# Martingale sequences: the concept, examples, and basic patterns  (Page 2/2)

 Page 2 / 2

Notation .  When we write $\left({X}_{\mathbf{N}},\phantom{\rule{0.166667em}{0ex}}{Z}_{\mathbf{N}}\right)$ is a martingale (submartingale, supermartingale), we are asserting X N is integrable, Z N is a decision sequence, ${X}_{\mathbf{N}}\sim {Z}_{\mathbf{N}}$ , and X N is a MG (SMG, SRMG) relative to Z N .

Definition .  If Y N is integrable and Z N is a decision sequence, then

1. Y N is absolutely fair relative to Z N iff
${Y}_{\mathbf{N}}\sim {Z}_{\mathbf{N}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{and}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}E\left[{Y}_{n+1}|{W}_{n}\right]=0\phantom{\rule{4pt}{0ex}}\mathrm{a}.\mathrm{s}.\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\forall \phantom{\rule{0.277778em}{0ex}}n\in \mathbf{N}$
2. Y N is favorable relative to Z N iff
${Y}_{\mathbf{N}}\sim {Z}_{\mathbf{N}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{and}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}E\left[{Y}_{n+1}|{W}_{n}\right]\ge 0\phantom{\rule{4pt}{0ex}}\mathrm{a}.\mathrm{s}.\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\forall \phantom{\rule{0.277778em}{0ex}}n\in \mathbf{N}$
3. Y N is unfavorable relative to Z N iff
${Y}_{\mathbf{N}}\sim {Z}_{\mathbf{N}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{and}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}E\left[{Y}_{n+1}|{W}_{n}\right]\le 0\phantom{\rule{4pt}{0ex}}\mathrm{a}.\mathrm{s}.\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\forall \phantom{\rule{0.277778em}{0ex}}n\in \mathbf{N}$

Notation .  When we write $\left({Y}_{\mathbf{N}},\phantom{\rule{0.166667em}{0ex}}{Z}_{\mathbf{N}}\right)$ is absolutely fair (favorable, unfavorable), we are asserting Y N is integrable, Z N is a decision sequence, ${Y}_{\mathbf{N}}\sim {Z}_{\mathbf{N}}$ , and Y N is absolutely fair (favorable, unfavorable) relative to Z N .    IXA2-2

## Ixa2-1

If X N is a basic sequence and Y N is the corresponding incremental sequence, then

1. $\left({X}_{\mathbf{N}},\phantom{\rule{0.166667em}{0ex}}{Z}_{\mathbf{N}}\right)$ is a martingale iff $\left({Y}_{\mathbf{N}},\phantom{\rule{0.166667em}{0ex}}{Z}_{\mathbf{N}}\right)$ is absolutely fair.
2. $\left({X}_{\mathbf{N}},\phantom{\rule{0.166667em}{0ex}}{Z}_{\mathbf{N}}\right)$ is a submartingale iff $\left({Y}_{\mathbf{N}},\phantom{\rule{0.166667em}{0ex}}{Z}_{\mathbf{N}}\right)$ is favorable.
3. $\left({X}_{\mathbf{N}},\phantom{\rule{0.166667em}{0ex}}{Z}_{\mathbf{N}}\right)$ is a supermartingale iff $\left({Y}_{\mathbf{N}},\phantom{\rule{0.166667em}{0ex}}{Z}_{\mathbf{N}}\right)$ is unfavorable.

Let * be any one of the symbols $=,\phantom{\rule{0.277778em}{0ex}}\ge$ , or $\le$ .  Then by linearity and (CE7)

$E\left[{X}_{n+1}|{W}_{n}\right]=E\left[{Y}_{n+1}|{W}_{n}\right]+E\left[{X}_{n}|{W}_{n}\right]=E\left[{Y}_{n+1}|{W}_{n}\right]+{X}_{n}\phantom{\rule{0.277778em}{0ex}}*\phantom{\rule{0.277778em}{0ex}}{X}_{n}\phantom{\rule{4pt}{0ex}}\mathrm{a}.\mathrm{s}.\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{4pt}{0ex}}\mathrm{iff}\phantom{\rule{4pt}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}E\left[{Y}_{n+1}|{W}_{n}\right]\phantom{\rule{0.277778em}{0ex}}*\phantom{\rule{0.277778em}{0ex}}0\phantom{\rule{4pt}{0ex}}\mathrm{a}.\mathrm{s}.\phantom{\rule{0.166667em}{0ex}}$

