Renormalization group approach to a generalization of the law of iterated logarithms for one-dimensional (non-Markovian) stochastic chains 2004.3.6

Renormalization group (RG) is, roughly speaking, a dynamical system determined by a map which represents a response of (a set of random) objects in consideration to a change of accuracy of observation, or ‘scale transformation’, on a parameter space of generating functions of quantities defined on the objects. The method is expected to analyze the asymptotic behaviors and critical phenomena of random objects. We can think of, and there have been deep works on, various objects, for which the RG approach may be effective. For the purpose of exhibiting the RG idea, we shall here focus on a simplest object for which the RGmethod is non-trivial, a class of probability measures on a set of paths (stochastic chains) on Z. The RG approach focuses on (stochastic and/or approximate) similarity of the object (paths, in our case), rather than on Markov and/or martingale properties, hence the RG has a (yet to be explored) possibility of being a complimentary tool to these well-established methods. One other point about introducing RG approaches to stochastic chains is that, like differential equations and stochastic differential equations, RG can be seen as a differential type equation which determine the object (stochastic chain, in our case) as a solution to a RG equation. In fact, we will see that, given an arbitrary one dimensional RG, we can uniquely construct a stochastic chain consistent with the equation. The RG approach to the simple random walk on Z has been known in mathematics [7]. Our standpoint is to place RG in the center, instead of regarding RG as another method of constructing wellknown stochastic processes, and to show that there is a large class of stochastic chains, including simple random walks and self-avoiding paths, for which RG acts naturally, and to show, in particular, that a generalization of the law of iterated logarithms hold for such chains.