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Conditional sub-Gaussian increments generate exponential test supermartingales

Statement

Assume AC and let $(Ω, F, (F_{t})_{t \in N_{0}}, P)$ , $(X_{t})_{t \geq 1}$ , $μ \in R$ , and $σ^{2} > 0$ satisfy Conditionally sub-Gaussian increments. For each fixed $λ \in R$ , set $S_{0} (μ) : = 0, S_{t} (μ) : = \sum_{i = 1}^{t} (X_{i} - μ), L_{t}^{λ} (μ) : = \exp ⁣ (λ S_{t} (μ) - \frac{λ^{2} t σ^{2}}{2}) .$ Then $(L_{t}^{λ} (μ))_{t \geq 0}$ is a nonnegative test supermartingale for the singleton null family ${P}$ , with $L_{0}^{λ} (μ) = 1$ .

Facts & Assumptions

Given: Assume AC, the conditional sub-Gaussian model, and a fixed real $λ$ .

[F1]

Conditionally sub-Gaussian increments: for each $t \geq 1$ and fixed $λ$ , $e^{λ (X_{t} - μ)}$ is integrable and its conditional expectation given $F_{t - 1}$ is at most $e^{λ^{2} σ^{2} / 2}$ almost surely.

[F2]

Taking out what is known: a bounded measurable factor can be taken out of a conditional expectation; the identity also holds for finite measurable factors when both products are integrable.

[F3]

Monotone convergence for the integral: an increasing sequence of nonnegative measurable functions has integrals increasing to the integral of its pointwise limit.

[F4]

Martingale submartingale and supermartingale: an adapted integrable process is a supermartingale when $E [L_{t} ∣ F_{t - 1}] \leq L_{t - 1}$ almost surely at every adjacent pair.

[F5]

Nonnegative test supermartingales: under AC, an adapted, integrable, almost surely nonnegative supermartingale with initial expectation at most one is a nonnegative test supermartingale.

[A1]

The Axiom of Choice: AC supplies the conditional-expectation convention used in the supermartingale and test-supermartingale definitions.

Proof

1.1givenF1F2F3construct

Fix $t \geq 1$ , put $Y_{t} : = \exp ⁣ (λ (X_{t} - μ) - λ^{2} σ^{2} / 2)$ and $Q_{t} : = E [Y_{t} ∣ F_{t - 1}]$ . By [F1], $Y_{t}$ is integrable and $0 \leq Q_{t} \leq 1$ . Starting from $L_{0}^{λ} (μ) = 1$ , suppose $L_{t - 1}^{λ} (μ)$ is integrable and let $Z_{n} : = L_{t - 1}^{λ} (μ) \land n$ . The bounded-factor clause of [F2] gives $E [Z_{n} Y_{t}] = E [Z_{n} Q_{t}] \leq E Z_{n} \leq E L_{t - 1}^{λ} (μ)$ . Since $Z_{n} Y_{t} ↑ L_{t}^{λ} (μ)$ , [F3] shows $L_{t}^{λ} (μ)$ is integrable and $E L_{t}^{λ} (μ) \leq E L_{t - 1}^{λ} (μ)$ . Thus induction establishes integrability at every time.

2.1F2F4F5A1step 1.1∎

The process is adapted and nonnegative by its definition. Since $L_{t - 1}^{λ} (μ) Y_{t} = L_{t}^{λ} (μ)$ and both products are integrable by step 1.1, the unbounded-factor clause of [F2] gives $E [L_{t}^{λ} (μ) ∣ F_{t - 1}] = L_{t - 1}^{λ} (μ) Q_{t} \leq L_{t - 1}^{λ} (μ)$ . Also $E L_{0}^{λ} (μ) = 1$ . Therefore [F4] and [F5] give the asserted test-supermartingale property.

Depends on

Used by

The Gaussian mixture is a sub-Gaussian test supermartingale Theorem

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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), §§2.1–2.2 (standard reference, not scraped)
Steven R. Howard, Aaditya Ramdas, Jon McAuliffe, and Jasjeet Sekhon, Time-uniform, nonparametric, nonasymptotic confidence sequences, §2 (standard reference, not scraped)