Suppose that a random function $g_T(\theta)$ converges in probability to a function $g(\theta)$ uniformly continuous over $\Theta$ as $T\rightarrow \infty$ $\forall \theta \in \Theta$. Is this condition sufficient for stochastic equicontinuity of the sequence of empirical processes $\{g_T(\cdot); T\geq 1\} $?
2026-03-28 06:48:10.1774680490
Is convergence in probability to a uniformly continuous function a sufficient condition for stochastic equicontinuity?
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