Μελέτη προδρομικών και αναδρομικών χρόνων εμφάνισης σε μία ανανεωτική ανέλιξη
Study of the backward and forward recurrence times in a renewal function
Δούκισσας, Λεωνίδας Α.
SubjectPoisson distribution ; Poisson processes ; Στοχαστικές ανελίξεις -- Μαθηματικά υποδείγματα ; Συναρτήσεις
To this study, we focus on the forward and backward recurrence time of a renewal process. Primarily, we find the renewal function that satisfies the density of forward and backward time and we also calculate the joint tale of the distribution of these two variables. Additionally, we study the covariance function between forward and backward recurrence time. Through the joint density function of the forward and backward recurrence time, we calculate a close type in the time depended state and a simple type for the steady state. Furthermore, we calculate the correlation function of the forward and backward recurrence time for the steady state. Finally, we examine a statistical problem which concerns the estimation of the density function for the forward recurrence time. Suppose, we have a random sample of interarrival times, that their common distribution is unknown, we find a sample estimator for the estimated function. With the assistance of the theorem Glivenko-Cantelli, we prove that this sample estimator, that is made of the empirical distribution function, it is consistent estimator. Specifically, we prove that the estimator converge uniformly to our estimated function, firstly in every interval of the type [a,b] and finally in all [0, ~).