \ end {matrix}} \ implies {\ begin {matrix} {\ text {convergence in}} \\ {\ text {distribution}} \ end {matrix}}}$

and

${\ displaystyle {\ begin {matrix} {\ text {convergence in}} \\ {\ text {p-th mean}} \ end {matrix}} \ implies {\ begin {matrix} {\ text {convergence in} } \\ {\ text {probability}} \ end {matrix}} \ implies {\ begin {matrix} {\ text {convergence in}} \\ {\ text {distribution}} \ end {matrix}}}$.

Almost certain convergence is thus one of the strong convergence concepts in probability theory. The relationships to the other types of convergence are detailed in the sections below.

### Convergence in probability

The convergence in probability follows from the almost certain convergence . To see this, one defines the quantities

${\ displaystyle B_ {N}: = \ {\ omega \ in \ Omega \ colon \ vert X_ {n} -X \ vert <\ epsilon \ quad \ forall n \ geq N \} {\ text {and}} B : = \ bigcup _ {i = 1} ^ {\ infty} B_ {i}}$.

They form a monotonically increasing sequence of sets , and the set contains the set ${\ displaystyle B_ {N}}$${\ displaystyle B}$

${\ displaystyle A: = \ {\ omega \ in \ Omega \ colon \ lim _ {n \ to \ infty} X_ {n} = X \}}$

of the elements on which the sequence converges point by point. According to prerequisite is and therefore also and accordingly . The statement then follows by forming a complement. ${\ displaystyle P (A) = 1}$${\ displaystyle P (B) = 1}$${\ displaystyle \ lim _ {N \ to \ infty} P (B_ {N}) = 1}$

However, the reverse is generally not true. An example of this is the consequence of Bernoulli distributed random variables for parameters , ie . Then ${\ displaystyle {\ tfrac {1} {n}}}$${\ displaystyle X_ {n} \ sim \ operatorname {Ber} _ {\ frac {1} {n}}}$

${\ displaystyle \ lim _ {n \ to \ infty} P (| X_ {n} | \ geq \ epsilon) = 0}$

for all and thus the sequence converges in probability to 0. The sequence does not converge almost definitely; this is shown with the sufficient criterion for almost certain convergence and the Borel-Cantelli lemma . ${\ displaystyle \ epsilon> 0}$

Conditions under which the probability of the convergence leads to an almost certain convergence:

• The convergence speed of the convergence in probability is sufficiently fast, i.e. it applies
${\ displaystyle \ sum _ {i = 1} ^ {\ infty} P (\ vert X_ {i} -X \ vert \ geq \ epsilon) <\ infty}$.
• The base space can be represented as a countable union of μ-atoms . This is always possible for probability spaces with at most a countable basic set.${\ displaystyle \ Omega}$
• If the sequence of the random variables is almost certainly strictly monotonically falling and converges in probability to 0, then the sequence almost certainly converges to 0.

More generally, any sequence that converges in probability has a subsequence that is almost certain to converge.

### Convergence in distribution

The scorochod representation makes a statement about the conditions under which the convergence in distribution can be used to infer the almost certain convergence.

### Convergence in the pth mean

In general, the convergence in the p-th mean does not result in an almost certain convergence. For example, if one considers a sequence of random variables with

${\ displaystyle P (X_ {n} = 0) = 1-P (X_ {n} = 1) = 1 - {\ tfrac {1} {n}}}$,

so is for everyone ${\ displaystyle p> 0}$

${\ displaystyle \ operatorname {E} (| X_ {n} | ^ {p}) = P (X_ {n} = 1) = {\ tfrac {1} {n}}}$,

what converges to zero. Thus the random variables converge on the -th mean to 0. However, the dependency structure of the one another (that is, the interaction of the carriers in in ) can be designed in such a way that they do not almost certainly converge. A similar but more detailed and concrete example can be found in the article Convergence (Stochastics) . ${\ displaystyle p}$${\ displaystyle X_ {n}}$${\ displaystyle X_ {n}}$${\ displaystyle \ Omega}$${\ displaystyle X_ {n}}$

However, if a sequence of random variables in the p-th mean converges to and holds ${\ displaystyle (X_ {n}) _ {n \ in \ mathbb {N}}}$${\ displaystyle X}$

${\ displaystyle \ sum _ {n = 1} ^ {\ infty} \ operatorname {E} (| X_ {n} -X | ^ {p}) <\ infty}$,

then the sequence will almost certainly converge against . The convergence must therefore be “fast enough”. (Alternatively, one can also use the fact that if Vitali 's convergence theorem is valid, the convergence according to probability and the almost certain convergence coincide. If the requirements of this theorem are fulfilled, then the convergence in the -th mean leads to the almost certain convergence, since the convergence in the -th mean automatically follows the convergence in probability.) ${\ displaystyle X}$${\ displaystyle p}$${\ displaystyle p}$

Conversely, the almost certain convergence does not result in convergence in the -th mean. For example, if we consider the random variables on the probability space${\ displaystyle p}$${\ displaystyle ([0,1], {\ mathcal {B}} ([0,1]), {\ mathcal {U}} _ {[0,1]})}$

${\ displaystyle X_ {n} (\ omega) = n ^ {2} \ cdot \ mathbf {1} _ {\ left [0, {\ tfrac {1} {n}} \ right]} (\ omega)}$,

so this converges point by point towards 0 and thus almost certainly towards 0 ( here denotes the uniform distribution on ). ${\ displaystyle \ omega \ in (0,1]}$${\ displaystyle [0,1]}$${\ displaystyle {\ mathcal {U}} _ {[0,1]}}$${\ displaystyle [0,1]}$

But it is and the consequence is therefore unlimited for all , so it cannot converge. ${\ displaystyle \ operatorname {E} (| X_ {n} | ^ {p}) = n ^ {2p-1}}$${\ displaystyle p \ geq 1}$

However, the theorem of majorized convergence provides a criterion under which this conclusion is correct. If the converges almost certainly and a random variable exists with and is almost certain, then the -th means converge against and also for holds . ${\ displaystyle X_ {n}}$${\ displaystyle Y}$${\ displaystyle \ operatorname {E} (\ vert Y \ vert ^ {p}) <\ infty}$${\ displaystyle X_ {n} \ leq Y}$${\ displaystyle X_ {n}}$${\ displaystyle p}$${\ displaystyle X}$${\ displaystyle X}$${\ displaystyle \ operatorname {E} (\ vert X \ vert ^ {p}) <\ infty}$