Improved Limit Theorem, The Incremental Modulus Of Continuity Of Random Processes

Let {Г(t),t∈ R} be a Banach space SB-valued stochastic process, let P be the probability measure generated by Г(). Assume Г()is P-almost surly continuous with respect to the norm||. ||of SB and that there exists a positive nondecreasing function σ(a), a>0,such that with some K,r,B>0 .Then assuming also that σ( )is a regularly varying function at zero, or atinfinity, with a positive exponent, [l]has proved large deviation results for increments like sup sup || Г(t+ s)-Г(t) ||, which then used toestablish module of continuity and large increment estimates for Г( ).In this article , we better the results in [1], and prove large deviation results for increments like sup sup || Гt + s)-Г(t) ||, Then wehave some limit theorems of Г(t) and module of continuity. One of the main application is to prove module of continuity estimates for l2 -valued Ornstein-Unlenbeck process.The main of this article has been publishied.