Uniform large and moderate deviations for functional empirical processes
Let \((X_i)\) be a sequence of i.i.d. random variables with the distribution \(\mu \) taking the values in a Polish space \(\Sigma \). The authors study the large and moderate deviation principles (LDP and MDP) for the processes \(L_n(t)=\frac 1n\sum _{i=1}^{[nt]}\delta _{X_i}\) and \(M\widehat {L}_n(t)=\frac {1}{\sqrt {n}}\sum _{i=1}^{[nt]}(\delta _{X_i}- \mu )\), resp., and \(t\geq 0\). They prove that \(L_n\) satisfies LDP with the good rate function \[ I_L(\nu )=\begin{cases} \int _0^\infty \int \frac {d\dot {\nu }}{d\mu } \log \left (\frac {d\dot {\nu }}{d\mu }\right ) d\mu dt & \text{for} \nu \in AC, \dot {\nu }\ll \mu, \\ \infty & \text{otherwise},\end{cases} \] where \(AC\) is the space of all maps \(\nu \) from \(\mathbf {R}_+\) into the space of bounded signed measures on \(\Sigma \) with \(\nu (0)=0\), \(\nu (t)\ll |\nu (t)|\), and such that the weak derivative \(\dot {\nu }\) exists for a.a. \(t\in \mathbf {R}_+\). Similarly \(\widehat {L}_n\) satisfies MDP with the good rate function \[ I_M(\nu )=\begin{cases} \frac 12 \int _0^\infty \int \left (\frac {d\dot {\nu }}{d\mu } \right )^2 d\mu dt & \text{for} \nu \in AC, \dot {\nu }\ll \mu ,\^^M\text{and}\^^M\nu (\Sigma )=0, \\ \infty & \text{otherwise.} \end{cases} \] Given a class \(\mathcal {F}\) of bounded measurable functions on \(\Sigma \), any bounded signed measure \(\nu \) naturally represents an element \(E_\nu \in l_\infty (\mathcal {F})\). With suitably chosen class \(\mathcal {F}\) the authors show the analogical statements for \(E_{L_n}\) and \(E_{\widehat {L}_n}\).
scientific article; zbMATH DE number 1134828
scientific article