# 1.1 Martingale sequences: examples and further patterns

 Page 1 / 3

## A4-1 sums of independent random variables

Suppose Y N is an independent, integrable sequence. Set ${X}_{n}=\sum _{k=0}^{n}{Y}_{k}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\forall \phantom{\rule{0.277778em}{0ex}}n\ge 0$ .

If $E\left[{Y}_{n}\right]\phantom{\rule{0.277778em}{0ex}}\left(\ge \right)\phantom{\rule{0.277778em}{0ex}}0\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\forall \phantom{\rule{0.277778em}{0ex}}n\ge 1$ , then X N is a (S)MG.

## A4-2 products of nonnegative random variables

Suppose ${Y}_{\mathbf{N}}\sim {Z}_{\mathbf{N}},{Y}_{n}\ge 0\mathrm{a}.\mathrm{s}.\forall n$ . Consider ${X}_{\mathbf{N}}:{X}_{n}=c\prod _{k=0}^{n}{Y}_{k},c>0$ .

If $E\left[{Y}_{n+1}|{W}_{n}\right]\phantom{\rule{0.277778em}{0ex}}\left(\ge \right)\phantom{\rule{0.277778em}{0ex}}1\phantom{\rule{4pt}{0ex}}\mathrm{a}.\mathrm{s}.\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\forall \phantom{\rule{0.277778em}{0ex}}n$ , then $\left({X}_{\mathbf{N}},\phantom{\rule{0.166667em}{0ex}}{Z}_{\mathbf{N}}\right)$ is a (S)MG

${X}_{n}\sim {W}_{n}$ and ${X}_{n+1}={Y}_{n+1}{X}_{n}$ . Hence, $E\left[{X}_{n+1}|{W}_{n}\right]={X}_{n}\phantom{\rule{3.33333pt}{0ex}}E\left[{Z}_{n+1}|{W}_{n}\right]\phantom{\rule{0.277778em}{0ex}}\left(\ge \right)\phantom{\rule{0.277778em}{0ex}}{X}_{n}\phantom{\rule{4pt}{0ex}}\mathrm{a}.\mathrm{s}.\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\forall \phantom{\rule{0.277778em}{0ex}}n$

## A4-3 discrete random walk

Consider ${Y}_{0}=0$ and $\left\{{Y}_{n}:1\le n\right\}$ iid. Set ${X}_{n}=\sum _{k=0}^{n}{Y}_{k}\forall n\ge 0$ . Suppose $P\left({Y}_{n}=k\right)={p}_{k}$ . Let

${g}_{Y}\left(s\right)=E\left[{s}^{{Y}_{n}}\right]=\sum _{k}{p}_{k}{s}^{k},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}s>0$

Now ${g}_{Y}\left(1\right)=1,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{g}_{Y}^{\text{'}}\left(1\right)=E\left[{Y}_{n}\right],\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{g}_{Y}^{\text{'}\text{'}}\left(s\right)=\sum _{k}k\left(k-1\right){p}_{k}{s}^{k-2}>0\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{for}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}s>0$ . Hence, ${g}_{Y}\left(s\right)=1$ has at most two roots, one of which is $s=1$ .

1. $s=1$ is a minimum point iff $E\left[{Y}_{n}\right]=0$ , in which case X N is a MG (see A4-1 )
2. If ${g}_{Y}\left(r\right)=1$ for $0 , then $E\left[{r}^{{Y}_{n}}\right]=1\forall n\ge 1$ . Let ${Z}_{0}=1,{Z}_{n}={r}^{{X}_{n}}=\prod _{k=1}^{n}{r}^{{Y}_{k}}$ . 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

$E\left[{Y}_{n+1}|{W}_{n}\right]=0\phantom{\rule{4pt}{0ex}}\mathrm{a}.\mathrm{s}.\phantom{\rule{0.166667em}{0ex}}$ The following is an extension of that pattern.

