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An incomplete U-statistic is obtained by sampling the terms of an U-statistic. This paper derives the asymptotic distribution (if the variance is finite). Depending on the number of sampled terms, the resulting distribution is either the same as for the U-statistic, a normal distribution, or something intermediate. Also the case of a non-random sampling of the terms is treated. As an example, a non-parametric test of the independence of two circular random variables is studied. The results are generalized to generalized U-statistics.

If a mean field model for spin glasses is generic in the sense that it satisfies the extended Ghirlanda–Guerra identities, and if the law of the overlaps has a point mass at the largest point q* of its support, we prove that one can decompose the configuration space into a sequence of sets (A k ) such that, generically, the overlap of two configurations is equal to q* if and only if they belong to the same set A k . For the study of the overlaps each set A k can be replaced by a single point. Combining this with a recent result of Panchenko (A connection between Ghirlanda–Guerra identities and ultrametricity. Ann Probab (2008, to appear)) this proves that if the overlaps take only finitely many values, ultrametricity occurs. We give an elementary, self-contained proof of this result based on simple inequalities and an averaging argument.

We investigate the percolative properties of the vacant set left by random interlacements on $${\mathbb{Z}^d}$$ , when d is large. A non-negative parameter u controls the density of random interlacements on $${\mathbb{Z}^d}$$ . It is known from Sznitman (Ann Math, 2010), and Sidoravicius and Sznitman (Comm Pure Appl Math 62(6):831–858, 2009), that there is a non-degenerate critical value u *, such that the vacant set at level u percolates when u < u *, and does not percolate when u > u *. Little is known about u *, however, random interlacements on $${\mathbb{Z}^d}$$ , for large d, ought to exhibit similarities to random interlacements on a (2d)-regular tree, where the corresponding critical parameter can be explicitly computed, see Teixeira (Electron J Probab 14:1604–1627, 2009). We show in this article that lim inf d  u */ log d ≥ 1. This lower bound is in agreement with the above mentioned heuristics.

Limit theorems with a non-Gaussian limiting distribution have been obtained, under appropriate conditions for partial sums of instantaneous nonlinear functions of stationary Gaussian sequences with long range dependence by a number of people. The normalization has typically been n α, with 1/2<α<1 where n is the sample size. Here examples of limit theorems are given for quadratic functions with long range memory (not instantaneous) with a normalization n α, 0<α<1/2.