20. Functional Limit Theorems

The chapter begins with Sect. 20.1 presenting the classical Functional Central Limit Theorem in the triangular array scheme. It establishes not only convergence of the distributions of the scaled trajectories of random walks to that of the Wiener process, but also convergence rates for Lipshchitz sets and distribution functions of Lipshchitz functionals in the case of finite third moments when the Lyapunov condition is met. Section 20.2 uses the Law of the Iterated Logarithm for the Wiener process to establish such a low for the trajectory of a random walk with independent non-identically distributed jumps. Section 20.3 is devoted to proving convergence to the Poisson process of the processes of cumulative sums of independent random indicators with low success probabilities and also that of the so-called thinning renewal processes.