# Law of Logarithm Type Result for Standard Brownian Motion

Let $$W_t$$ be a standard Brownian motion. Show that almost surely we have:

$$\limsup_{t \to \infty} \frac{W_t}{\sqrt{t \log t}}<\infty$$

The approach suggested by the writer of the problem is to consider sets: $$V_{n,r} = \left \{\max_{0 \leq t \leq 2^n} W_t \geq r\cdot 2^{n/2} \cdot n^{1/2} \right\}$$ And showing that for $$r$$ sufficiently large but finite, we have that: $$\sum_{n = 1}^\infty \mathbb{P}(V_n) < \infty$$ Applying the Borel-Cantelli lemma to the events $$V_n$$ finishes the proof.

This is all well and good, and I understand how the hint proves the result. However, does anyone have any idea how I may get any kinds of bounds on $$\mathbb{P}(V_{n,r})$$?

## 1 Answer

Hint: By the reflection principle, $$\sup_{0 \leq t \leq T} W_t$$ equals in distribution $$|W_T|$$ for each $$T>0$$. Apply Markov's inequality to get suitable bounds for $$\mathbb{P}(V_{n,r})$$.

Remark: In case that you are not familiar with the reflection principle, you can combine instead Markov's inequality with Doob's maximal inequality for martingales; it allows you to get rid of the supremum: $$\mathbb{E} \left( \sup_{t \leq T} |W_t|^p \right) \leq c_p \mathbb{E}(|W_T|^p), \qquad p>1.$$