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XIV International EM Conference on Eventological Mathematics and Related Fields, Krasnoyarsk, Siberia, Russia, 2010

Composition of Poisson Processes

Enzo Orsingher Federico Polito Dipartimento di Scienze Statistiche Dipartimento di Scienze Statistiche Sapienza University of Rome Sapienza University of Rome Italy Italy [email protected] [email protected]

Abstract. The main topic of this paper is the composi- of the process Nˆ(t)= N1(N2(t)) turns out to be tion of independent Poisson processes of which we study λ(u−1)− the probability law (involving Bell polynomials), the gov- Gˆ(u,t)= eβt(e 1), t> 0, |u|≤ 1, erning equations, and its representation as a random sum. Some compositions of Poisson processes with the we establish the useful representation: inverses of classical and fractional Poisson processes, d and with the inverse of the fractional linear birth process, N1(N2(t)) = X1 + ··· + XN2(t), (2) are examined and their pricipal features analysed. where the Xj are i.i.d. Poisson-distributed random vari- Keywords. Poisson process, Iterated processes, Bell ables with parameter λ. The iterated Poisson process is polynomials, Subordination, Fractional Poisson process, thus, a compound Poisson process with Poisson compo- Fractional Yule–Furry process. nents. The difference-differential equations solved by the state 1 Introduction probabilities is also derived and reads

k m The subordination of processes, first introduced in [1], d −λ λ pˆ (t)= −βpˆ (t)+ βe pˆ − (t). (3) dt k k m! k m is a technique which permits us to introduce in various m=0 stochastic models, further randomness. The so-called X Iterated Brownian motion, for instance, is a process In Section 3 a Poisson process N (t), t> 0, is subordi- which was extensively studied in the past years (see for 1 nated to the inverse process of a second Poisson process example [2]). N2(t) (Erlang process), independent of N1(t), i.e. Recently, the study of subordinated processes has been extended to point processes. It has been proved that N1(τk), k ≥ 0, (4) the one-dimensional distributions of some subordinated point processes satisfy fractional difference-differential where equations [3, 4]. This suggests us to study different kinds τ = inf(t: N (t)= k), k ≥ 0, (5) of subordinated point processes. In particular, in this k 2 paper, we study in details the iterated Poisson process and the explicit form of the state probabilities is given as well as some other subordinated Poisson processes. and discussed. In Section 2, the iterated Poisson process is presented The last two sections are devoted to the study of Pois- and analysed. Let N (t) and N (t), t > 0, be two 1 2 son processes subordinated to inverse processes of the independent homogeneous Poisson processes with rates fractional Poisson and fractional Yule process. λ > 0 and β > 0, respectively. The iterated Posson process is defined as Nˆ(t)= N1(N2(t)), t> 0. The state probabilities pˆk(t) = Pr{Nˆ(t) = k}, k ≥ 0, have the 2 First results on subordinated Poisson form processes

k −λ λ −βt(1−e ) −λ pˆk(t)= e Bk βte , (1) Consider two independent homogeneous Poisson pro- k! cesses, say N1(t), t > 0, with rate λ > 0, and N2(t),
t> 0, with rate β > 0. We are interested in the subor- with k ≥ 0, t > 0, and where B (x) is the nth order k dinated process Nˆ(t)= N (N (t)), t> 0. The subordi- Bell polynomials. 1 2 nation (see [1]) permits us to introduce in the system By considering that the probability generating function further randomness, thus allowing to model phenomena 214 XIV International EM’2010 Conference

which exhibit either a speeded up or a slowed down be- 0.20 haviour. In particular, the type of subordination consid- ` p HtL ered in this section is a little different. Poisson processes 1 ` 0.15 p HtL have a countable state space, therefore the inner Poisson 2 ` process N2(t), t> 0, in addition to randomize the time, p3HtL ` operates a sampling of the external process N1(t), t> 0. 0.10 p4HtL Note that this sampling allows the subordinated process to have jumps of arbitrary positive size. In addition, 0.05 Nˆ(t), t> 0, can be regarded as a Poisson process N1(t), t> 0, running on the sample paths of the independent Poisson process N2(t), t> 0. 2 4 6 t 8 10 12 14 The state probabilities pˆk(t) = Pr{N1(N2(t)) = k}, k ≥ 0, t> 0, can be determined as follows. Figure 1: Graphs of the first four state probabilities for ∞ e−λr(λr)k e−βt(βt)r the Bell process. The parameters here have the values pˆ (t)= · (6) k k! r! λ = 1, β = 1. r=0 X ∞ k −λr k r ∞ ∞ λ −βt e r (βt) −βt r k = e . e (βt) − (λr) k! r! = e λr k r=0 r! k! X r=0 k=0 X∞ X∞ By considering that −βt r k e (βt) −λr (λr) ∞ = e λr k r r! k! −x r x r=0 k=0 Bk(x)= e , (7) X ∞ X r r! − λr(βt) r=0 = e βt X r! r=0 is the nth order Bell polynomial (for a review of Bell X ∞ r polynomials, the reader can consult for example [5]), we − (βt) = e βtλβt obtain that r! r=0 k X λ − − −λ − = λβt, t> 0. pˆ (t)= e βt(1 e )B βte λ , k ≥ 0, t> 0. k k! k
(8) To determine the variance we first calculate the second order moment: The following alternative form of the state probabilities ∞ ∞ e−λr(λr)k e−βt(βt)r is immediately calculated: µ = k2 (12) 2 k! r! k=0 r=0 pˆk(t) (9) X ∞X ∞ r k ∞ k −λr k r −βt (βt) −λr 2 (λr) λ − e r (βt) = e e k . = e βt r! k! k! r! r=0 k=0 r=0 X X X ∞ k Considering that λt −λr r = Pr{N1(t)= k}e e Pr{N2(t)= r}. ∞ ∞ ∞ t k k k+1 r=0 (λr) (λr) (λr) X   k2 = k = (k + 1) k! (k − 1)! k! k k k In Figure 1, we can appreciate the plots of the first four X=0 X=1 X=0 ˆ (13) state probabilities for N(t), t> 0, with λ = 1 and β = 1. ∞ ∞ By recalling that the exponential generating function (λr)k+1 (λr)k+1 = k + for Bell polynomials is k! k! k=1 k=0 ∞ X∞ X k k+2 z z − (λr) λr B (x)= ex(e 1), (10) = + λre k! k k! k=0 k=0 X Xλr ∞ = λre (λr + 1), it is immediate to check that k=0 pˆk(t) = 1. The mean value is directly calculated as follows. we have P ∞ ∞ ∞ −λr k −βt r r e (λr) e (βt) − λr(λr + 1)(βt) E N (N (t)) = k (11) µ = e βt (14) 1 2 k! r! 2 r! k=0 r=0 r=0   X X X Enzo Orsingher, Federico Polito 153

