Law of iterated logarithm for quadratic variation of Brownian motion

Question

Let $(\Omega, \mathcal{F}, \mathbb{P})$ denote a probability space supporting a standard Brownian motion $B$. Let $\Pi=\{\pi_n : n \ge 0\}$ denote the sequence of dyadic uniform partitions of the interval $[0,1]$; that is, $|\pi_n|=2^{-n}$. For each $t$ and $n$, define $$V^n_t(B)=\sum_{[t_i,t_{i+1}]\cap [0,t]\in \pi_n} (B_{t_{i+1}}-B_{t_i})^2.$$ Is there a law of iterated logarithm type $\mathbb{P}$-a.s. convergence rate (in n) for $\sup_{t\le 1}|V^n_t(B)-t|$? I believe I can prove that for each fixed $t$, $\mathbb{P}$-a.s., $$V^n_t(B)-t=\mathcal{O}(2^{-n}\sqrt{2\lfloor t2^n\rfloor \ln \ln \lfloor t 2^n\rfloor})$$ using Proposition 1 of (Arcones 1999 https://mathscinet.ams.org/mathscinet/relay-station?mr=MR1702915), but this doesn't seem to imply the result I am asking for since $V^n_t(B)$ is not continuous in $t$. I would need to find a version of Arcones 1999 for triangular arrays. However, the result itself seems so fundamental that I feel it must be known. Thanks in advance.

Answer 1

In "Bounded Laws of the Iterated Logarithm for Quadratic Forms in Gaussian Random Variables", they at least prove the upper bound in Theorem 3.1.

$$\limsup_{n}|V^{n}_{1}-1|/\phi_{n}\leq 1,$$

for $\phi_{n}:=\sqrt{2}E[(V^{n}_{1})^{2}]\log\log(1/E[(V^{n}_{1})^{2})$. And as mentioned in remark 3, we also have

$$\limsup_{n}|V^{n}_{1}-1|/\phi_{n}>0,$$

for some special partitions (fast decay $t_{i}=\frac{i}{2^{3n}}$).

Since you are interested in various divisions, here are some more references

• "On almost sure convergence of the quadratic variation of Brownian motion"
• "Sufficient and Necessary Conditions for Limit Theorems for Quadratic Variations of Gaussian Sequences"
• "Quadratic variation for Gaussian processes and application to time deformation"