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Limit theorems for $$U$$-processes. (English) Zbl 0789.60031
Let $$(S,{\mathcal S},P)$$ be a probability space, $$\{X_ i\}$$ an i.i.d. sequence of $$S$$-valued random variables with distribution $$P$$ and $$F$$ a class of measurable real functions on $$S^ m$$, $$m \geq 2$$. The paper gives a systematic study of the law of large numbers and the central limit theorem for $$U$$-processes of order $$m$$, $U^ n_ m (f)=(n-m)!/n! \sum_{I_{mn}} f(X_{i_ 1},\dots,X_{i_ m}),$ indexed by kernels $$f \in F$$, where $$I_{mn}=\{(i_ 1,\dots,i_ m):1 \leq i_ j \leq n$$, $$i_ j \neq i_ k$$ if $$j \neq k\}$$. There are given necessary and sufficient conditions for LLN and sufficient conditions for CLT both for nondegenerate and degenerate $$U$$-processes. These conditions are in terms of random metric entropies. The results are relatively complete: the CLT and LLN for measurable VC subgraph classes $$F$$ and for classes $$F$$ satisfying bracketing conditions are corollaries of the general results. As applications there are considered some particular $$U$$-processes including Liu’s simplicial depth process. The authors use various techniques as: a decoupling inequality, exponential inequalities including a new Bernstein type inequality for degenerate $$U$$-statistics, integrability of Gaussian and Rademacher chaos, etc.

##### MSC:
 60F17 Functional limit theorems; invariance principles 60B12 Limit theorems for vector-valued random variables (infinite-dimensional case) 62E20 Asymptotic distribution theory in statistics 60F15 Strong limit theorems 60F05 Central limit and other weak theorems
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