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Finite-horizon crossing bound for a nonnegative test supermartingale

Statement

Assume AC. Let $P_{0}$ be a family of probability laws and let $M = (M_{t})_{t \in N_{0}}$ be a nonnegative test supermartingale for $P_{0}$ . For every $P \in P_{0}$ , every $a > 0$ , and every $N \in N_{0}$ , $P ⁣ (\max_{0 \leq t \leq N} M_{t} \geq a) \leq \frac{E_{P} M_{0}}{a} \leq \frac{1}{a} .$

Facts & Assumptions

Given: Assume AC. Let $P_{0}$ be a null family and let $M = (M_{t})_{t \geq 0}$ be a nonnegative test supermartingale for that family. Fix $P \in P_{0}$ , $a > 0$ , and a finite horizon $N \in N_{0}$ .

[F1]

Nonnegative test supermartingales: under every null law, $M$ is adapted and integrable, is nonnegative almost surely at every time, is a supermartingale, and satisfies $E_{P} M_{0} \leq 1$ .

[F2]

First hitting time of an adapted process is a stopping time: for an adapted process and a Borel set $B$ , its first time in $B$ is a stopping time.

[F3]

Discrete stopping time: a discrete stopping time is a map into $N_{0} \cup {\infty}$ whose events ${τ \leq n}$ belong to $F_{n}$ ; it is bounded when it is bounded almost surely by a deterministic integer.

[F4]

Optional sampling for bounded stopping times: for bounded stopping times $σ \leq τ$ , optional sampling for a supermartingale gives $E_{P} M_{τ} \leq E_{P} M_{σ}$ .

[F5]

Linearity, monotonicity, and the modulus bound for expectation and The expectation of an indicator is the probability of the event: expectation is monotone, and $E_{P} [1_{A}] = P (A)$ for measurable $A$ .

[A1]

AC is assumed by the test-supermartingale definition and the bounded optional-sampling theorem (The Axiom of Choice); it is used only through those suppliers.

Proof

1.1givenF1F2F3A1choose

Define $τ_{a} = \inf {t \geq 0 : M_{t} \geq a}$ , with $\inf \emptyset = \infty$ , and $A_{N} = {τ_{a} \leq N} = {\max_{0 \leq t \leq N} M_{t} \geq a}$ . By [F1], $M$ is adapted; applying [F2] with the Borel set $[a, \infty)$ makes $τ_{a}$ a stopping time. Thus $τ_{a} \land N$ and the constant time $0$ are bounded stopping times, with $0 \leq τ_{a} \land N$ .

2.1F1F4A1step 1.1

Apply [F4] to $0 \leq τ_{a} \land N$ to get $E_{P} M_{τ_{a} \land N} \leq E_{P} M_{0} \leq 1$ .

2.2F1step 1.1

On $A_{N}$ , $τ_{a} \land N = τ_{a}$ and $M_{τ_{a}} \geq a$ ; on $A_{N}^{c}$ , $τ_{a} \land N = N$ and $M_{N} \geq 0$ almost surely by [F1]. Hence $M_{τ_{a} \land N} \geq a 1_{A_{N}}$ almost surely.

3.1F5step 2.1step 2.2algebra∎

By [F5] and step 2.1, $a P (A_{N}) = E_{P} [a 1_{A_{N}}] \leq E_{P} M_{τ_{a} \land N} \leq E_{P} M_{0} \leq 1$ . Dividing by $a > 0$ proves both bounds.

Remarks

The finite-horizon event is exactly the first-crossing event by time $N$ . No assumption that an infinite-horizon supremum is attained is used here.

Depends on

Used by

Discrete Ville inequality and anytime threshold validity Theorem

Dependency tree · two levels

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Sources

Steven R. Howard, Aaditya Ramdas, Jon McAuliffe, and Jasjeet Sekhon, Time-uniform Chernoff bounds via nonnegative supermartingales, Probability Surveys 17 (2020), 257–317 (standard reference, not scraped)
R. L. Wolpert, STA 205: Martingales, Duke University lecture notes (2010) (standard reference, not scraped)