Remarks

1. $\left({X}_{\mathbf{N}},\phantom{\rule{0.166667em}{0ex}}{Z}_{\mathbf{N}}\right)$ is a SMG iff $\left(-{X}_{\mathbf{N}},\phantom{\rule{0.166667em}{0ex}}{Z}_{\mathbf{N}}\right)$ is a SRMG
2. We write (S)MG to indicate the same statement can be made for a MG or a SMG with the appropriate choice of = or $\ge$
3. We write $\left(\ge \right)$ to indicate simultaneously two cases:
• $\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\left(\ge \right)$ read as = in all places (for a MG)
• $\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\left(\ge \right)$ read as $\ge$ in all places (for a SMG)

## Ixa3-1

If $\left({X}_{\mathbf{N}},\phantom{\rule{0.166667em}{0ex}}{Z}_{\mathbf{N}}\right)$ is a (S)MG and ${X}_{\mathbf{N}}\sim {H}_{\mathbf{N}}$ , with ${H}_{\mathbf{N}}\sim {Z}_{\mathbf{N}}$ ,  then $\left({X}_{\mathbf{N}},\phantom{\rule{0.166667em}{0ex}}{H}_{\mathbf{N}}\right)$ is a (S)MG.

Let ${K}_{n}=\left({H}_{0},\phantom{\rule{0.166667em}{0ex}}{H}_{1},\phantom{\rule{0.166667em}{0ex}}\cdots ,\phantom{\rule{0.166667em}{0ex}}{H}_{n}\right)$ .  By (CE9) , the (S)MG definition, monotonicity, and (CE7)

$E\left[{X}_{n+1}|{K}_{n}\right]=E\left\{E\left[{X}_{n+1}|{W}_{n}\right]|{K}_{n}\right\}\phantom{\rule{0.277778em}{0ex}}\left(\ge \right)\phantom{\rule{0.277778em}{0ex}}E\left[{X}_{n}|{K}_{n}\right]={X}_{n}\phantom{\rule{4pt}{0ex}}\mathrm{a}.\mathrm{s}.\phantom{\rule{0.166667em}{0ex}}$

## Ixa3-2

For integrable ${X}_{\mathbf{N}}\sim {Z}_{\mathbf{N}}$ , the following are equivalent

$\begin{array}{cccc}\text{a}& \left({X}_{\mathbf{N}},\phantom{\rule{0.166667em}{0ex}}{Z}_{\mathbf{N}}\right)& & \text{is a (S)MG}\\ \text{b}& E\left[{X}_{n+k}|{W}_{n}\right]\phantom{\rule{0.277778em}{0ex}}\left(\ge \right)\phantom{\rule{0.277778em}{0ex}}{X}_{n}\phantom{\rule{4pt}{0ex}}\mathrm{a}.\mathrm{s}.& \forall & n,\phantom{\rule{0.277778em}{0ex}}k\in \mathbf{N}\\ \text{c}& E\left[{I}_{C}{X}_{n+1}\right]\phantom{\rule{0.277778em}{0ex}}\left(\ge \right)\phantom{\rule{0.277778em}{0ex}}E\left[{I}_{C}{X}_{n}\right]& \forall & C\in \sigma \left({W}_{n}\right)& \forall & n\in \mathbf{N}\\ \text{d}& E\left[{I}_{C}{X}_{n+k}\right]\phantom{\rule{0.277778em}{0ex}}\left(\ge \right)\phantom{\rule{0.277778em}{0ex}}E\left[{I}_{C}{X}_{n}\right]& \forall & C\in \sigma \left({W}_{n}\right)& \forall & n,\phantom{\rule{0.277778em}{0ex}}k\in \mathbf{N}\end{array}$

