<< Chapter < Page Chapter >> Page >

Convergence of random variables

We include some background material for the course. Let us recall some notions of convergence of random variables (RV's).

  • A sequence of RV's { x n } n 1 converges in probability if ϵ 0 , lim n sup Pr ( | x n - x ¯ | > ϵ ) = 0 . We denote this by x n P . x ¯ .
  • A sequence of RV's { x n } n 1 converges to x ¯ with probability 1 if Pr { x 1 , x 2 , ... : lim n x n = x ¯ } = 1 . We denote this by x n w . p . 1 x ¯ or x n a . s . x ¯ .
  • A sequence of RV's { x n } n 1 converges to x ¯ in the p sense if E [ | x n - x ¯ | p ] 0 . We denote this by x n p x ¯ .

For example, for p = 2 we have mean square convergence, x n m . s . x ¯ . For p 2 ,

E x n - x ¯ p - 1 = E x n - x ¯ p p - 1 p ( E | x n - x ¯ | p ) p - 1 p .

Therefore, x n p x ¯ yields x n p - 1 x ¯ . Note that for convergence in 1 sense, we have

Pr ( | x n - x ¯ | > ϵ ) E | x n - x ¯ | ϵ 0 .

Typical sequences

The following material appears in most textbooks on information theory (c.f., Cover and Thomas  [link] and references therein). We include the highlights in orderto make these notes self contained, but skip some details and the proofs. Consider a sequence x = x n = ( x 1 , x 2 , . . . , x n ) , where x i α , α is the alphabet, and the cardinality of α is r , i.e., | α | = r .

Definition 1 The type of x consists of the empirical probabilities of symbols in x ,

P x ( a ) = n x ( a ) n , a α ,

where n x ( a ) is the empirical symbol count , which is the the number of times that a α appears in x .

Definition 2 The set of all possible types is defined as P n .

For an alphabet α = { 0 , 1 } we have P n = { ( 0 n , n n ) , ( 1 n , n - 1 n ) , . . . , ( n n , 0 n ) } . In this case, | P n | = n + 1 .

Definition 3 A type class T x contains all x ' α n , such that P x = P x ' ,

T x = T ( P x ) = { x ' α n : P x ' = P x } .

Consider α = 1 , 2 , 3 and x = 11321 . We have n = 5 and the empirical counts are n x = ( 3 , 1 , 1 ) . Therefore, the type is P x = ( 3 5 , 1 5 , 1 5 ) , and the type class T x contains all length-5 sequences with 3 ones, 1 two, and 1 three. That is, T x = { 11123 , 11132 , . . . , 32111 } . It is easy to see that | T x | = 5 ! 3 ! 1 ! 1 ! = 20 .

Theorem 1 The cardinality of the set of all types satisfies | P n | ( n + 1 ) r - 1 .

The proof is simple, and was given in class. We note in passing that this bound is loose, but it is good enough for our discussion.

Next, consider an i.i.d. source with the following prior,

Q ( x ) = i = 1 n Q ( x i ) .

We note in passing that i.i.d. sources are sometimes called memoryless. Let the entropy be

H ( P x ) = - Σ a α n x ( a ) n log n x ( a ) n ,

where we use base-two logarithms throughout. We are studying the entropy H ( P x ) in order to show that it is the fundamental performance limit in lossless compression. Σ find me

We also define the divergence as

D ( P x Q x ) = Σ a α P x log P x Q x .

It is well known that the divergence is non-negative,

D ( P x Q x ) 0 .

Moreover, D ( P Q ) = 0 only if the distributions are identical.

Claim 1 The following relation holds,

Q ( x ) = 2 - n [ H ( P x ) + D ( P x Q ( x ) ) ] .

The derivation is straightforward,

Q ( x ) = Π a α Q ( a ) n x ( a ) = 2 Σ a α n x ( a ) log Q ( a ) = 2 n Σ P x ( a ) ( log Q P + log P ) = 2 - n [ H ( P x ) + D ( P x Q ( x ) ) ] .

Seeing that the divergence is non-negative [link] , and zero only if the distributions are equal,we have Q ( x ) P x ( x ) . When P x = Q the divergence between them is zero, and we have that P x ( x ) = Q x = 2 - n H ( P x ) .

The proof of the following theorem was discussed in class.

Theorem 2 The cardinality of the type class T ( P x ) obeys,

( n + 1 ) - ( r - 1 ) · 2 n H ( P x ) | T ( P x ) | 2 n H ( P x ) .

Having computed the probability of x and cardinality of its type class, we can easily compute the probability of the type class.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Universal algorithms in signal processing and communications. OpenStax CNX. May 16, 2013 Download for free at http://cnx.org/content/col11524/1.1
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Universal algorithms in signal processing and communications' conversation and receive update notifications?

Ask