Abstract
In 1970 Kaijser showed in some particular but typical cases that the contractive action of the random products Sn = Yn...Y1 on P(ℝd) implies that Log ∥Snx∥ suitably normalized converges in distribution to a gaussian law (see [38], [39], [40]). This idea was later fully developed by Le Page in [49] where he proved that, loosely speaking, Log ∥Snx∥ behaves like a sum of i.i.d. real random variables and satisfies analogues of the main classical limit theorems. We shall give here a detailed introduction to the important work of Le Page and present, with some improvements, his main results (namely, the central limit theorem with speed of convergence and an estimate of the large deviations of Log ∥Snx∥). We also study the tightness of {Snx, n ≧ 1} without moment assumption. This chapter ends with an application to linear stochastic differential equations.
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