Functional limit results in probability theory




Scott, David John

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Various methods are used to obtain functional limit results. Interest is centred on the methods as well as the results themselves. The starting point of the thesis is the monograph "Convergence of Probability Measures" by Patrick Billingsley« Chapter 1 is an introduction« In Chapter 2 we extend known functional limit results to obtain limit results for functionals arising, in particular, in queueing theory. Attention is also given in this chapter to the possible uses of functional results to obtain various distribution convergence results. Chapter 3 is concerned with reversed martingales. The main result is a funcational central limit theorem for reversed martingales obtained using the standard method of first showing convergence of finite-dimensional distributions and then tightness. In Chapter U we depart from methods considered by Billingsley. The main results of this chapter are a functional central limit theorem for martingales and a functional central limit theorem for triangular arrays in which each row is a martingale sequence. The proof of these results is based on the use of the Skorokhod representation. The results of Chapter U are then used in Chapter 5 to obtain two functional central limit theorems for processes with stationary ergodic increments. The results in Chapters 2,3,H and 5 are all weak limit results, i.e. convergence in distribution. In Chapter 6 we use the Skorokhod representation more fully to obtain strong functional limit results; three functional laws of the iterated logarithm for martingales.






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