## Journal of Mathematics of Kyoto University

- J. Math. Kyoto Univ.
- Volume 43, Number 3 (2003), 567-583.

### A conditional limit theorem for generalized diffusion processes

Zenghu Li, Tokuzo Shiga, and Matsuyo Tomisaki

#### Abstract

Let $\mathbf{X} = \{ X(t) :t \leq 0\}$ be a one-dimensional generalized diffusion process with initial state $X(0) > 0$, hitting time $\tau _{\mathbf{X}}(0)$ at the origin and speed measure m which is regularly varying at infinity with exponent $1/\alpha -1 > 0$. It is proved that, for a suitable function $u(c)$, the probability law of $\{ u(c)^{-1}X(ct) : 0 < t \leq 1\}$ conditioned by $\{\tau _{\mathbf{X}}(0) > c \}$ converges as $c \to \infty$ to the conditioned $2(1-\alpha )$-dimensional Bessel excursion on natural scale and that the latter is equivalent to the $2(1-\alpha )$-dimensional Bessel meander up to a scale transformation. In particular, the distribution of $u(c)^{-1}X(c)$ converges to the Weibull distribution. From the conditional limit theorem we also derive a limit theorem for some of regenerative process associated with $\mathbf{X}$.

#### Article information

**Source**

J. Math. Kyoto Univ., Volume 43, Number 3 (2003), 567-583.

**Dates**

First available in Project Euclid: 14 August 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.kjm/1250283695

**Digital Object Identifier**

doi:10.1215/kjm/1250283695

**Mathematical Reviews number (MathSciNet)**

MR2028667

**Zentralblatt MATH identifier**

1061.60080

**Subjects**

Primary: 60J60: Diffusion processes [See also 58J65]

Secondary: 60J25: Continuous-time Markov processes on general state spaces

#### Citation

Li, Zenghu; Shiga, Tokuzo; Tomisaki, Matsuyo. A conditional limit theorem for generalized diffusion processes. J. Math. Kyoto Univ. 43 (2003), no. 3, 567--583. doi:10.1215/kjm/1250283695. https://projecteuclid.org/euclid.kjm/1250283695