Asymptotic formula for probabilistic representations of Bi-continuous C-semigroups
Yue Tia · Journal of China University of Mining and Technology · 2013
Based on the definition and properties of the Bi-continuous C-semigroups on a Banach space endowed with an additional locally convex topologyτ,the probabilistic representations for Bi-continuous C-semigroups on Banach space were discussed in the light of operator-valued mathematical expectation and Riemann-Stieltjes integral.Then,with Riemann-Stieltjes integral,Taylor expansion of the Bi-continuous C-semigroups,Hlder's inequality and estimations of moment-generating functions of some suitable random variables being used,some probabilistic approximations and estimations of convergent rates were presented for Bi-continuous C-semigroups,and the general conclusion of probabilistic approximation was drawn.Finally,considering several kinds of common probability distributions,and with these given asymptotic formulas,some results for strongly continuous semigroups,such as Kendall formula and Chung formula,were generalized to the Bi-continuous C-semigroups.The results show that the central moment of a random variable plays an important role in convergent rates of asymptotic formula and they present a negative relationship.