• as a  special case
• By (CE9) , (a), and monotonicity
$E\left[{X}_{n+k}|{W}_{n}\right]=E\left\{E\left[{X}_{n+k}|{W}_{n+k-1}\right]|{W}_{n}\right\}\phantom{\rule{3.33333pt}{0ex}}\left(\ge \right)\phantom{\rule{0.277778em}{0ex}}E\left[{X}_{n+k-1}|{W}_{n}\right]\phantom{\rule{4pt}{0ex}}\mathrm{a}.\mathrm{s}.\phantom{\rule{0.166667em}{0ex}}$
$k-1$ iterations yield   $E\left[{X}_{n+k}|{W}_{n}\right]\phantom{\rule{0.277778em}{0ex}}\left(\ge \right)\phantom{\rule{0.277778em}{0ex}}{X}_{n}\phantom{\rule{4pt}{0ex}}\mathrm{a}.\mathrm{s}.\phantom{\rule{0.166667em}{0ex}}$
• as a  special case
• By (CE1) and (c),   $E\left[{I}_{C}{X}_{n+1}\right]=E\left\{{I}_{C}E\left[{X}_{n+1}|{W}_{n}\right]\right\}\phantom{\rule{0.277778em}{0ex}}\left(\ge \right)\phantom{\rule{0.277778em}{0ex}}E\left[{I}_{C}{X}_{n}\right]\phantom{\rule{4pt}{0ex}}\mathrm{a}.\mathrm{s}.\phantom{\rule{0.166667em}{0ex}}$ .   Since ${X}_{n}\sim {W}_{n}\phantom{\rule{4pt}{0ex}}\mathrm{a}.\mathrm{s}.\phantom{\rule{0.166667em}{0ex}}$ and $E\left[{X}_{n+1}|{W}_{n}\right]\sim {W}_{n}\phantom{\rule{4pt}{0ex}}\mathrm{a}.\mathrm{s}.\phantom{\rule{0.166667em}{0ex}}$ , the result follows from the uniqueness property (E5)
• By (CE1) , (b), and monotonicity $E\left[{I}_{C}{X}_{n+k}\right]=E\left\{{I}_{C}E\left[{X}_{n+k}|{W}_{n}\right]\right\}\phantom{\rule{0.277778em}{0ex}}\left(\ge \right)\phantom{\rule{0.277778em}{0ex}}E\left[{I}_{C}{X}_{n}\right]$

We thus have $d⇒c⇒a⇔b⇒d$

## Ixa3-3

If $\left({X}_{\mathbf{N}},\phantom{\rule{0.166667em}{0ex}}{Z}_{\mathbf{N}}\right)$ is a (S)MG, then $E\left[{X}_{n+k}\right]\phantom{\rule{0.277778em}{0ex}}\left(\ge \right)\phantom{\rule{0.277778em}{0ex}}E\left[{X}_{n}\right]\phantom{\rule{0.277778em}{0ex}}\left(\ge \right)\phantom{\rule{0.277778em}{0ex}}E\left[{X}_{0}\right]$

## Ixa3-4

$\left({X}_{\mathbf{N}},\phantom{\rule{0.166667em}{0ex}}{Z}_{\mathbf{N}}\right)$ is a (S)MG iff $E\left[{X}_{q}-{X}_{p}|{W}_{n}\right]\phantom{\rule{0.277778em}{0ex}}\left(\ge \right)\phantom{\rule{0.277778em}{0ex}}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\le p

EXERCISE.  Note ${X}_{q}-{X}_{p}={Y}_{p+1}+\phantom{\rule{0.277778em}{0ex}}\cdots \phantom{\rule{0.277778em}{0ex}}+{Y}_{q}$

## Ixa3-5

If $\left({X}_{\mathbf{N}},\phantom{\rule{0.166667em}{0ex}}{Z}_{\mathbf{N}}\right)$ is an ${\mathbf{L}}^{2}$ MG, then

$\begin{array}{ccc}E\left[{X}_{q}-{X}_{p}\right]=0& \forall & p

1. $E\left[{X}_{q}-{X}_{p}\right]=E\left\{E\left[{X}_{q}-{X}_{p}|{W}_{n}\right]\right\}=0\phantom{\rule{3.33333pt}{0ex}}$ by (CE1b) and Thm IXA3-4
2. $E\left[{X}_{n}\left({X}_{q}-{X}_{p}\right)\right]=E\left\{{X}_{n}E\left[{X}_{q}-{X}_{p}|{W}_{n}\right]\right\}=0$ by (CE1b) , (CE8) , and Thm IXA3-4
3. As in b, since ${X}_{n}-{X}_{m}\sim {W}_{n}$
4. Suppose $p .  Then, since ${X}_{p}\sim {W}_{p}$ , $E\left[{X}_{p}{X}_{q}\right]=E\left\{{X}_{p}E\left[{X}_{q}|{W}_{p}\right]\right\}=E\left[{X}_{p}^{2}\right]$ by definition of MG.   For $q , interchange $p,\phantom{\rule{0.277778em}{0ex}}q$ in the argument above.
5. $E\left[{\left({X}_{q}-{X}_{p}\right)}^{2}\right]=E\left[{X}_{q}^{2}\right]-2E\left[{X}_{p}{X}_{q}\right]+E\left[{X}_{p}^{2}\right]=E\left[{X}_{q}^{2}\right]-2E\left[{X}_{p}^{2}\right]+E\left[{X}_{p}^{2}\right]$ by d, above
6. By c, $E\left[{Y}_{j}{Y}_{k}\right]=0$ for $j\ne k$ .  Hence, $E\left[{X}_{p}^{2}\right]=E\left[{\left(\sum _{k=0}^{p}{Y}_{k}\right)}^{2}\right]=\sum _{j}\sum _{k}E\left[{Y}_{j}{Y}_{k}\right]=\sum _{k=0}^{p}E\left[{Y}_{k}^{2}\right]$

A variety of weighted sums of increments are useful.

