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Uniform convergence in probability and stochastic equicontinuity. (English) Zbl 0743.60012
Pointwise convergence and equicontinuity characterize uniform convergence to a continuous function on a compact set. The author provides a convergence-in-probability analogue for random functions. Let \(\{\hat Q_ n(\theta)\}\) be random functions indexed by \(\theta\in\Theta\), where \(\Theta\) is compact, and let \(\{\bar Q_ n(\theta)\}\) be deterministic functions. It is shown that \(\sup_{\theta\in\Theta}|\hat Q_ n(\theta)-\bar Q_ n(\theta)|\) converging to 0 in probability is equivalent to convergence of \(| \hat Q_ n(\theta)-\bar Q_ n(\theta)|\) to 0 in probability for each \(\theta\), together with a suitable stochastic equicontinuity property. If \(\hat Q_ n(\theta)\) is continuous and regarded as a random element of \(C\), the space of continuous functions on \(\Theta\), then the stochastic equicontinuity property is related to tightness of the sequence \(\{\hat Q_ n(\theta)\}\) in \(C\). Examples from \(U\)-statistics and non-parametric least squares are discussed.

60B10 Convergence of probability measures
60F05 Central limit and other weak theorems
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