Let $B\_t$ be a standard Brownian motion. Does there with probability one exist a sequence of partitions $\{t\_{k, n} : k = 0, 1, \dots, k\_n\}$ $$0 = t\_{0, n} < t\_{1, n} < \dots < t\_{k\_n, n} = 1,$$with$$\lim\_{n \to \infty} \max\{t\_{j, n} - t\_{j - 1, n} : j = 1, \dots, k\_n\} = 0$$and$$\liminf\_{n \to \infty} \sum\_{j=1}^{k\_n} (B(t\_{j, n}) - B(t\_{j - 1 n}))^2 > 1?$$



Brownian motion, quadratic variation, existence of partitions?
If the sequence of partitions is fixed/deterministic, the answer is no by an old result of Paul Lévy (P. Levy , Theorie de l'addition des variables aleatoires, Paris, 1937).


If one allows the partitions to depend on the element of the probability space than the quantity will get as large as $2\log \log n$ almost surely where the partitions are restricted to be $1/n$ separated. This is closely related to the law of the iterated logarithm for Brownian motion and was proved by S. J. Taylor in 1972 (S. J. Taylor, Exact asymptotic estimates of Brownian path variation, Duke Math. J.
Volume 39, Number 2 (1972), 219-241.) 


The introduction to Taylor's paper is a good source for a discussion of this topic including Lévy's result.


