Abstract
Limit distributions of scaled sums of p-adic valued i.i.d. are characterized as semistable laws, and a condition to assure the weak convergence of a scaled sum is verified. The limit supremum of the norm of the weakly convergent scaled sum is divergent in fact, and the exact growth rate of the sum is given. It is also shown that, if a scaled sum including a time parameter in the number of the added i.i.d. is considered, the semigroup of the limit distributions corresponds to a p-adic valued Markov process having right continuous sample paths with left limits. These are generalizations of the former results for rotation-symmetric i.i.d., with some necessary modifications.
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References
S. Albeverio and W. Karwowski, “A random walk on p-adics–the generator and its spectrum,” Stoch. Process. Appl. 53, 1–22 (1994).
V. I. Bogachev, Measure Theory, Vol. II (Springer-Verlag, Berlin, Heidelberg, 2007).
M. Del Muto and A. Figa-Talamanka, “Diffusion on locally compact ultrametric spaces,” Expo. Math. 22, 197–211 (2004).
S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, Wiley Ser. Probab. Stat. (2005).
H. Kaneko, “A class of spatially inhomogeneous Dirichlet spaces on the p-adic number field,” Stoch. Process. Appl. 88, 161–174 (2000).
N. Koblitz, p-adic numbers, p-adic analysis, and zeta-functions, Grad. Texts in Math. 58 (Springer-Verlag, New York, 1984).
A. N. Kochubei, “Limit theorems for sums of p-adic random variables,” Expo. Math. 16, 425–439 (1998).
A. N. Kochubei, Pseudo-Differential Equations and Stochastics over Non-Archimedean Fields, Pure Appl. Math. (Boca Raton), CRC Press (2001).
S. Lang, Algebraic Number Theory, Grad. Texts in Math. 110 (Springer-Verlag, New York, 1994).
A. E. Mikhailov, “Extimates of convergence rates to stables distributions on Qp,” J. Math. Sci.(N.Y.) 206, 207–211 (2015).
K. R. Parthasarathy, Probability Measures on Metric Spaces (AMS Chelsea Publishing, 2005).
K. Yasuda, “On infinitely divisible distributions on locally compact Abelian groups,” J. Theor. Probab. 13, 635–657 (2000).
K. Yasuda, “On the asymptotic growth rate of the sum of p-adic-valued independent identically distributed random variables,” Arab. J. Math. (Springer) 3, 449–460 (2014).
V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, p-Adic Analysis and Mathematical Physics, Series on Soviet and East EuropeanMathematics 1 (World Scientific, 1994).
A. Weil, Basic Number Theory, Classics inMathematics (Springer-Verlag, Berlin, Heidelberg, 1995).
E. I. Zelenov, “The p-adic law of large numbers,” Izv. Math. 80, 489–499 (2016).
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Yasuda, K. Limit theorems for p-adic valued asymmetric semistable laws and processes. P-Adic Num Ultrametr Anal Appl 9, 62–77 (2017). https://doi.org/10.1134/S207004661701006X
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DOI: https://doi.org/10.1134/S207004661701006X