Perron–Frobenius theory for kernels and Crump–Mode–Jagers processes with macro-individuals

Perron–Frobenius theory developed for irreducible non-negative kernels deals with so-called R-positive recurrent kernels. If the kernel M is R-positive recurrent, then the main result determines the limit of the scaled kernel iterations $R^nM^n$ as $n\to\infty$ . In Nummelin (1984) this important result is proven using a regeneration method whose major focus is on M having an atom. In the special case when $M=P$ is a stochastic kernel with an atom, the regeneration method has an elegant explanation in terms of an associated split chain. In this paper we give a new probabilistic interpretation of the general regeneration method in terms of multi-type Galton–Watson processes producing clusters of particles. Treating clusters as macro-individuals, we arrive at a single-type Crump–Mode–Jagers process with a naturally embedded renewal structure.

EVOLUTIONARY TREND OF FOREIGN INVESTMENT IN CHINA: A COMBINED DECOMPOSITION AND TRANSITIONAL DYNAMICS APPROACH

This paper examines the pattern and evolution trend of foreign investment in China through combining decomposition analysis and framework of transitional dynamics. It is recognized that inter-regional disparity contributes the most to China’s disparity in foreign investment. Stochastic kernel analyses are then performed for the country and the economic zones regarding the foreign investment trend. It is concluded that convergence of foreign investment to the country’s mean cannot be attained and continues to locate at the lower end. This analysis offers illuminating insights on the evolution of foreign investment in China across time.