Tohoku Mathematical Journal

Semi-stable processes on local fields

Kumi Yasuda

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Some characters of semi-stable stochastic processes on local fields such as epochs, spans, and indices are given, and differences in nature from the corresponding objects for Euclidean spaces are clarified. Criteria for the recurrence and for the polarity of one point sets are given, and it is shown that semi-stable processes are characterized as limits of suitably scaled sums of independent identically distributed random variables.

Article information

Tohoku Math. J. (2), Volume 58, Number 3 (2006), 419-431.

First available in Project Euclid: 17 November 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G52: Stable processes
Secondary: 60G50: Sums of independent random variables; random walks

Semi-stable processes local field limit theorem


Yasuda, Kumi. Semi-stable processes on local fields. Tohoku Math. J. (2) 58 (2006), no. 3, 419--431. doi:10.2748/tmj/1163775138.

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  • S. Albeverio and W. Karwowski, A random walk on $p$-adics---the generator and its spectrum, Stochastic Process. Appl. 53 (1994), 1--22.
  • S. N. Ethier and T. G. Kurtz, Markov processes, Characterization and convergence, Wiley Ser. Prob. Math. Stat. Prob. Math. Stat., John Wiley & Sons, Inc., New York, 1986.
  • A. N. Kochubei, Limit theorems for sums of $p$-adic random variables, Exposition. Math. 16 (1998), 425--439.
  • A. N. Kochubei, Pseudo-differential equations and stochastics over non-Archimedean fields, Monogr. Textbooks Pure Appl. Math. 244, Marcel Dekker, Inc., New York, 2001.
  • S. Lang, Algebraic number theory, Grad. Texts in Math. 110, Springer-Verlag, New York, 1994.
  • M. Maejima and Y. Naito, Semi-selfdecomposable distributions and a new class of limit theorems, Probab. Theory Related Fields 112 (1998), 13--31.
  • M. Maejima and K. Sato, Semi-selfsimilar processes, J. Theoret. Probab. 12 (1999), 347--373.
  • M. Maejima and R. Shah, Moments and projections of semistable probability measures on $p$-adic vector spaces, to appear in J. Theoret. Prob.
  • M. Maejima, K. Sato and T. Watanabe, Distributions of selfsimilar and semi-selfsimilar processes with independent increments, Statist. Probab. Lett. 47 (2000), 395--401.
  • K. R. Parthasarathy, Probabability measures on metric spaces, Probab. Math. Statist. 3, Academic Press, Inc., New York-London, 1967.
  • W. Rudin, Fourier analysis on groups, Wiley Classics Lib., A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1990.
  • K. Sato, Lévy processes and infinitely divisible distributions, Cambridge Stud. Adv. Math. 68, Cambridge Univ. Press, 1999.
  • K. Sato and K. Yamamuro, On selfsimilar and semi-selfsimilar processes with independent increments, J. Korean Math. Soc. 35 (1998), 207--224.
  • V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, $p$-adic analysis and mathematical physics, Ser. Soviet East European Math. 1, World Scientific Publishing Co., Inc., River Edge, N. J., 1994.
  • K. Yasuda, Additive processes on local fields, J. Math. Sci. Univ. Tokyo 3 (1996), 629--654.
  • K. Yasuda, On infinitely divisible distributions on locally compact abelian groups, J. Theoret. Prob. 13 (2000), 635--657.