## A4-4 more general sums

Consider integrable ${Y}_{\mathbf{N}}\sim {Z}_{\mathbf{N}}$ and bounded ${H}_{\mathbf{N}}\sim {Z}_{\mathbf{N}}$ . Let ${W}_{n}=$ a constant for $n<0$ and ${H}_{n}=1$ for $n<0$ . Set

${X}_{n}=\sum _{k=0}^{n}\left\{{Y}_{k}-E\left[{Y}_{k}|{W}_{k-1}\right]\right\}{H}_{k-1}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\forall \phantom{\rule{0.277778em}{0ex}}n\ge 0$

Then $\left({X}_{\mathbf{N}},\phantom{\rule{0.166667em}{0ex}}{Z}_{\mathbf{N}}\right)$ is a MG.

${X}_{n}\sim {W}_{n};\forall n\ge 0$ and $E\left[{X}_{n+1}|{W}_{n}\right]={X}_{n}+{H}_{n}E\left\{{Y}_{n+1}-E\left[{Y}_{n+1}|{W}_{n}\right]|{W}_{n}\right\}={X}_{n}+0\mathrm{a}.\mathrm{s}.$

IXA4-2

## A4-5 sums of products

Suppose Y N is absolutely fair relative to Z N , with $E\left[|{Y}_{n}{|}^{k}\right]<\infty \forall n$ , fixed $k>0$ . For $n\ge k$ , set

${X}_{n}=\sum _{0\le {i}_{1}<\cdots \le n}{Y}_{{i}_{1}}{Y}_{{i}_{2}}\cdots {Y}_{{i}_{k}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\sim {\mathbf{G}}_{n}$

Then $\left({X}_{{\mathbf{N}}_{k}},\phantom{\rule{0.166667em}{0ex}}{Z}_{{\mathbf{N}}_{k}}\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{\mathbf{N}}_{k}=\left\{k,\phantom{\rule{0.166667em}{0ex}}k+1,\phantom{\rule{0.166667em}{0ex}}k+2,\phantom{\rule{0.166667em}{0ex}}\cdots \phantom{\rule{0.277778em}{0ex}}\right\}$ is a MG,

${X}_{n+1}={X}_{n}+{K}_{n+1}$ , where

${K}_{n+1}={Y}_{n+1}\sum _{0\le {i}_{1}<\cdots \le n}{Y}_{{i}_{1}}{Y}_{{i}_{2}}\cdots {Y}_{{i}_{k-1}}={Y}_{n+1}{K}_{n}^{*}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{K}_{n}^{*}\sim {W}_{n}$
$E\left[{K}_{n+1}|{W}_{n}\right]={K}_{n}^{*}E\left[{Y}_{n+1}|{W}_{n}\right]=0\phantom{\rule{4pt}{0ex}}\mathrm{a}.\mathrm{s}.\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\forall \phantom{\rule{0.277778em}{0ex}}n\ge k$

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

Suppose $\left({X}_{\mathbf{N}},\phantom{\rule{0.166667em}{0ex}}{Z}_{\mathbf{N}}\right)$ is a homogeneous Markov sequence with finite state space $E=\left\{1,\phantom{\rule{0.166667em}{0ex}}2,\phantom{\rule{0.166667em}{0ex}}\cdots ,\phantom{\rule{0.166667em}{0ex}}M\right\}$ and transition matrix $P=\left[p\left(i,\phantom{\rule{0.166667em}{0ex}}j\right)\right]$ . A function f on E is represented by a column matrix $f={\left[f\left(1\right),\phantom{\rule{0.166667em}{0ex}}f\left(2\right),\phantom{\rule{0.166667em}{0ex}}\cdots ,\phantom{\rule{0.166667em}{0ex}}f\left(M\right)\right]}^{T}$ . Then $f\left({X}_{n}\right)$ has value $f\left(k\right)$ when ${X}_{n}=k$ . $Pf$ is an $m×1$ column matrix and $Pf\left(j\right)$ is the j th element of that matrix. Consider $E\left[f\left({X}_{n+1}\right)|{W}_{n}\right]=E\left[f\left({X}_{n+1}\right)|{X}_{n}\right]\phantom{\rule{4pt}{0ex}}\mathrm{a}.\mathrm{s}.\phantom{\rule{0.166667em}{0ex}}$ . Now

$E\left[f\left({X}_{n+1}\right)|{X}_{n}=j\right]=\sum _{k\in E}f\left(k\right)p\left(j,\phantom{\rule{0.166667em}{0ex}}k\right)=Pf\left(j\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{so}\phantom{\rule{4.pt}{0ex}}\text{that}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}E\left[f\left({X}_{n+1}\right)|{W}_{n}\right]=Pf\left({X}_{n}\right)$

A nonnegative function f on E is called (super)harmonic for P iff $Pf\phantom{\rule{0.277778em}{0ex}}\left(\le \right)\phantom{\rule{0.277778em}{0ex}}f$ .

