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

Mariusz Mirek. "Convergence to stable laws and a local limit theorem for stochastic recursions." Colloquium Mathematicae 118.2 (2010): 705-720. <http://eudml.org/doc/283412>.

@article{MariuszMirek2010,
abstract = {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.},
author = {Mariusz Mirek},
journal = {Colloquium Mathematicae},
keywords = {asymptotics of the stationary solutions; normalized Birkhoff sums; -stable law},
language = {eng},
number = {2},
pages = {705-720},
title = {Convergence to stable laws and a local limit theorem for stochastic recursions},
url = {http://eudml.org/doc/283412},
volume = {118},
year = {2010},
}

TY - JOUR
AU - Mariusz Mirek
TI - Convergence to stable laws and a local limit theorem for stochastic recursions
JO - Colloquium Mathematicae
PY - 2010
VL - 118
IS - 2
SP - 705
EP - 720
AB - 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.
LA - eng
KW - asymptotics of the stationary solutions; normalized Birkhoff sums; -stable law
UR - http://eudml.org/doc/283412
ER -