3
$\begingroup$

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 $\mu = E[h(X,\theta_0)]$, $X\sim F$.

$\endgroup$
1
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.