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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.

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