1.1 Martingale sequences: examples and further patterns  (Page 3/3)

Note that $V_{n+1}(\ge )V_n$ iff $r(\ge )X_{n+1}/V_n$ . Set $Y_n=E\left[V_n\right|U_n]$ . Then $Y_{n+1}=(1+r)E\left[V_n\right|U_{n+1}]-X_{n+1}\mathrm{a}.\mathrm{s}.$ so that

Thus, $(Y_{\mathbf{N}},X_{\mathbf{N}})$ is a (S)MG iff

Summary: convergence of submartingales

The submartingale convergence theorem

If $(X_{\mathbf{N}},Z_{\mathbf{N}})$ is a SMG with ${\displaystyle \underset{n}{lim}E\left[X_n^{+}\right]<\infty }$ , then there exists $X_{\infty }\sim W_{\infty }$ such that $X_n\to X_{\infty }\mathrm{a}.\mathrm{s}.$

Uniform integrability and some convergence conditions

Definition . The class $\{X_t:t\in T\}$ is uniformly integrable iff

Any of the following conditions ensures uniform integrability:

1. The class is dominated by an integrable random variable Y .
2. The class is finite and integrable.
3. There is a u.i. class $\{Y_t:t\in T\}$ such that $|X_t|\le |Y_t|\mathrm{a}.\mathrm{s}.$ for all $t\in T$ .
4. X integrable implies Doob's MG $\{X_n=E\left[X\right|W_n]:n\in \mathbf{N}\}$ is u.i.

Definition . The class $\{X_t:t\in T\}$ is uniformly absolutely continuous iff for each $ϵ>0$ there is a $\delta >0$ such that $P\left(A\right)<\delta$ implies ${\displaystyle \underset{T}{sup}\left\{E\right[I_A|X_t\left|\right]:t\in T\}<ϵ}$ .

$X_T=\{X_t:t\in T\}$ is u.i. iff both (i) X _T is u.a.c., and (ii) ${\displaystyle \underset{T}{sup}\left\{E\right[|X_t\left|\right]\}<\infty }$ .

Definition . $X_n\stackrel{P}{\to }X$ iff $P\left(\right|X_n-X|>ϵ)\to$ as $n\to \infty$ for all $ϵ>0$ .

$X_n\to X\mathrm{a}.\mathrm{s}.$ implies $X_n\stackrel{P}{\to }X$

If (i) $X_n\stackrel{{\mathbf{L}}^p}{\to }X$ , (ii) $X_n\to X\mathrm{a}.\mathrm{s}.$ and (iii) ${\displaystyle \underset{n}{lim}E\left[X_n\right|Z]}$ exists a.s., then

Suppose $(X_{\mathbf{N}},Z_{\mathbf{N}})$ is a (S)MG. Consider the following

• (A) ${\displaystyle \underset{n}{lim}E\left[X_n^{+}\right]<\infty }$ or, equivalently, ${\displaystyle \underset{n}{sup}E\left[\right|X_n\left|\right]<\infty }$ - - - - - - - - - - - -
• (a) X _N is uniformly integrable.
• (a ${}^{+}$ ) $X_{\mathbf{N}}^{+}$ is uniformly integrable.
• (b) $X_n\stackrel{{\mathbf{L}}^1}{\to }X$ .
• (b ${}^{+}$ ) $X_n^{+}\stackrel{{\mathbf{L}}^1}{\to }X$ .
• (c) There is an integrable $X_{\infty }\sim W_{\infty }$ such that
  $$X_n\to X_{\infty }\mathrm{a}.\mathrm{s}.\text{and}E\left[X_{\infty }\right|W_n](\ge )X_n\mathrm{a}.\mathrm{s}.\forall n\in \mathbf{N}$$
• (c ${}^{\text{'}}$ ) Condition (c) with $\ge$ even for a MG.
• (d) There is an integrable X with $E\left[X\right|W_n](\ge )X_n\mathrm{a}.\mathrm{s}.\forall n\in \mathbf{N}$
• (d ${}^{\text{'}}$ ) Condition (d) with $\ge$ even for a MG.

