Central Limit Theorem , long time

The first result here is known as the elementary renewal theorem. Simply put, it says that if the average time between arrivals is a half-hour, then on average there wiU be two arrivals per hour (A = 2) in the long run. To enhance the results (2.18), we may invoke an extension of the central limit theorem of Subsection 1.3, stating that N f) is asymptotically normally distributed with mean At and variance XcH for large times t. This fact is useM in understanding queues in heavy traffic see Subsection 5.2. 

According to the central limit theorem for renewals, is asymptotically N(f//r, normal. In the case of long enough counting times and constant dead time, therefore, the expected value and the variance of the counts are... 

To this date, no stable simulation methods are known which are successful at obtaining quantum dynamical properties of arbitrary many-particle systems over long times. However, significant progress has been made recently in the special case where a low-dimensional nonlinear system is coupled to a dissipative bath of harmonic oscillators. The system-bath model can often provide a realistic description of the effects of common condensed phase environments on the observable dynamics of the microscopic system of interest. A typical example is that of an impurity in a crystalline solid, where the harmonic bath arises naturally from the small-amplitude lattice vibrations. The harmonic picture is often relevant even in situations where the motion of individual solvent atoms is very anhaimonic in such cases validity of the linear response approximation can lead to Gaussian behavior of appropriate effective modes by virtue of the central limit theorem. ... 

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