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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.

MSC:

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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References:

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