Is there a version of the law of large numbers for random functions of the type: $h(X_j,\hat{\theta}_n)$, where $X_1,\dots,X_n$ are i.i.d. random variables, with distribution $F$, and $\hat{\theta}_n = \hat{\theta}_n(X_1,\dots,X_n)$ such that $\hat{\theta}_n\stackrel{P}{\rightarrow}\theta_0$, as $n\rightarrow \infty$?

This is, I am interested in a result of the type

$$\left\vert\frac{1}{n}\sum_{j=1}^n h(X_j,\hat{\theta}_n) -\mu\right\vert\stackrel{P}{\rightarrow} 0,$$ where $\mu = E[h(X,\theta_0)]$, $X\sim F$.


The sequence

$$ \{ h(X_1, \hat \theta_n), \ldots, h(X_n, \hat \theta_n) \} $$

is a row-wise exchangeable triangular array. This appears to be the result you need http://www.tandfonline.com/doi/pdf/10.1080/07362998508809059?needAccess=true


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