Helmut H. Pitters

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math.PR2025

A Poisson representation of the positive sojourn time of LÃ©vy processes

Helmut H. Pitters

We study the distribution of the positive sojourn time $A_{t} := \int_{0}^{t} 1 {X_{s} > 0} d s$ of an arbitrary LÃ©vy process $X := (X_{t})_{t \geq 0}$ . For an exponential random va…

math.PR2025

Levy's second arcsine law via the ballot theorem

Helmut H. Pitters

We provide a new and elementary proof of Levy's second arcsine law for Brownian motion. The only tools required are basic properties of Brownian motion and Poisson processes, and t…

math.PR2024

Occupation times and areas derived from random sampling

Frank Aurzada, Leif DÃ¶ring, Helmut H. Pitters

We consider the occupation area of spherical (fractional) Brownian motion, i.e. the area where the process is positive, and show that it is uniformly distributed. For the proof, we…