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Limit theorems for hitting times of 1-dimensional generalized diffusions

Published online by Cambridge University Press:  22 January 2016

Matsuyo Tomisaki  and
Makoto Yamazato
Matsuyo Tomisaki
Affiliation:
Department of Mathematics, Faculty of Science, Nara Women’s UniversityNara, 630-8506, Japan, tomisaki@cc.nara-wu.ac.jp
Makoto Yamazato
Affiliation:
Department of Mathematical Sciences, College of Science, University of the Ryukyus, Nishihara, Okinawa 903-0213, Japan, yamazato@math.u-ryukyu.ac.jp
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Abstract.

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Limit theorems are obtained for suitably normalized hitting times of single points for 1-dimensional generalized diffusion processes as the hitting points tend to boundaries under an assumption which is slightly stronger than that the existence of limits γ + 1 of the ratio of the mean and the variance of the hitting time. Laplace transforms of limit distributions are modifications of Bessel functions. Results are classified by the one parameter {γ}, each of which is the degree of corresponding Bessel function. In case the limit distribution is degenerate to one point, by changing the normalization, we obtain convergence to the normal distribution. Regarding the starting point as a time parameter, we obtain convergence in finite dimensional distributions to self-similar processes with independent increments under slightly stronger assumption.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 152 , December 1998 , pp. 1 - 37
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1998

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