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.
2025-01-13 02:21:34.1736734894
convergence in distribution
53 Views Asked by john https://math.techqa.club/user/john/detail AtRelated Questions in PROBABILITY
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