## Ixa3-6

Suppose $\left({X}_{\mathbf{N}},\phantom{\rule{0.166667em}{0ex}}{Z}_{\mathbf{N}}\right)$ is a (S)MG and Y N is the incremental sequence. Let H 0 be a (nonnegative) constant and let ${H}_{n}\sim {W}_{n-1},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}n\ge 1$ , be bounded (nonnegative).  Set

${X}_{n}^{*}=\sum _{k=0}^{n}{H}_{k}{Y}_{k}=\sum _{k=0}^{n}{Y}_{k}^{*}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\forall \phantom{\rule{0.277778em}{0ex}}n\in \mathbf{N}$

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

$E\left[{Y}_{n+1}^{*}|{W}_{n}\right]=E\left[{H}_{n+1}{Y}_{n+1}|{W}_{n}\right]={H}_{n+1}E\left[{Y}_{n+1}|{W}_{n}\right]\phantom{\rule{4pt}{0ex}}\mathrm{a}.\mathrm{s}.\phantom{\rule{0.166667em}{0ex}}$ by (CE8)

For MG case: $E\left[{Y}_{n+1}^{*}|{W}_{n}\right]=0\phantom{\rule{4pt}{0ex}}\mathrm{a}.\mathrm{s}.\phantom{\rule{0.166667em}{0ex}}$ for arbitrary bounded H n

For SMG case:   $E\left[{Y}_{n+1}^{*}|{W}_{n}\right]\ge 0\phantom{\rule{4pt}{0ex}}\mathrm{a}.\mathrm{s}.\phantom{\rule{0.166667em}{0ex}}$ for ${H}_{n}\ge 0$ , bounded

Remark .  This result extends the pattern in the introductory gambling example.   [link]  IXA3-3

## Ixa3-7

In Theorem IXA3-6 , if $E\left[{X}_{0}\right]\ge 0$ and $0\le {H}_{n}\le 1\mathrm{a}.\mathrm{s}.\forall n\in \mathbf{N}$ , then $0\le E\left[{X}_{n}^{*}\right]\le E\left[{X}_{n}\right]\forall n\in \mathbf{N}$

$E\left[{Y}_{n+1}|{W}_{n}\right]\ge \phantom{\rule{3.33333pt}{0ex}}{H}_{n+1}E\left[{Y}_{n+1}|{W}_{n}\right]\phantom{\rule{0.277778em}{0ex}}\left(\ge \right)\phantom{\rule{0.277778em}{0ex}}0\phantom{\rule{4pt}{0ex}}\mathrm{a}.\mathrm{s}.\phantom{\rule{0.166667em}{0ex}}$ , by hypothesis, and ${H}_{n+1}E\left[{Y}_{n+1}|{W}_{n}\right]=E\left[{Y}_{n+1}^{*}|{W}_{n}\right]\phantom{\rule{4pt}{0ex}}\mathrm{a}.\mathrm{s}.\phantom{\rule{0.166667em}{0ex}}$ , by (CE8) .   Thus, by monotonicity and (CE1b)

$E\left[{Y}_{n+1}\right]\phantom{\rule{0.277778em}{0ex}}\left(\ge \right)\phantom{\rule{0.277778em}{0ex}}E\left[{Y}_{n+1}^{*}\right]\phantom{\rule{0.277778em}{0ex}}\left(\ge \right)\phantom{\rule{0.277778em}{0ex}}0\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\forall \phantom{\rule{0.277778em}{0ex}}n\in \mathbf{N}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{and}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}E\left[{Y}_{0}\right]=E\left[{X}_{0}\right]\ge {H}_{0}E\left[{Y}_{0}\right]=E\left[{Y}_{0}^{*}\right]$

Hence

$E\left[{X}_{n}\right]=\sum _{k=0}^{n}E\left[{Y}_{k}\right]\ge \sum _{k=0}^{n}E\left[{Y}_{k}^{*}\right]=E\left[{X}_{n}^{*}\right]\ge 0$

Some important special cases

## Ixa3-8

Suppose integrable ${X}_{\mathbf{N}}\sim {Z}_{\mathbf{N}}$ . If ${X}_{n+1}-{X}_{n}\left(\ge \right)0\mathrm{a}.\mathrm{s}.\forall n\in \mathbf{N}$ , then $\left({X}_{\mathbf{N}},{Z}_{\mathbf{N}}\right)$ is a (S)MG.

