Convergence rate for a class of supercritical superprocesses

Suppose X={Xt,t≥0} is a supercritical superprocess. Let ϕ be the non-negative eigenfunction of mean semigroup X corresponding to principal eigenvalue λ>0. Then Mt(ϕ)=e−λt〈ϕ,Xt〉,t≥0, martingale with almost sure limit M∞(ϕ). In this paper we study rate at which Mt(ϕ)−M∞(ϕ) converges 0 as t→∞ when process may not have finite variance. Under some conditions on semigroup, provide sufficient and necessary for in sense. Some results convergence Lp p∈(1,2) are also obtained.