Backward Martingale

Backword Martingale

Theorems

Def: A series of r.v of index \(n\in\{-1,-2,\cdots\}\) is a Backward Martingale if

\[\mathbb{E}(X_n|\mathcal{F}_{n-1})=X_{n-1} \]

In short, it is martingale looking backward.

In some sense, a martingale is "divergent", and backward martingale is "convergent" if we look at its sigma algebra. The following theorems explain my idea.

Thm: \(X_n\) convergence in a.s and \(L_1\).

Pf: To prove a.s. Use Doob's upwards inequality.

Define

\[U_N=\text{The times that } X_n \text{ goes from below }a \text{ over }b \]

We have

\[\mathbb{E}(U_N)\leq \frac{1}{b-a}\mathbb{E}(X_0^{+}) \]

Note that these two sets

\[\{\limsup_n X_n \geq b \text{and} \liminf_n X_n \leq a\} \supset \{U_\infty =\infty \} \]

but we already bounded \(U_\infty=\lim_N U_N\) by taking the limit in the Doob's upwards inequality. So this set could not happen. By the arbitary of \(a,b\) we finish the proof.

The \(L_1\) is symple, since it is UI.

Thm: \(X_{-\infty}=\lim_{n \to -\infty}X_n,\ \mathcal{F}_{-\infty}=\cap\mathcal{F}_n\), then \(X_{-\infty}=\mathbb{E}(X_0|\mathcal{F}_{\infty})\)

It is obvious.

Thm: \(\mathcal{F}_n \downarrow \mathcal{F}_{-\infty}\), \(Y\) integrable, then \(\mathbb{E}(Y|\mathcal{F}_n)\to \mathbb{E}(Y|\mathcal{F}_{-\infty})\) a.s. and \(L_1\)

Pf First, it is a backward martingale, then convegence got. Use the second theorem, we know the form of limits.

Applications

Setings

Let

\[\Omega=\{(\omega_1,\omega_2,\cdots) \omega_i \in S \} \]

\[\mathcal{F}=\mathcal{S}\times \mathcal{S}\cdots \]

\[X_n(\omega)=\omega_n \]

In settings, we first give a probability space and make a product for infinately times. Then we constract a r.p. But in application, we first get r.p. then consider its probability space.

\(\mathcal{E}_n\) is the sigma field generate by events invarient undet permutations in \(S_n\) acting on its indexs. its limit is \(\mathcal{E}\)

Exmp: Simple random walk:

\[\xi_1,\xi_2,\cdots \]

S is canonical space and so is sigma algebra. It is easy to give \(\mathcal{E}_n\) which is

\[\sigma(\cup_{t \in S_n}\{\xi_{t(1)}=i_1,\cdots,\xi_{t(n)}=i_n\}) \]

It hard to give \(\mathcal{E}\),but we know the sets like this

\[\{\frac{\sum_{i=1}^n \xi_i}{n} \in A\} \]

is in it. And this gives ideas to the following applications.