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Examples and further patterns

A4-1 sums of independent random variables

Suppose Y N is an independent, integrable sequence. Set X n = k = 0 n Y k n 0 .

If E [ Y n ] ( ) 0 n 1 , then X N is a (S)MG.

A4-2 products of nonnegative random variables

Suppose Y N Z N , Y n 0 a . s . n . Consider X N : X n = c k = 0 n Y k , c > 0 .

If E [ Y n + 1 | W n ] ( ) 1 a . s . n , then ( X N , Z N ) is a (S)MG

X n W n and X n + 1 = Y n + 1 X n . Hence, E [ X n + 1 | W n ] = X n E [ Z n + 1 | W n ] ( ) X n a . s . n

A4-3 discrete random walk

Consider Y 0 = 0 and { Y n : 1 n } iid. Set X n = k = 0 n Y k n 0 . Suppose P ( Y n = k ) = p k . Let

g Y ( s ) = E [ s Y n ] = k p k s k , s > 0

Now g Y ( 1 ) = 1 , g Y ' ( 1 ) = E [ Y n ] , g Y ' ' ( s ) = k k ( k - 1 ) p k s k - 2 > 0 for s > 0 . Hence, g Y ( s ) = 1 has at most two roots, one of which is s = 1 .

  1. s = 1 is a minimum point iff E [ Y n ] = 0 , in which case X N is a MG (see A4-1 )
  2. If g Y ( r ) = 1 for 0 < r < 1 , then E [ r Y n ] = 1 n 1 . Let Z 0 = 1 , Z n = r X n = k = 1 n r Y k . By A4-2, Z N is a MG

For the MG case in Theorem IXA3-6 , the Y n are centered at conditional expectation; that is

E [ Y n + 1 | W n ] = 0 a . s . The following is an extension of that pattern.

A4-4 more general sums

Consider integrable Y N Z N and bounded H N Z N . Let W n = a constant for n < 0 and H n = 1 for n < 0 . Set

X n = k = 0 n { Y k - E [ Y k | W k - 1 ] } H k - 1 n 0

Then ( X N , Z N ) is a MG.

X n W n ; n 0 and E [ X n + 1 | W n ] = X n + H n E { Y n + 1 - E [ Y n + 1 | W n ] | W n } = X n + 0 a . s .

IXA4-2

A4-5 sums of products

Suppose Y N is absolutely fair relative to Z N , with E [ | Y n | k ] < n , fixed k > 0 . For n k , set

X n = 0 i 1 < n Y i 1 Y i 2 Y i k G n

Then ( X N k , Z N k ) N k = { k , k + 1 , k + 2 , } is a MG,

X n + 1 = X n + K n + 1 , where

K n + 1 = Y n + 1 0 i 1 < n Y i 1 Y i 2 Y i k - 1 = Y n + 1 K n * K n * W n
E [ K n + 1 | W n ] = K n * E [ Y n + 1 | W n ] = 0 a . s . n k

We consider, next, some relationships with homogeneous Markov sequences .

Suppose ( X N , Z N ) is a homogeneous Markov sequence with finite state space E = { 1 , 2 , , M } and transition matrix P = [ p ( i , j ) ] . A function f on E is represented by a column matrix f = [ f ( 1 ) , f ( 2 ) , , f ( M ) ] T . Then f ( X n ) has value f ( k ) when X n = k . P f is an m × 1 column matrix and P f ( j ) is the j th element of that matrix. Consider E [ f ( X n + 1 ) | W n ] = E [ f ( X n + 1 ) | X n ] a . s . . Now

E [ f ( X n + 1 ) | X n = j ] = k E f ( k ) p ( j , k ) = P f ( j ) so that E [ f ( X n + 1 ) | W n ] = P f ( X n )

A nonnegative function f on E is called (super)harmonic for P iff P f ( ) f .

A4-6 positive supermartingales and superharmonic functions.

Suppose ( X N , Z N ) is a homogeneous Markov sequence with finite state space E = { 1 , 2 , , M } and transition matrix P = [ p ( i , j ) ] . For nonnegative f on E , let Y n = f ( X n ) n N . Then ( Y N , Z N ) is a positive (super)martingale P(SR)MG iff f is (super)harmonic for P .

As noted above E [ f ( X n + 1 ) | W n ] = P f ( X n ) .

  1. If f is (super)harmonic P f ( X n ) ( ) f ( X n ) = Y n , so that
    E [ Y n + 1 | W n ] ( ) Y n a . s .
  2. If ( Y N , Z N ) is a P(SR)MG, then
    Y n = f ( X n ) ( ) E [ f ( X n + 1 ) | W n ] = P f ( X n ) a . s . , so that f is (super)harmonic

IX A4-3

An eigenfunction f and associated eigenvalue λ for P satisfy P f = λ f (i.e., ( λ I - P ) f = 0 ). In most cases, | λ | < 1 . For real λ , 0 < λ < 1 , the eigenfunctions are superharmonic functions. We may use the construction of Theorem IXA3-12 to obtain the associated MG.

A4-7 martingales induced by eigenfunctions for homogeneous markov sequences

Let ( Y N , Z N ) be a homogenous Markov sequence, and f be an eigenfunction with eigenvalue λ . Put X n = λ - n f ( Y n ) . Then, by Theorem IAXA3-12 , ( X N , Z N ) is a MG.

A4-8 a dynamic programming example.

We consider a horizon of N stages and a finite state space E = { 1 , 2 , , M } .

  • Observe the system at prescribed instants
  • Take action on the basis of previous states and actions.

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
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da
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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
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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
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Alexandre
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Brian Reply
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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
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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
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Damian Reply
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Renato
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Stoney Reply
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Adin Reply
?
Kyle
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Adin
why?
Adin
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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
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Source:  OpenStax, Topics in applied probability. OpenStax CNX. Sep 04, 2009 Download for free at http://cnx.org/content/col10964/1.2
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