RenewalTheory

# RenewalTheory - STAT 333 Renewal Theory Basics Let be a...

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STAT 333 Renewal Theory Basics Let λ be a possible event of a stochastic process X 1 ,X 2 3 ,... . Let T λ = waiting time until λ Frst occurs in the sequence, i.e. T λ = n means λ Frst occurs on trial n , while T λ = means that λ never occurs. Let T ( k,k +1) λ = waiting time between the k th and ( k +1)th occurrences of λ , conditional on λ having occurred k times. λ is called a renewal event if T λ ,T (1 , 2) λ (2 , 3) λ ,...,T ( k,k +1) λ are independent and identically-distributed (i.i.d.) random variables. λ is called a delayed renewal event if it satisFes the deFnition of a renewal event with one exception: the distribution of the Frst waiting time T λ is diﬀerent from that of the between waiting times T ( k,k +1) λ . Let λ be a renewal event and let f λ = P ( T λ < ). Then λ is called recurrent if f λ = 1 and transient if f λ < 1. Thus a recurrent event must occur at least once in the sequence, whereas a transient event has positive probability of never occurring. Equivalently, λ is recurrent if T λ is a proper waiting time variable, and is transient if T λ is an improper waiting time variable. A recurrent event λ is called positive recurrent if E ( T λ ) < (short proper waiting time) and null recurrent if E ( T λ )= (null or long proper waiting time). Let V λ = the total number of occurrences (visits) of a renewal event λ in the sequence. V λ is a counting variable with potential range { 0 , 1 , 2 }∪{∞} . Then we have the following important theorem: 1. the probability mass function of V λ is given by

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RenewalTheory - STAT 333 Renewal Theory Basics Let be a...

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