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Some estimators of maximum likelihood type are constructed for estimating functionals of one-dimensional Gibbs states. We also show that those estimators are strongly consistent, asymptotically normal and asymptotically efficient.

We compute the expected values of certain random variables associated with a random process of manifolds in R n by inserting certain general formulas of integral geometry into the definition of the moment measures of a point process.

Let ℳ be a geometrically finite hyperbolic surface with infinite volume, having at least one cusp. We obtain the limit law under the Patterson-Sullivan measure on T 1ℳ of the windings of the geodesics of ℳ around the cusps. This limit law is stable with parameter 2δ− 1, where δ is the Hausdorff dimension of the limit set of the subgroup Γ of Möbius isometries associated with ℳ. The normalization is t −1/(2δ−1), for geodesics of length t. Our method relies on a precise comparison between geodesics and diffusion paths, for which we need to approach the Patterson-Sullivan measure mentioned above by measures that are regular along the stable leaves.

If C is a distribution function on (0, ∞) then the harmonic renewal function associated with C is the function $$G(x) = \sum\limits_1^\infty {n^{ - 1} } C^{(n)} (x)$$ . We link the asymptotic behaviour of G to that of 1−C. Applications to the ladder index and the ladder height of a random walk are included.

Let {W(t), t≧0} be a standard Wiener process, and let L(x, t) be its jointly continuous local time. Define $$T_r = inf\{ t \geqq 0;L(0,t \geqq r)\} .$$ The upper and lower class behaviour of inf L(y, T r) is investigated, where the infimum is taken on an interval, which is an appropriately chosen function of r.