Abstract
This work studies identications of values of probability limits based on trajectories of convergent (random) sequences. The key concept is the so called Probability Limit Identication Function (PLIF).
The main concern is focused on the existence of PLIFs, mainly those, which are measurable and adapted. We also study in more detail special cases, when the convergence in probability and the convergence almost surely coincide.
Furthermore, possible applications of the PLIF concept in stochastic analysis (path-wise representations of stochastic integrals and weak solutions of the stochastic dierential equations), as well as in estimation theory (the existence of strongly consistent estimators) are outlined. The achieved results are based on analyses of the topologies on spaces of measures, spaces of random variables and spaces of real-valued functions.