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The Gaussian mixture is a sub-Gaussian test supermartingale

Statement

Assume AC. Let $(Ω, F, (F_{t})_{t \in N_{0}}, P)$ and $(X_{t})_{t \geq 1}$ satisfy Conditionally sub-Gaussian increments for $μ \in R$ with known proxy $σ^{2} > 0$ . Fix $ρ > 0$ . Define $S_{0} (μ) = 0$ , $S_{t} (μ) = \sum_{i = 1}^{t} (X_{i} - μ)$ , $V_{t} = t σ^{2}$ , and $M_{t} (μ) : = \sqrt{\frac{ρ}{V_{t} + ρ}} \exp ⁣ (\frac{S_{t} (μ)^{2}}{2 (V_{t} + ρ)}) (t \in N_{0}) .$ Then $M (μ)$ is a nonnegative test supermartingale for ${P}$ , with $M_{0} (μ) = 1$ .

Facts & Assumptions

Given: Assume AC and the conditional sub-Gaussian model for fixed $μ$ , $σ^{2} > 0$ , and $ρ > 0$ .

[F1]

Conditional sub-Gaussian increments generate exponential test supermartingales: for each fixed $λ \in R$ , $L_{t}^{λ} (μ) = \exp (λ S_{t} (μ) - λ^{2} t σ^{2} / 2)$ is a nonnegative test supermartingale with initial value one.

[F2]

Mixtures of null test supermartingales: if each component is a nonnegative test supermartingale under every null law and $(λ, ω) \mapsto L_{t}^{λ} (ω)$ is product-measurable, its probability mixture is a test supermartingale; the theorem uses a real-valued modification only where the integral is infinite.

[F3]

Tonelli's theorem for nonnegative measurable functions on a sigma-finite product: for sigma-finite measure spaces and nonnegative product-measurable functions, section integrals are measurable and equal the corresponding iterated integrals.

[F4]

Closed form for a Gaussian exponential mixture: for $v \geq 0$ , $s \in R$ , and $ρ > 0$ , the mixture of $\exp (λ s - v λ^{2} / 2)$ under $N (0, 1 / ρ)$ equals $\sqrt{ρ / (v + ρ)} \exp (s^{2} / [2 (v + ρ)])$ .

[F5]

Standard normal and normal laws: $N (0, 1 / ρ)$ is a probability law under the affine-image normal-law convention.

[A1]

The Axiom of Choice: AC is required by the conditional-expectation and test-supermartingale conventions in [F1] and [F2].

Proof

1.1givenF1construct

For each fixed $λ$ , [F1] supplies the component process $L_{t}^{λ} (μ)$ . At fixed $t$ , $S_{t} (μ)$ is $F_{t}$ -measurable, so $(λ, ω) \mapsto \exp (λ S_{t} (μ) (ω) - λ^{2} t σ^{2} / 2)$ is measurable on $B (R) \otimes F_{t}$ : it is the composition of the measurable coordinate maps with continuous arithmetic and exponential operations. Each component is nonnegative.

2.1F2F3F4F5A1step 1.1∎

Mix these components using the probability law $N (0, 1 / ρ)$ from [F5]. Both this prior and $P$ are probability measures, hence sigma-finite. The product-measurability in step 1.1 and the component conditions in [F1] meet [F2]; [F3] gives the needed measurable nonnegative section integrals. Therefore the mixture process is a nonnegative test supermartingale. By [F4] its value is the displayed finite formula for every finite $S_{t} (μ)$ , so the mixture theorem's infinite-integral modification is never applied. At $t = 0$ the formula gives $M_{0} (μ) = 1$ .

Depends on

Used by

A Gaussian mixture confidence sequence for conditional sub-Gaussian means Theorem

Dependency tree · two levels

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Sources

Steven R. Howard, Aaditya Ramdas, Jon McAuliffe, and Jasjeet Sekhon, Time-uniform, nonparametric, nonasymptotic confidence sequences, Lemma 2 and Appendix A.3 (standard reference, not scraped)
Aaditya Ramdas, Martingales 1: Concentration Inequalities — Conjugate Mixtures, 36-771 lecture notes, Carnegie Mellon University, Fall 2018 (standard reference, not scraped)