Asymptotic inference for Markov step processes: Observation up to a random time

Consider a Markov step process whose generator depends on an unknown one-dimensional parameter [theta]. Under a 'homogeneity' assumption concerning the family of information processes I[theta], [theta] [set membership, variant] [Theta], which does not require exact knowledge of the asymptotics of I[theta] under P[theta], there is an increasing sequence of bounded stopping times Un such that, observing X continuously over the random time interval [[0, Un]], the sequence of resulting statistical models is LAN as n --> [infinity], at every point [theta] [set membership, variant] [Theta], with local scale which does not depend on the parameter.