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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Marginal sub-Gaussianity does not justify an anytime confidence sequence

Statement refuted

For every $α \in (0, 1)$ and $ρ > 0$ , marginal centered sub-Gaussianity with proxy $1$ alone makes the intervals in Gaussian-mixture confidence-sequence boundary a $(1 - α)$ confidence sequence for the common mean, regardless of dependence.

Facts & Assumptions

Given: Fix $α \in (0, 1)$ and $ρ > 0$ . Let $Ω = {- 1, + 1}$ with equal probabilities, let $Z (ω) = ω$ , let $F_{0} = {\emptyset, Ω}$ and $F_{t} = σ (Z)$ for every $t \geq 1$ , and set $X_{t} = Z$ , $μ = 0$ , and $σ^{2} = 1$ .

[F1]

Conditional Hoeffding bound for bounded martingale differences: if $D$ is conditionally centered and lies between measurable endpoints $A \leq D \leq B$ of deterministic width at most $c$ , then $E [e^{λ D} ∣ G] \leq e^{λ^{2} c^{2} / 8}$ for every real $λ$ .

[F2]

Sub-Gaussian variables and vectors, and sub-exponential variables: a centered real random variable is sub-Gaussian with proxy $1$ when $E e^{λ X} \leq e^{λ^{2} / 2}$ for every $λ \in R$ .

[F3]

Conditioning a known variable and an independent variable: an integrable variable independent of a sigma-algebra has conditional expectation equal almost surely to its ordinary expectation.

[F4]

Conditional expectation given a sigma algebra: a conditional-expectation version is a measurable integrable function with the same integral as the input on every event in the conditioning sigma-algebra.

[F5]

Conditional expectation is unique almost surely: any two versions of the same conditional expectation agree almost surely.

[F6]

Conditionally sub-Gaussian increments: the required sequential premise bounds the conditional mgf of each increment given $F_{t - 1}$ for every fixed real $λ$ .

[F7]

Gaussian-mixture confidence-sequence boundary: for $σ^{2} = 1$ , the radius at time $t \geq 1$ is $t^{- 1} \sqrt{(t + ρ) \log ((t + ρ) / (α^{2} ρ))}$ .

[F8]

The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t: for $x > 0$ , $\log x = \int_{1}^{x} d u / u$ .

[A1]

The Axiom of Choice: AC is assumed by the conditional-expectation convention in [F1].

Counterexample

1.1givenF1F2F3A1

We have $E Z = 0$ and $- 1 \leq Z \leq 1$ . Since $F_{0}$ is trivial, [F3] gives $E [Z ∣ F_{0}] = 0$ and, for every $λ$ , $E [e^{λ Z} ∣ F_{0}] = E e^{λ Z}$ ; the latter variable is integrable because $Z$ has finite support. Apply [F1] with $D = Z$ , $A = - 1$ , $B = 1$ , and $c = 2$ to get $E [e^{λ Z} ∣ F_{0}] \leq e^{λ^{2} / 2}$ , hence $E e^{λ Z} \leq e^{λ^{2} / 2}$ for every $λ \in R$ . Thus every marginal $X_{t} = Z$ is centered and sub-Gaussian with proxy $1$ by [F2].

1.2givenF4F5F6algebra

At time $t = 2$ , $X_{2} = Z$ is $F_{1}$ -measurable. Its own version satisfies [F4], so [F5] gives $E [e^{X_{2}} ∣ F_{1}] = e^{Z}$ almost surely. On the event ${Z = 1}$ , which has probability $1 / 2$ , this equals $e > e^{1 / 2}$ . Hence the conditional sub-Gaussian premise [F6] with proxy $1$ fails at $t = 2$ and $λ = 1$ .

1.3givenF7F8algebra

Every sample mean is ${\overset{ˉ}{X}}_{t} = Z$ . For all sufficiently large $t$ , $(t + ρ) / (α^{2} ρ) \geq 1$ ; using [F8] and $u^{- 1} \leq u^{- 3 / 4}$ for $u \geq 1$ gives $(u_{α} (t) / t)^{2} \leq 4 (t + ρ)^{5 / 4} / ((α^{2} ρ)^{1 / 4} t^{2}) = 4 t^{- 3 / 4} (1 + ρ / t)^{5 / 4} / (α^{2} ρ)^{1 / 4} \to 0$ . Thus the radius is eventually less than $1$ .

2.1givenstep 1.3F7algebra∎

Since $∣ {\overset{ˉ}{X}}_{t} ∣ = ∣ Z ∣ = 1$ , step 1.3 implies that $0$ is outside every boundary interval for all sufficiently large $t$ , on both outcomes. Therefore $P (0 \in C_{t} for all t \geq 1) = 0 < 1 - α$ , refuting the statement.

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