Limit theorems for some critical superprocesses

Let X = {X_t, t ≥ 0; P_μ} be a critical superprocess starting from a finite measure μ. Under some conditions, we first prove that lim_t→∞ tP_μ(∥X_t≠ = 0) = ν⁻¹〈φ₀, μ〉, where φ₀ is the eigenfunction corresponding to the first eigenvalue of the infinitesimal generator L of the mean semigroup of X, and ν is a positive constant. Then we show that, for a large class of functions f, conditioning on ∥X_t≠ = 0, t⁻¹〈f, X_t〉 converges in distribution to 〈f, ψ₀〉_mW, where W is an exponential random variable, and ψ₀ is the eigenfunction corresponding to the first eigenvalue of the dual of L. Finally, if 〈f, ψ₀〉_m = 0, we prove that, conditioning on ∥X_t∥ = 0, (t⁻¹〈φ₀, X_t〉, t^−1/2〈f, X_t〉) converges in distribution to (W, G(f) √ W), where G(f) ∼ N (0, σ²_f) is a normal random variable, and W and G(f) are independent.