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Independent Gaussian increments satisfy the conditional proxy

Example

Let $Θ \subseteq R$ , let $(Ω, F, (F_{t})_{t \in N_{0}})$ be a common discrete filtered measurable space, let $(P_{θ})_{θ \in Θ}$ be probability laws on it, and let $(X_{t})_{t \geq 1}$ be a common real-valued process adapted under every $P_{θ}$ . Under each $P_{θ}$ , assume the residuals $Y_{t}^{(θ)} : = X_{t} - θ$ have law $N (0, τ^{2})$ for a known $τ^{2} > 0$ and, for every $t \geq 1$ , every Borel $B \subseteq R$ , and every $A \in F_{t - 1}$ , $P_{θ} ({Y_{t}^{(θ)} \in B} \cap A) = P_{θ} (Y_{t}^{(θ)} \in B) P_{θ} (A) .$ Let the reported proxy be any known $σ^{2} \geq τ^{2}$ , and fix $ρ > 0$ and $α \in (0, 1)$ . The Gaussian-mixture intervals with proxy $σ^{2}$ are a $(1 - α)$ confidence sequence for $θ$ .

Facts & Assumptions

Given: Assume AC, $τ^{2} > 0$ , $σ^{2} \geq τ^{2}$ , $ρ > 0$ , $α \in (0, 1)$ , a common discrete filtered measurable space with laws $(P_{θ})_{θ \in Θ}$ , a common real-valued process $(X_{t})$ adapted under every $P_{θ}$ , and, under each $P_{θ}$ , residuals $Y_{t}^{(θ)} = X_{t} - θ \sim N (0, τ^{2})$ satisfying the displayed event-factorization identity for every $t \geq 1$ , Borel $B$ , and $A \in F_{t - 1}$ .

[F1]

Standard normal and normal laws: $N (0, τ^{2})$ is the law of an affine scaling of the standard normal law.

[F2]

Closed form for a Gaussian exponential mixture: for $ρ^{'} > 0$ , $s \in R$ , and $v \geq 0$ , the Gaussian mixture integral equals $\sqrt{ρ^{'} / (v + ρ^{'})} e^{s^{2} / [2 (v + ρ^{'})]}$ .

[F3]

Conditioning a known variable and an independent variable: if Borel events of an integrable random variable factor with every event in $G$ , its conditional expectation given $G$ is its unconditional expectation almost surely.

[F4]

Conditionally sub-Gaussian increments: conditional sub-Gaussianity requires the conditional mgf bound for every fixed real $λ$ at every time.

[F5]

A Gaussian mixture confidence sequence for conditional sub-Gaussian means: under that conditional proxy at every time and for every $λ$ , the stated closed intervals cover the true parameter simultaneously with probability at least $1 - α$ .

[A1]

The Axiom of Choice: AC is used by the normal-law, conditional-expectation, and confidence-sequence suppliers.

Verification

1.1givenF1F2algebra

If $Θ = \emptyset$ the conclusion is vacuous. Otherwise fix $θ \in Θ$ , $t \geq 1$ , and $λ \in R$ , and write $Y_{t} = Y_{t}^{(θ)}$ . Apply [F2] with $ρ^{'} = 1 / τ^{2}$ , $v = 0$ , and $s = λ$ . Since $N (0, τ^{2}) = N (0, 1 / ρ^{'})$ by [F1], this gives $E e^{λ Y_{t}} = e^{λ^{2} τ^{2} / 2}$ under $P_{θ}$ , finite for every real $λ$ .

2.1givenF3F4step 1.1

For the fixed $θ, t, λ$ , the map $y \mapsto e^{λ y}$ is Borel, so for each Borel $D \subseteq R$ its preimage is Borel; the stated factorization for $Y_{t}^{(θ)}$ therefore gives the same factorization for $e^{λ Y_{t}}$ . This variable is integrable by step 1.1, so [F3] yields $E [e^{λ Y_{t}} ∣ F_{t - 1}] = E [e^{λ Y_{t}}]$ under $P_{θ}$ . Since $X_{t} - θ = Y_{t}$ , step 1.1 now gives $E_{P_{θ}} [e^{λ (X_{t} - θ)} ∣ F_{t - 1}] = e^{λ^{2} τ^{2} / 2} \leq e^{λ^{2} σ^{2} / 2}$ almost surely. As $t$ and $λ$ were arbitrary, [F4] applies for each $θ$ , time, and fixed tuning value.

3.1F5A1step 2.1∎

For each $θ$ , the common filtered model and adapted process together with the conditional proxy verified in step 2.1 and [A1] meet the hypotheses of [F5]. Therefore the intervals have simultaneous coverage at least $1 - α$ under each $P_{θ}$ .

Depends on

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