extbf {Last Time} extbf {Markov Property of Poisson Process} extbf {Compensated Poisson Process} extbf {Law of Large Numbers for Poisson Process} extbf {Poisson Approximation} extbf {Conditional Distribution of Arrival Times} extbf {Thinning a Poisson Process} extbf {Superposition of Poisson Processes} extbf {Nonhomogeneous Poisson Process} extbf {Compound Poisson Process} extbf {Markov Jump Process}Last Time•Counting process•Poisson process•Equivalent definitionsToday’s lecture: Sections 6.2, 6.3MATH136/STAT219 Lecture 26, December 3, 2008 – p. 1/11Markov Property of Poisson Process•Let {N(t), t ≥ 0} be a Poisson process with rate λ•Then {N(t), t ≥ 0} is a homogenous Markov process with•State space: S = {0, 1, 2, . . .}•Initial distribution: π({0}) = 1•Stationary transition probability function:pt(n + k|n) =e−λt(λt)kk!, t ≥ 0, n, k = 0, 1, 2, . . .•A Poisson process is a strong Markov processMATH136/STAT219 Lecture 26, December 3, 2008 – p. 2/11Compensated Poisson Process•Let {N(t), t ≥ 0} be a Poisson process with rate λ•Let {Ft, t ≥ 0} be the canonical filtration of N•Define M(t) = N(t) − λt•The process {M(t), t ≥ 0} is called a compensated Poissonprocess•{M(t), Ft} is a martingale•{M2(t) − λt, Ft} is a martingaleMATH136/STAT219 Lecture 26, December 3, 2008 – p. 3/11Law of Large Numbers for Poisson Process•Let {N(t), t ≥ 0} be a Poisson process with rate λ > 0•Then as t → ∞N(t)t→ λ almost surelyMATH136/STAT219 Lecture 26, December 3, 2008 – p. 4/11Poisson ApproximationSuppose that for each n, Z(n)kare independent, nonnegativeinteger valued random variables that satisfy:•IP (Z(n)k= 1) = p(n)k,•IP (Z(n)k≥ 2) = ǫ(n)k,•Pnk=1p(n)k→ λ as n → ∞ for some λ ∈ (0, ∞),•Pnk=1ǫ(n)k→ 0 as n → ∞,•maxk=1,...,n{p(n)k} → 0 as n → ∞Then Sn=Pnk=1Z(n)kconverges in distribution as n → ∞ to aPoisson(λ) random variableMATH136/STAT219 Lecture 26, December 3, 2008 – p. 5/11Conditional Distribution of Arrival Times•Let {N(t)} be a Possion process with arrival times T1, T2, . . .•Conditional on the event {N(t) = n}, the first n arrival times{Tk: k = 1, . . . , n} have the joint probability density:fT1,...,Tn|N(t)=n(t1, . . . , tn) =n!tn, for 0 < t1< t2< · · · < tn≤ t•That is, conditional on the event {N (t) = n}, the first narrival times {Tk: k = 1, . . . , n} have the same distributionas the order statistics of a sample of n i.i.d. Uniform([0, t])random variables•Corollary: if s < t and 0 ≤ m ≤ n thenIP (N(s) = m|N(t) = n) =nmstm1 −stn−mMATH136/STAT219 Lecture 26, December 3, 2008 – p. 6/11Thinning a Poisson Process•Let {N(t)} be a Poisson process with rate λ•Let Ykbe a sequence of i.i.d random variables taking valuesin {1, 2, . . .} that are independent of {N(t)}•For j = 1, 2, . . . let {Nj(t)} be the counting process thatcounts only the events for which Yk= j•Then {Nj(t)} is a Poisson process with rate λIP (Yk= j)and the processes {Nj(t)} are independent of each otherMATH136/STAT219 Lecture 26, December 3, 2008 – p. 7/11Superposition of Poisson Processes•Let {Nj(t)}, j = 1, 2, . . . be independent Poisson processeswith rates λj, respectively•For each t ≥ 0 letN(t) =∞Xj=1Nj(t)•Then {N(t)} is a Poisson process with rate λ =P∞j=1λjMATH136/STAT219 Lecture 26, December 3, 2008 – p. 8/11Nonhomogeneous Poisson Process•For each time u ≥ 0, let λ(u) denote the instantaneousarrival rate of some process at that time•DefineΛ(t) =Zt0λ(u)du•A counting process {X(t)} is called a nonhomogeneousPoisson processif◦{X(t)} has independent increments◦For all t > s ≥ 0, X(t) − X(s) has a Poisson distributionwith mean Λ(t) − Λ(s)•If {N(t), t ≥ 0} is a Poisson process with rate 1, then {X(t)}is given by the time-changed processX(t) = N(Λ(t))MATH136/STAT219 Lecture 26, December 3, 2008 – p. 9/11Compound Poisson Process•Let {N(t)} be a Poisson process with rate λ•Let Ykbe a sequence of i.i.d random variables with finitevariance that are independent of {N(t)}•For t ≥ 0 letS(t) =N(t)Xi=1Yi•Then {S(t)} is called a compound Poisson process•{S(t)} has stationary and independent increments and is ahomogeneous Markov processMATH136/STAT219 Lecture 26, December 3, 2008 – p. 10/11Markov Jump Process•Let {N(t)} be a Poisson process with rate λ and jump timesT1, T2, . . .•Let X1, X2, . . . be a sequence of random variables withZn=Pnk=1Xk•Assume that◦{Zn, n = 0, 1, 2, . . .} is a discrete time Markov chain◦{Zn} is independent of {Tn}•For t ≥ 0 letX(t) =N(t)Xj=1Xj•Then {X(t)} is called a jump Markov processMATH136/STAT219 Lecture 26, December 3, 2008 – p.