Limit theorems for the empirical distribution function of scaled increments of It\^(o) semimartingales at high frequencies

We derive limit theorems for the empirical distribution function of "devolatilized" increments of an It\^{o} semimartingale observed at high frequencies. These "devolatilized" increments are formed by suitably rescaling and truncating the raw increments to remove the effects of stochastic volatility and "large" jumps. We derive the limit of the empirical c.d.f. of the adjusted increments for any It\^{o} semimartingale whose dominant component at high frequencies has activity index of $1<\beta\le2$, where $\beta=2$ corresponds to diffusion. We further derive an associated CLT in the jump-diffusion case. We use the developed limit theory to construct a feasible and pivotal test for the class of It\^{o} semimartingales with nonvanishing diffusion coefficient against It\^{o} semimartingales with no diffusion component.