Limit theorems for a class of critical superprocesses with stable branching

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Abstract: We consider a critical superprocess

X; m a t h b f P_{m} u

with general spatial motion and spatially dependent stable branching mechanism with lowest stable index

g a m m a_{0} > 1

. We first show that, under some conditions,

m a t h b f P_{m u} (| X_{t} | e q 0)

converges to

0

as

t o i n f t y

and is regularly varying with index

(g a m m a_{0} - 1)^{- 1}

. Then we show that, for a large class of non-negative testing functions

f

, the distribution of

X_{t} (f); m a t h b f P_{m} u (c d o t | | X_{t} | e q 0)

, after appropriate rescaling, converges weakly to a positive random variable

m a t h b f z^{(g a m m a_{0} - 1)}

with Laplace transform

E [e^{- u m a t h b f z^{(g a m m a_{0} - 1)}}] = 1 - (1 + u^{- (g a m m a_{0} - 1)})^{- 1 / (g a m m a_{0} - 1)} .

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