For grids (p/3^n)Z+(r/3^n) with p rational and r irrational, the Peano curve's horizontal coordinate x has quadratic variation 3^{floor(-log_3 p)}p(1-(3/4)3^{floor(-log_3 p)}p)t.
Local times of deterministic paths and self-similar processes with stationary increments as normalized numbers of interval crossings
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abstract
We prove a general result on a relationship between a limit of normalized numbers of interval crossings by a c\`adl\`ag path and an occupation measure associated with this path. Using this result we define local times of fractional Brownian motions (classically defined as densities of relevant occupation measure) as weak limits of properly normalized numbers of interval crossings. We also discuss a similar result for c\`adl\`ag semimartingales, in particular for alpha-stable processes, and for Rosenblatt processes, and provide natural examples of deterministic paths which possess quadratic or higher order variation but no local times.
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Quadratic variation and local times of the horizontal component of the Peano curve (square filling curve)
For grids (p/3^n)Z+(r/3^n) with p rational and r irrational, the Peano curve's horizontal coordinate x has quadratic variation 3^{floor(-log_3 p)}p(1-(3/4)3^{floor(-log_3 p)}p)t.