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

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Mirek M.

Colloquium Mathematicum

2010

tom 118

nr 2

705-720

We consider the random recursion $Xₙ^{x} = MₙX_{n-1}^{x} + Qₙ + Nₙ(X_{n-1}^{x})$, 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 $Nₙ(X_{n-1}^{x})$ is smaller at infinity, than $MₙX_{n-1}^{x}$. Using the asymptotics of the stationary solutions we show that properly normalized Birkhoff sums $Sₙ^{x} = ∑_{k=0}^{n} X_{k}^{x}$ converge weakly to an α-stable law for α ∈ (0,2]. The related local limit theorem is also proved.

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1 Colloquium Mathematicum

1 Mirek M.

1 2010

Wyniki wyszukiwania

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