## A4-6 positive supermartingales and superharmonic functions.

Suppose $\left({X}_{\mathbf{N}},\phantom{\rule{0.166667em}{0ex}}{Z}_{\mathbf{N}}\right)$ is a homogeneous Markov sequence with finite state space $E=\left\{1,\phantom{\rule{0.166667em}{0ex}}2,\phantom{\rule{0.166667em}{0ex}}\cdots ,\phantom{\rule{0.166667em}{0ex}}M\right\}$ and transition matrix $P=\left[p\left(i,\phantom{\rule{0.166667em}{0ex}}j\right)\right]$ . For nonnegative f on E , let ${Y}_{n}=f\left({X}_{n}\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\forall \phantom{\rule{0.277778em}{0ex}}n\in \mathbf{N}$ . Then $\left({Y}_{\mathbf{N}},\phantom{\rule{0.166667em}{0ex}}{Z}_{\mathbf{N}}\right)$ is a positive (super)martingale P(SR)MG iff f is (super)harmonic for P .

As noted above $E\left[f\left({X}_{n+1}\right)|{W}_{n}\right]=Pf\left({X}_{n}\right)$ .

1. If f is (super)harmonic $Pf\left({X}_{n}\right)\phantom{\rule{0.277778em}{0ex}}\left(\le \right)\phantom{\rule{0.277778em}{0ex}}f\left({X}_{n}\right)={Y}_{n}$ , so that
$E\left[{Y}_{n+1}|{W}_{n}\right]\phantom{\rule{0.277778em}{0ex}}\left(\le \right)\phantom{\rule{0.277778em}{0ex}}{Y}_{n}\phantom{\rule{4pt}{0ex}}\mathrm{a}.\mathrm{s}.\phantom{\rule{0.166667em}{0ex}}$
2. If $\left({Y}_{\mathbf{N}},\phantom{\rule{0.166667em}{0ex}}{Z}_{\mathbf{N}}\right)$ is a P(SR)MG, then
${Y}_{n}=f\left({X}_{n}\right)\phantom{\rule{0.277778em}{0ex}}\left(\ge \right)\phantom{\rule{0.277778em}{0ex}}E\left[f\left({X}_{n+1}\right)|{W}_{n}\right]=Pf\left({X}_{n}\right)\phantom{\rule{4pt}{0ex}}\mathrm{a}.\mathrm{s}.\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{so}\phantom{\rule{4.pt}{0ex}}\text{that}\phantom{\rule{4.pt}{0ex}}f\phantom{\rule{4.pt}{0ex}}\text{is}\phantom{\rule{4.pt}{0ex}}\text{(super)harmonic}$

IX A4-3

An eigenfunction f and associated eigenvalue λ for P satisfy $Pf=\lambda f$ (i.e., $\left(\lambda I-P\right)f=0$ ). In most cases, $|\lambda |<1$ . For real λ , $0<\lambda <1$ , 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 $\left({Y}_{\mathbf{N}},\phantom{\rule{0.166667em}{0ex}}{Z}_{\mathbf{N}}\right)$ be a homogenous Markov sequence, and f be an eigenfunction with eigenvalue λ . Put ${X}_{n}={\lambda }^{-n}f\left({Y}_{n}\right)$ . Then, by Theorem IAXA3-12 , $\left({X}_{\mathbf{N}},\phantom{\rule{0.166667em}{0ex}}{Z}_{\mathbf{N}}\right)$ is a MG.

## A4-8 a dynamic programming example.

We consider a horizon of N stages and a finite state space $\mathbf{E}=\left\{1,\phantom{\rule{0.166667em}{0ex}}2,\phantom{\rule{0.166667em}{0ex}}\cdots ,\phantom{\rule{0.166667em}{0ex}}M\right\}$ .

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

what is Nano technology ?
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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!