On the typical level crossing time and path

Let Y1, Y2, … be a stochastic process and M a positive real number. Define the level crossing time TM = inf{nYn > M} (TM) = + ∞ if Yn ≤ M for n = 1, 2, …). We study the process with the condition that the high level M is crossed. Using the techniques of large deviations theory we describe roughly when and how the level crossing typically occurs. The main hypotheses required are stated in terms of the generating functions associated with the process (Yn).