Stable fractal sums of pulses: the cylindrical case
Mandelbrot, Benoit B.
Taqqu, Murad S.
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Citation (published version)Renata Cioczek-Georges, Benoit B Mandelbrot, Gennady Samorodnitsky, Murad S Taqqu. 1995. "Stable fractal sums of pulses: the cylindrical case." Bernoulli, Volume 1, pp. 201 - 216. https://doi.org/10.3150/bj/1193667815
A class of α-stable, 0\textlessα\textless2, processes is obtained as a sum of ’up-and-down’ pulses determined by an appropriate Poisson random measure. Processes are H-self-affine (also frequently called ’self-similar’) with H\textless1/α and have stationary increments. Their two-dimensional dependence structure resembles that of the fractional Brownian motion (for H\textless1/2), but their sample paths are highly irregular (nowhere bounded with probability 1). Generalizations using different shapes of pulses are also discussed.
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