Apply monotonicity and Theorem IXA3-4

## Ixa3-9

Suppose X N has independent increments.

1. If $E\left[{X}_{n}\right]=c$ , invariant with n , then X N is a MG.
2. If $E\left[{X}_{n+1}-{X}_{n}\right]\left(\ge \right)0,\forall n\in \mathbf{N}$ ,   then $\left({X}_{\mathbf{N}}$ is a (S)MG.
1. For any n , consider any $C\in \sigma \left({U}_{n}\right)$ .  By independent increments, $\left\{{I}_{C},\left({X}_{n+1}-{X}_{n}\right)\right\}$ is independent.  Hence, $E\left[{I}_{C}{X}_{n+1}\right]-E\left[{I}_{C}{X}_{n}\right]=E\left[{I}_{C}\left({X}_{n+1}-{X}_{n}\right)\right]=E\left[{I}_{C}\right]E\left[\left({X}_{n+1}-{X}_{n}\right)\right]\left(\ge \right)0$ . The desired result follows from Theorem IXA3-2(c) .

## Ixa3-10

Suppose g is a convex Borel function on an interval I which contains the range of all X n and

$E\left[|g\left({X}_{n}\right)|\right]<\infty \forall n\in \mathbf{N}$ ,  Let ${H}_{n}=g\left({X}_{n}\right)\forall n\in \mathbf{N}$ ,

1. If $\left({X}_{\mathbf{N}},\phantom{\rule{0.166667em}{0ex}}{Z}_{\mathbf{N}}\right)$ is a MG, then $\left({H}_{\mathbf{N}},\phantom{\rule{0.166667em}{0ex}}{Z}_{\mathbf{N}}\right)$ is a SMG.
2. If g is nondecreasing and $\left({X}_{\mathbf{N}},\phantom{\rule{0.166667em}{0ex}}{Z}_{\mathbf{N}}\right)$ is a SMG, then so is $\left({H}_{\mathbf{N}},\phantom{\rule{0.166667em}{0ex}}{Z}_{\mathbf{N}}\right)$
• By Jensen's inequality and the definition of a MG
$E\left[g\left({X}_{n+1}\right)|{W}_{n}\right]\ge g\left(E,\left[,{X}_{n+1},|,{W}_{n},\right]\right)=g\left({X}_{n}\right)\phantom{\rule{4pt}{0ex}}\mathrm{a}.\mathrm{s}.\phantom{\rule{0.166667em}{0ex}}$
• By Jensen's inequality
$E\left[g\left({X}_{n+1}\right)|{W}_{n}\right]\ge g\left(E,\left[,{X}_{n+1},|,{W}_{n},\right]\right)\phantom{\rule{4pt}{0ex}}\mathrm{a}.\mathrm{s}.\phantom{\rule{0.166667em}{0ex}}$
Since $E\left[{X}_{n+1}|{W}_{n}\right]\ge {X}_{n}\phantom{\rule{4pt}{0ex}}\mathrm{a}.\mathrm{s}.\phantom{\rule{0.166667em}{0ex}}$ and g is nondecreasing, we have
$g\left(E,\left[,{X}_{n+1},|,{W}_{n},\right]\right)\ge g\left({X}_{n}\right)\phantom{\rule{4pt}{0ex}}\mathrm{a}.\mathrm{s}.\phantom{\rule{0.166667em}{0ex}}$

Some commonly utilized convex functions

1. $g\left(t\right)=|t|$
2. $g\left(t\right)={t}^{2}$ g is increasing for $t\ge 0$
3. $g\left(t\right)=u\left(t\right)t$ $\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}g\left({X}_{n}\right)={X}_{n}^{+}$ g nondecreasing for all t
4. $g\left(t\right)=\phantom{\rule{3.33333pt}{0ex}}-u\left(-t\right)t$ $g\left({X}_{n}\right)={X}_{n}^{-}$ g nonincreasing for all t
5. $g\left(t\right)={e}^{at},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}a>0$ g is increasing for all t

## Ixa3-11

Consider integrable ${X}_{\mathbf{N}}\sim {Z}_{\mathbf{N}}$ .

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

The restrictionl $a>0$ is needed in the $\ge$ case.

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