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Brownian law of the iterated logarithm at zero

Statement

Let $B$ be a standard Brownian motion Brownian motion. Then almost surely ${lim sup}_{t ↓ 0} \frac{B_{t}}{\sqrt{2 t \log \log (1 / t)}} = 1, {lim inf}_{t ↓ 0} \frac{B_{t}}{\sqrt{2 t \log \log (1 / t)}} = - 1,$ the normalizer being taken for $0 < t < e^{- 1}$ so that $\log \log (1 / t) > 0$ .

Facts & Assumptions

Given: AC and a standard Brownian motion $B$ .

[F1]

Almost surely ${lim sup}_{s \to \infty} Y_{s} / \sqrt{2 s \log \log s} = 1$ and the corresponding limit inferior is $- 1$ for every standard Brownian motion $Y$ . Brownian law of the iterated logarithm at infinity

[F2]

Time inversion: the process $Y_{0} = 0$ , $Y_{s} : = s B_{1 / s}$ for $s > 0$ , is again a standard Brownian motion, with continuity at $s = 0$ part of the conclusion. Brownian time inversion

[F3]

AC is the ambient assumption of the Brownian interfaces. The Axiom of Choice

Proof

technique · direct

1.1givenF1F2

Let $Y$ be the time-inverted process of [F2]; by [F1] applied to $Y$ there is a probability-one event on which ${lim sup}_{s \to \infty} Y_{s} / \sqrt{2 s \log \log s} = 1$ and ${lim inf}_{s \to \infty} Y_{s} / \sqrt{2 s \log \log s} = - 1$ .

2.1step 1.1F2

On that event, substituting $s = 1 / t$ with $t ↓ 0$ and using $Y_{1 / t} = B_{t} / t$ gives $\frac{Y_{1 / t}}{\sqrt{2 t^{- 1} \log \log (1 / t)}} = \frac{B_{t} / t}{\sqrt{2 t^{- 1} \log \log (1 / t)}} = \frac{B_{t}}{\sqrt{2 t \log \log (1 / t)}}$ , so ${lim sup}_{t ↓ 0} \frac{B_{t}}{\sqrt{2 t \log \log (1 / t)}} = 1$ and ${lim inf}_{t ↓ 0} \frac{B_{t}}{\sqrt{2 t \log \log (1 / t)}} = - 1$ almost surely.

3.1step 2.1F2F3given∎

The cases are covered: the substitution $t \mapsto 1 / t$ is a bijection of $(0, \infty)$ onto itself, so $t ↓ 0$ corresponds to $s ↑ \infty$ ; the normalizer is positive exactly for $0 < t < e^{- 1}$ ; the endpoint $t = 0$ is not evaluated, continuity at zero being part of [F2]; and AC enters only through [F3].

Source notes

Time inversion converts the law of the iterated logarithm at infinity, Theorem 8.5.1 of Durrett, into the corresponding statement at zero, with the normalizing factor transforming exactly as displayed.

Depends on

Used by

The critical Hölder boundary at zero Corollary
LIL rules out a square-root-time bound Example
The Brownian p-variation threshold Example

Dependency tree · two levels

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Sources

Rick Durrett, Probability: Theory and Examples, fifth edition, Theorem 8.5.1 with Brownian time inversion (standard reference, not scraped)