Simple bounds on the most predictable component of a stochastic model
Abstract
Special combinations of variables can have more predictability than any single variable in the combination. What is the maximum possible predictability that can be achieved through such combinations? Recently, this question was answered in the context of a linear stochastic model with fixed dynamics, where a sharp upper bound on predictability time was derived. However, the precise maximum is a complicated function of the entire spectrum of dynamical eigenvalues, obscuring any simple relation between predictability and eigenmodes. Based on numerical solutions of specific cases, it is conjectured here that the predictability of a stochastic model with a given least damped mode is bounded above by the predictability of a model in which all dynamical eigenvalues coalesce to the value corresponding to the least damped mode. Furthermore, it is shown that in this limit the maximum predictability time is determined by the largest root of a Laguerre polynomial. This result is used to prove the following simple bound: The maximum predictability time in the limit of coalesced eigenvalues is at most 4D-6 times the predictability time of the least damped mode, where D is the number of dynamical eigenmodes and D exceeds 3.
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