Inverse Exponential Decay: Stochastic Fixed Point Equation and ARMA Models

Bartosz Ko{\l}odziejek; Krzysztof Burdzy; Tvrtko Tadi\'c

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Inverse Exponential Decay: Stochastic Fixed Point Equation and ARMA Models

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arxiv 1801.00149 v4 pith:VEPPLO5H submitted 2017-12-30 math.PR math.DSmath.STstat.TH

Inverse Exponential Decay: Stochastic Fixed Point Equation and ARMA Models

Krzysztof Burdzy , Bartosz Ko{\l}odziejek , Tvrtko Tadi\'c This is my paper

classification math.PR math.DSmath.STstat.TH

keywords armacoefficientsdecayequationexponentialfixedinversenonnegative

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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" (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 IED class. We describe the lower envelope for these ARMA processes.

Inverse Exponential Decay: Stochastic Fixed Point Equation and ARMA Models

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