Let $U_{t}$ be iid Uniformly distributed on (0,1). Suppose $\hat{\theta}_{T}\stackrel{d}\rightarrow \theta^{*}$ with $\theta^{*}$ some random variable on (0,1). I believe $\sum_{t=1}^{T}I(U_{t}\leq \hat{\theta}_{T})/T \stackrel{d}\rightarrow \theta^{*}$ should hold but am having trouble formalizing the result. Any help would be appreciated.
2026-03-08 01:00:34.1772931634
convergence in distribution
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