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Convergence to stable laws and a local limit theorem for stochastic recursions

Mariusz Mirek — 2010

Colloquium Mathematicae

We consider the random recursion X x = M X n - 1 x + Q + N ( X n - 1 x ) , where x ∈ ℝ and (Mₙ,Qₙ,Nₙ) are i.i.d., Qₙ has a heavy tail with exponent α > 0, the tail of Mₙ is lighter and N ( X n - 1 x ) is smaller at infinity, than M X n - 1 x . Using the asymptotics of the stationary solutions we show that properly normalized Birkhoff sums S x = k = 0 n X k x converge weakly to an α-stable law for α ∈ (0,2]. The related local limit theorem is also proved.

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