1.1 Martingale sequences: examples and further patterns

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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.

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