Open Access
November 2019 Inverse exponential decay: Stochastic fixed point equation and ARMA models
Krzysztof Burdzy, Bartosz Kołodziejek, Tvrtko Tadić
Bernoulli 25(4B): 3939-3977 (November 2019). DOI: 10.3150/19-BEJ1116


We study solutions to the stochastic fixed point equation $X\stackrel{d}{=}AX+B$ when the coefficients are nonnegative and $B$ is an “inverse exponential decay” ($\operatorname{IED}$) random variable. We provide theorems on the left tail of $X$ which complement well-known tail results of Kesten and Goldie. We generalize our results to ARMA processes with nonnegative coefficients whose noise terms are from the $\operatorname{IED}$ class. We describe the lower envelope for these ARMA processes.


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Krzysztof Burdzy. Bartosz Kołodziejek. Tvrtko Tadić. "Inverse exponential decay: Stochastic fixed point equation and ARMA models." Bernoulli 25 (4B) 3939 - 3977, November 2019.


Received: 1 May 2018; Revised: 1 January 2019; Published: November 2019
First available in Project Euclid: 25 September 2019

zbMATH: 07110161
MathSciNet: MR4010978
Digital Object Identifier: 10.3150/19-BEJ1116

Keywords: ARMA models , inverse-gamma distribution , iterated random sequences , Stochastic fixed point equation , tail estimates , time series

Rights: Copyright © 2019 Bernoulli Society for Mathematical Statistics and Probability

Vol.25 • No. 4B • November 2019
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