Large deviations of products of random topical operators. (English) Zbl 1073.60027

Toomey, Fergal

Ann. Appl. Probab. 12, No. 1, 317-333 (2002).

Summary: A topical operator on \(\mathbb{R}^d\) is one which is isotone and homogeneous. Let \(\{A(n) : n \geq 1\}\) be a sequence of i.i.d. random topical operators such that the projective radius of \(A(n) \dots A(1)\) is almost surely bounded for large \(n\). If \(\{x(n) : n \geq 1\}\), is a sequence of vectors given by \(x(n) = A(n) \dots A(1)x_0\), for some fixed initial condition \(x_0\), then the sequence \(\{x(n)/n : n \geq 1\}\) satisfies a weak large deviation principle. As corollaries of this result we obtain large deviation principles for products of certain random aperiodic max-plus and min-plus matrix operators and for products of certain random aperiodic nonnegative matrix operators.

MSC:

60F10	Large deviations
37H99	Random dynamical systems
47H07	Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
47H40	Random nonlinear operators

Keywords:

discrete event systems; max-plus algebra; nonnegative matrices

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