Lagmental Vicfred

Doob's Maximal Inequality Controls a Submartingale Peak by Vicfred

tags: mathematics, martingales, branching processes, optional stopping

The expected positive terminal value bounds the probability of crossing a positive threshold. I want the notation, the mechanism, and the failure mode visible at the same time.

Notation

A process \((M_n,\mathcal F_n)\) is a martingale when \(\mathbf E[M_{n+1}\mid\mathcal F_n]=M_n\). Its baseline \(M_0\) models fair evolution relative to the information currently available.

$$ (X_k)_{k=0}^{n}\ \text{nonnegative submartingale},\qquad\lambda>0 $$

There are two layers here: the object \(\mathsf D\) and the law \(\mathsf C\). Writing them separately makes the direction of \(\Longrightarrow\) visible and keeps an accidental converse from slipping in.

$$ \lambda\,\mathbf P\!\left(\max_{k\le n}X_k\ge\lambda\right)\le\mathbf E[X_n] $$

Stress the formula

The middle display is intentionally dense: it is where signs, bounds, multiplicities, or normalising factors are most likely to be lost.

$$ \lambda\,\mathbf1_{\{\max_{k\le n}X_k\ge\lambda\}}\le X_\tau\mathbf1_{\{\tau\le n\}},\qquad\tau=\min\{k:X_k\ge\lambda\} $$

Interpretation

What survives the example is not its particular numbers but the relation encoded by the two rows below. That relation is the part worth transporting to a new setting.

$$ \begin{aligned} \mathsf{D}\;&:\quad (X_k)_{k=0}^{n}\ \text{nonnegative submartingale},\qquad\lambda>0,\\[5pt] \mathsf{C}\;&:\quad \lambda\,\mathbf P\!\left(\max_{k\le n}X_k\ge\lambda\right)\le\mathbf E[X_n]. \end{aligned} $$

Limit of the argument

Optional stopping is not valid for every stopping time. Boundedness, integrability, or uniform-integrability conditions prevent hidden mass from escaping at infinity.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \lambda\,\mathbf P\!\left(\max_{k\le n}X_k\ge\lambda\right)\le\mathbf E[X_n] \end{gathered}} $$

The final box is a summary, not a new assumption; the proof still lives in the definitions and the intervening calculation. The source keeps each scope delimiter visible for later inspection.

This article was posted on Sat 06 February 2016. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.