Μελέτη της εξάρτησης τυχαίων μεταβλητών στην ανανεωτική θεωρία, με εφαρμογές σε τυχαίους περιπάτους
Study of the dependence among random variables in renewal theory, with applications in random walks
Λοσίδης, Σωτήριος Ι.
KeywordsΑνανεωτική θεωρία ; Ανανεωτικές συναρτήσεις ; Κατανομές ; Τυχαίες μεταβλητές ; Ασυμπτωτική συνδιακύμανση ; Διακύμανση υπολειπόμενης ζωής ; Απόδοση φραγμάτων
Chapter 1 makes reference to certain results in renewal theory herein used as primary tools for completion of this thesis. Studying the renewal function and creating bounds for this quantity are in the core of renewal theory studies. Improvements of these bounds are discussed in Chapter 2. The number of renewals occurring within a period (t, t + x) is a quantity of special interest in renewal theory. The bounds presented for this quantity in Chapter 2 serve to improve the already existing ones. Chapter 9 studies these same quantities in a more generic application context, namely for random walks. As the number of renewals occurring within a period (t, t+x) is linked to the right tail of the remaining life, it makes sense to study the right tail of the remaining life as well as the joint right tail of the current and the remaining life. We use right tails to study the higher moments for those times studied by renewal theory, which we in turn discuss in Chapter 7. The formulae given in this chapter enable us to calculate the moments of the recurrence times in any class and to create bounds under specific distribution classes. These results are then used in the context of random walks as discussed in Chapter 8. Studying the right tails for the aforementioned quantities helps study the covariance between the current and the remaining life as per chapters 3,4 and 5 of this thesis. Both the sign and the monotonicity of the covariance are linked to specific distribution classes and are relation- dependent for both finite and infinite times.