Limit theorems for nondegenerate \(U\)-statistics of continuous semimartingales. (English) Zbl 1308.60051

For a sequence of observations from a one-dimensional continuous Itō semimartingale, a U-statistic is defined by extending the similar definition based on a sequence of independent and identically distributed observations. The uniform convergence in probability (i.e., the law of large numbers) of this \(U\)-statistic is first proved. Then, a stable central limit theorem is established. With an estimate for the conditional variance, a standard central limit theorem is also proved for a standardized version of the \(U\)-statistic. These general results are applied to derive the limit distributions for Gini’s mean difference, which may be viewed as an alternative measure of price variability in mathematical finance, an \(L^p\)-type test statistic for constant volatility, and the Wilcoxon test statistic for structure breaks.


60G48 Generalizations of martingales
60F05 Central limit and other weak theorems
60F15 Strong limit theorems
60F17 Functional limit theorems; invariance principles
60H05 Stochastic integrals
62M09 Non-Markovian processes: estimation
62P05 Applications of statistics to actuarial sciences and financial mathematics
91G80 Financial applications of other theories
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