Conditions of the uniform convergence of empirical averages to their expectations

Series: 
Stochastics Seminar
Thursday, January 15, 2009 - 15:00
1 hour (actually 50 minutes)
Location: 
Skiles 269
,  
Russian Academy of Sciences and Royal Holloway University of London
The uniform convergence of empirical averages to their expectations for a set of bounded test functions will be discussed. In our previous work, we proved a necessary and sufficient condition for the uniform convergence that can be formulated in terms of the epsilon-entropy of certain sets associated to the sample. In this talk, I will consider the case where that condition is violated. The main result is that in this situation strong almost sure oscillations take place. In fact, with probability one, for a given oscillation pattern, one can find an admissible test function that realizes this pattern for any positive prescribed precision level.