Process convergence of fluctuations of linear eigenvalue statistics of random circulant matrices

We discuss the process convergence of the time dependent fluctuations of linear eigenvalue statistics of random circulant matrices with independent Brownian motion entries, as the dimension of the matrix tends to [Formula: see text]. Our derivation is based on the trace formula of circulant matrix, method of moments and some combinatorial techniques.

Demographic situation in China: Convergence or divergence?

The purpose of this study is to locate the presence of convergence in the demographic development of Chinese provinces during the end of the demographic transition at the turn of the millennium. We have estimated sigma and beta convergence in fertility, mortality, urbanization, and population ageing basing on the official Chinese statistics for 31 provinces of China. Our results show that the regional convergence in the above indicators has not been sustainable. It was observed only in certain periods, except for the urbanization process. Convergence was accompanied by a catching-up effect in such periods when “lagging” provinces were passing the demographic transition relatively quickly. The paper can serve as a contribution to the regional demographic and economic policy of China, since the issue of the dynamics of the regional demographic development differentiation is the basis for demographic and economic projections and development of local policy measures. The demographic divergence that we discovered in the last decade can determine an obstacle to the sustainable development of the country in the near future.

Gumbel and Fréchet convergence of the maxima of independent random walks

We consider point process convergence for sequences of independent and identically distributed random walks. The objective is to derive asymptotic theory for the largest extremes of these random walks. We show convergence of the maximum random walk to the Gumbel or the Fréchet distributions. The proofs depend heavily on precise large deviation results for sums of independent random variables with a finite moment generating function or with a subexponential distribution.