One-step asymptotics of hyperplane crossings by a mixed stable-Brownian L´evy process
DOI:
https://doi.org/10.31471/2304-7399-2026-22(83)-105-111Keywords:
L´evy process, stable process, Brownian motion, hyperplane crossing, one-step asymptoticsAbstract
We consider a d-dimensional L´evy process which is the sum of independent processes: a rotationally invariant alpha-stable process with alpha between 1 and 2 and a Brownian motion. For a fixed hyperplane we study the one-step crossing probability. The main result states that for any smooth compactly supported test function, as the time tends to zero, the leading contribution is determined by the Brownian component and has order square root
of the time step
References
1. Sato K. L´evy Processes and Infinitely Divisible Distributions / K. Sato. – Cambridge: Cambridge University Press, 1999. – 486 p.
2. Samorodnitsky G. Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance / G. Samorodnitsky, M. S. Taqqu. – New York: Chapman & Hall, 1994. – 632 p.
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Copyright (c) 2026 Ihor Melnyk
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