Birkhoff's pointwise ergodic theorem with the shift operator over [0,1]11yields a new practical method to compute expectations of functionals of stochastic process. Indeedconverges toasNconverges toward infinity, almost surely. By numerical simulations we will explain the efficiency of this method especially when compared to the classic Monte-Carlo one. It will furthermore be proven that under suitable assumptions a. central limit theorem holds. These assumptions are satisfied in most encountered practical problems. It will precisely be fulfilled whenwith a stopping timeThaving a moment of orderp, p > 2.Moreover, under this assumptions a “weak” law of iterated logarithm applies. Such that:Numerical simulations were processed.