Then

1. Each of the propositions (a) through (d ${}^{\text{'}}$ ) implies (A), hence SMG convergence.
2. (a) $\Rightarrow$ (a ${}^{+}$ )
3. (a) $\iff$ (b) $\Rightarrow$ (c) $\Rightarrow$ (d)
4. (a ${}^{+}$ ) $\iff$ (b ${}^{+}$ ) $\iff$ (c ${}^{\text{'}}$ ) $\iff$ (d ${}^{\text{'}}$ )
5. For a MG, (d) $\Rightarrow$ (a), so that (a) $\iff$ (b) $\iff$ (c) $\iff$ (d)

The notion of regularity is characterized in terms of the conditions in the theorem.

Definition . A martingale $(X_{\mathbf{N}},Z_{\mathbf{N}})$ is said to be martingale regular iff the equivalent conditions (a), (b), (c), (d) in the theorem hold.

A submartingale $(X_{\mathbf{N}},Z_{\mathbf{N}})$ is said to be submartingale regular iff the equivalent conditions (a ${}^{+}$ ), (b ${}^{+}$ ), (c ${}^{\text{'}}$ ), (d ${}^{\text{'}}$ ) in the theorem hold.

Remarks

1. Since a MG is a SMG, a martingale regular MG is also submartingale regular.
2. It is not true, in general that a submartingale regular SMG is martingale regular. We do have for SMG (a) $\iff$ (b) $\Rightarrow$ (c) $\Rightarrow$ (d).
3. Regularity may be viewed in terms of membership of X _∞ in the (S)MG. The condition $E\left[X_{\infty }\right|W_n](\ge )X_n\mathrm{a}.\mathrm{s}.$ is indicated by saying X _∞ belongs to the (S)MG or by saying the (S)MG is closed (on the right) by X _∞ .

Summary

For a martingale $(X_{\mathbf{N}},Z_{\mathbf{N}})$

1. If martingale regular, then $X_n\to X_{\infty }\sim W_{\infty }\mathrm{a}.\mathrm{s}.$ and $X_n=E\left[X_{\infty }\right|W_n]\mathrm{a}.\mathrm{s}.\forall n\in \mathbf{N}$ $E\left[X_{n+k}\right|W_n]=E\left\{E\left[X_{\infty }\right|W_{n+k}]\right|W_n\}=E\left[X_{\infty }\right|W_n]=X_n\mathrm{a}.\mathrm{s}.$ and $E\left[X_0\right]=E\left[X_n\right]=E\left[X_{\infty }\right]\forall n\in \mathbf{N}$
2. If submartingale regular, but not martingale regular, then $X_n\to X_{\infty }\sim W_{\infty }\mathrm{a}.\mathrm{s}.$ but $E\left[X_{\infty }\right|W_n]\ge X_n\mathrm{a}.\mathrm{s}.\forall n\in \mathbf{N}$ and $E\left[X_0\right]=E\left[X_n\right]\le E\left[X_{\infty }\right]<\infty \forall n\in \mathbf{N}$

For a submartingale $(X_{\mathbf{N}},Z_{\mathbf{N}})$

Either martingale regularity or submartingale regularity implies

$X_n\to X_{\infty }\sim W_{\infty }\mathrm{a}.\mathrm{s}.$ and $X_n\le E\left[X_{n+1}\right|W_n]\le E\left[X_{\infty }\right|W_n]\mathrm{a}.\mathrm{s}.\forall n\in \mathbf{N}$

and $E\left[X_0\right]\le E\left[X_n\right]\le E\left[X_{\infty }\right]<\infty \forall n\in \mathbf{N}$

If X _N is uniformly integrable , then $E\left[X_n\right]\to E\left[X_{\infty }\right]$ .

If $(X_{\mathbf{N}},Z_{\mathbf{N}})$ is a MG with $E\left[X_n^2\right]<K\forall nı\mathbf{N}$ , then the proceess is MG regular, with