Asymptotic Suprema of Averaged Random Functions
Asymptotic Suprema of Averaged Random Functions
Suppose $X_i$ are i.i.d. random variables taking values in $\mathscr{X}, \Theta$ is a parameter space and $y: \mathscr{X} \times \Theta \rightarrow \mathbf{R}$ is a map. Define the averages $S_n(y, \theta) = (1/n)\sum^n_{i = 1}y(X_i, \theta)$ and the truncated expectations $T_m(y, \theta) = \mathbf{E} \max(y(X_1, \theta), - m)$. Under the hypothesis …