1. A multiplicative version of the Lindley recursion.
- Author
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Boxma, Onno, Löpker, Andreas, Mandjes, Michel, and Palmowski, Zbigniew
- Subjects
- *
STOCHASTIC analysis , *AUTOREGRESSIVE models , *RECURSION theory , *BOUNDARY value problems , *ORDER picking systems , *PROBABILITY theory , *RANDOM variables - Abstract
This paper presents an analysis of the stochastic recursion W i + 1 = [ V i W i + Y i ] + that can be interpreted as an autoregressive process of order 1, reflected at 0. We start our exposition by a discussion of the model's stability condition. Writing Y i = B i - A i , for independent sequences of nonnegative i.i.d. random variables { A i } i ∈ N 0 and { B i } i ∈ N 0 , and assuming { V i } i ∈ N 0 is an i.i.d. sequence as well (independent of { A i } i ∈ N 0 and { B i } i ∈ N 0 ), we then consider three special cases (i) V i equals a positive value a with certain probability p ∈ (0 , 1) and is negative otherwise, and both A i and B i have a rational LST, (ii) V i attains negative values only and B i has a rational LST, (iii) V i is uniformly distributed on [0, 1], and A i is exponentially distributed. In all three cases, we derive transient and stationary results, where the transient results are in terms of the transform at a geometrically distributed epoch. [ABSTRACT FROM AUTHOR]
- Published
- 2021
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