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  • Convergent iff there exists a number L such that for any ϵ > 0 there is an N such that
    | L - a n | ϵ for all n N
  • Fundamental iff for any ϵ > 0 there is an N such that
    | a n - a m | ϵ for all n , m N

As a result of the completeness of the real numbers, it is true that any fundamental sequence converges (i.e., has a limit). And such convergence has certain desirableproperties. For example the limit of a linear combination of sequences is that linear combination of the separate limits; and limits of products are the products of the limits.

The notion of convergent and fundamental sequences applies to sequences of real-valued functions with a common domain. For each x in the domain, we have a sequence

{ f n ( x ) : 1 n } of real numbers. The sequence may converge for some x and fail to converge for others.

A somewhat more restrictive condition (and often a more desirable one) for sequences of functions is uniform convergence. Here the uniformity is over values of the argument x . In this case, for any ϵ > 0 there exists an N which works for all x (or for some suitable prescribed set of x ).

These concepts may be applied to a sequence of random variables , which are real-valued functions with domain Ω and argument ω . Suppose { X n : 1 n } is a sequence of real random variables. For each argument ω we have a sequence { X n ( ω ) : 1 n } of real numbers. It is quite possible that such a sequence converges for some ω and diverges (fails to converge) for others. As a matter of fact, in many important cases the sequence converges for all ω except possibly a set (event) of probability zero. In this case, we say the seqeunce converges almost surely (abbreviated a.s.). The notion of uniform convergence also applies. In probability theory we have the notion of almost uniform convergence. This is the case that the sequence converges uniformly for all ω except for a set of arbitrarily small probability.

The notion of convergence in probability noted above is a quite different kind of convergence. Rather than deal with the sequence on a pointwise basis, it deals withthe random variables as such. In the case of sample average, the “closeness” to a limit is expressed in terms of the probability that the observed value X n ( ω ) should lie close the the value X ( ω ) of the limiting random variable. We may state this precisely as follows:

A sequence { X n : 1 n } converges to X in probability , designated X n P X iff for any ϵ > 0 ,

lim n P ( | X - X n | > ϵ ) = 0

There is a corresponding notion of a sequence fundamental in probability.

The following schematic representation may help to visualize the difference between almost-sure convergence and convergence in probability. In setting up the basic probability model, we thinkin terms of “balls” drawn from a jar or box. Instead of balls, consider for each possible outcome ω a “tape” on which there is the sequence of values X 1 ( ω ) , X 2 ( ω ) , X 3 ( ω ) , .

  • If the sequence of random variable converges a.s. to a random variable X , then there is an set of “exceptional tapes” which has zero probability. For all other tapes, X n ( ω ) X ( ω ) . This means that by going far enough out on any such tape, the values X n ( ω ) beyond that point all lie within a prescribed distance of the value X ( ω ) of the limit random variable.
  • If the sequence converges in probability, the situation may be quite different. A tape is selected. For n sufficiently large, the probability is arbitrarily near one that the observed value X n ( ω ) lies within a prescribed distance of X ( ω ) . This says nothing about the values X m ( ω ) on the selected tape for any larger m . In fact, the sequence on the selected tape may very well diverge.

It is not difficult to construct examples for which there is convergence in probability but pointwise convergence for no ω . It is easy to confuse these two types of convergence. The kind of convergence noted for the sample average is convergence inprobability (a “weak” law of large numbers). What is really desired in most cases is a.s. convergence (a “strong” law of large numbers). It turns out that for a samplingprocess of the kind used in simple statistics, the convergence of the sample average is almost sure (i.e., the strong law holds). To establishthis requires much more detailed and sophisticated analysis than we are prepared to make in this treatment.

The notion of mean convergence illustrated by the reduction of Var [ A n ] with increasing n may be expressed more generally and more precisely as follows. A sequence { X n : 1 n } converges in the mean of order p to X iff

E [ | X - X n | p ] 0 as n designated X n L p X ; as n

If the order p is one, we simply say the sequence converges in the mean. For p = 2 , we speak of mean-square convergence .

The introduction of a new type of convergence raises a number of questions.

  1. There is the question of fundamental (or Cauchy) sequences and convergent sequences.
  2. Do the various types of limits have the usual properties of limits? Is the limit of a linear combination of sequences the linear combination of the limits? Isthe limit of products the product of the limits?
  3. What conditions imply the various kinds of convergence?
  4. What is the relation between the various kinds of convergence?

Before sketching briefly some of the relationships between convergence types, we consider one important condition known as uniform integrability . According to the property (E9b) for integrals

X is integrable iff E [ I { | X t | > a } | X t | ] 0 as a

Roughly speaking, to be integrable a random variable cannot be too large on too large a set. We use this characterization of the integrability of a single random variable to define thenotion of the uniform integrability of a class.

Definition . An arbitrary class { X t : t T } is uniformly integrable (abbreviated u.i.) with respect to probability measure P iff

sup t T E [ I { | X t | > a } | X t | ] 0 as a

This condition plays a key role in many aspects of theoretical probability.

The relationships between types of convergence are important. Sometimes only one kind can be established. Also, it may beeasier to establish one type which implies another of more immediate interest. We simply state informally some of the important relationships. A somewhat more detailed summaryis given in PA, Chapter 17. But for a complete treatment it is necessary to consult more advanced treatments of probability and measure.

Relationships between types of convergence for probability measures

Consider a sequence { X n : 1 n } of random variables.

  1. It converges almost surely iff it converges almost uniformly.
  2. If it converges almost surely, then it converges in probability.
  3. It converges in mean, order p , iff it is uniformly integrable and converges in probability.
  4. If it converges in probability, then it converges in distribution (i.e. weakly).

Various chains of implication can be traced. For example

  • Almost sure convergence implies convergence in probability implies convergence in distribution.
  • Almost sure convergence and uniform integrability implies convergence in mean p .

We do not develop the underlying theory. While much of it could be treated with elementary ideas, a complete treatment requires considerable development of the underlying measuretheory. However, it is important to be aware of these various types of convergence, since they are frequently utilized in advanced treatments of applied probability and of statistics.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
A fair die is tossed 180 times. Find the probability P that the face 6 will appear between 29 and 32 times inclusive
Samson Reply

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Source:  OpenStax, Applied probability. OpenStax CNX. Aug 31, 2009 Download for free at http://cnx.org/content/col10708/1.6
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