Factorization identities

in the theory of random walk

A system of multi-parameter identities giving a connection between different characteristics of a random walk. As characteristics, the so-called boundary functionals — random variables related to the attainment of a boundary by the walk, such as the supremum of wandering, the moment of first attaining this supremum, the size of the first overshoot, etc. — are used.

The name factorization identities is related to the fact that they are obtained by factorization, i.e. by a representation of the function $1-zf(\lambda)$, where $f(\lambda)$ is the characteristic function of the random variable generating the walk, $|z|\leq1$, for real $\lambda$ in the form of a product of two factors, one of which is analytic, bounded, does not vanish, and is continuous up to the boundary in the upper half-plane $\operatorname{Im}\lambda>0$, while the second has the same properties in the lower half-plane. Such a representation is unique up to a constant factor (cf. Wiener–Hopf method), which allows one to identify the corresponding factors of the different factorizations of the type indicated which are obtained by probabilistic consideration and are written in terms of characteristic functions of the joint distributions of the boundary functionals of the random walk.

From factorization identities a number of both new and already-known results related to the theory of random walk can be obtained. For example, the strong law of large numbers and the arcsine law.

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For related questions concerning spectral factorization in linear filtering theory see [a1].

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