A conditional limit theorem for generalized diffusion processes (Q1890293)

Consider a one-dimensional generalized diffusion process \( X=\{X(t):t\geq 0\} \). Suppose that \(X(0)>0\), and define \(\tau_ X(0)=\inf\{t\geq 0:X(t)=0\}\). The authors show that, for a suitable scale function \(u(c)\), the distribution of \(\{u(c)^{-1}X(ct):0<t\leq 1\}\) conditioned on \(\{\tau_X(0)>c\}\) converges as \(c\to\infty\); the limit can be characterized in terms of the Bessel excursion and the Bessel meander. The conditional distribution of \(u(c)^ {-1}X(c)\) is shown to converge to the Weibull distribution. The authors also prove a limit theorem for some regenerative processes associated with \(X\).