∞ r − (λr + 1)(βt) N1 (t) = e βtλ (r − 1)! r=1 X∞ r+1 − (λr + λ + 1)(βt) 10 = e βtλ r! r=0 X ∞ ∞ r+1 r+1 5 − (βt) (βt) = e βtλ (λ + 1) + λ r r! r! " r=0 r=1 # X ∞ X 0 r+2 1 2 3 4 5 6 7 t −βt 2 (βt) = λ(λ + 1)βt + e λ Time r! r=0 X = λ(λ + 1)βt +(λβt)2. Figure 2: A possible path of the external Poisson process N1(t), t> 0. Therefore, the variance of the process reads k −βt λ −λ Var N1(N2(t)) = λ(λ + 1)βt, t> 0. (15) = − βpˆ (t)+ βe e k k! ∞   k r The probability generating function k! − (βt) × e λrrm ∞ m!(k − m)! r! m=0 r=0 ˆ k ˆ X X G(u,t)= u N(t) (16) k −m − λ k=0 = − βpˆ (t)+ βλke λ pˆ (t) X k (k − m)! m m=0 can be found in the following way: X k m ∞ ∞ −λ λ −λr k −βt r = − βpˆk(t)+ βe pˆk−m(t). k e (λr) e (βt) m! Gˆ(u,t)= u (17) m=0 k! r! X k=0 r=0 X X ∞ k From result (19), the differential equation governing − − −λ λ − = e βt(1 e ) uk B βte λ the probability generating function is easily derived and k! k=0 reads −λ X−λ uλ
−βt(1−e ) βte (e −1) = e e ∂ − Gˆ(u,t)= −βGˆ(u,t)+ βe λGˆ(u,t)eλu. (21) λ(u−1)− = eβt(e 1), t> 0, |u|≤ 1. ∂t ˆ Result (17) implies that With initial condition G(u, 0) = 1 this yields result (17). Remark 1 (A note on paths). It is interesting to de- d scribe the path behaviour of the subordinated process N1(N2(t)) = X1 + ··· + XN2(t), (18) Nˆ(t) = N1(N2(t)), t > 0. In Figure 2, a possible path where the random variables Xj are i.i.d. with Poisson of the external Poisson process N1(t), t> 0, is depicted. distribution with parameter λ. In other words, the Since the internal process has discrete state space, it subordinated process Nˆ(t), t> 0, is a compound Poisson produces a discretisation of the time argument of the process with Poisson components. external process. Furthermore the process N1(N2(t)) ad- mits arbitrarily high jumps. After the appearance of The difference-differential equations solved by the state the first component, the process behaves as a pure birth probabilitiesp ˆ (t), t> 0, is k process with a random number of offsprings. k m d −λ λ pˆ (t)= −βpˆ (t)+ βe pˆ − (t), (19) dt k k m! k m 3 Poisson process subordinated to the m=0 X inverse of a second independent and can be derived with the following steps. Poisson process d pˆ (t) (20) We now consider the inverse process (hitting time) of dt k ∞ the Poisson process N2(t), t> 0: k k r−1 λ −βt −λr r (βt) = − βpˆk(t)+ β e e k! (r − 1)! τk = inf(t: N2(t)= k), k ≥ 0. (22) r=1 X∞ k k r λ − − (r + 1) (βt) = − βpˆ (t)+ β e βt e λ(r+1) The process τk, k ≥ 0, is a discrete time process with con- k k! r! r=0 tinuous state-space. It is a non-decreasing process with X 416 XIV International EM’2010 Conference

N(t) (r ≥ 0, k ≥ 0) which is a negative binomial distribution 13 12 with parameter λ/(λ + β). We can, therefore, immedi- 11 10 ately obtain that 9 8 7 λ 6 EN˜(k)= k, (26) 5 4 β 3 2 1 and 0 t λ(λ + β) VarN˜(k)= k. (27) β2 N2 (t) 7 6 Note that the subordinated process N1(τk), k ≥ 0, can 5 4 be interpreted as follows: 3 2 1 d 0 N1(τk) = X1 + ··· + XN , (28) t1 t2 t3 t 4 t 5 t 6 t 7 t 8 Time t where N is a Poisson random variable with parameter Figure 3: Construction of the path of the subordinated equal to process Nˆ(t)= N1(N2(t)), t> 0, starting from the path k of the internal process N2(t), t> 0. λ + β µ = log , (29) β   scattered jumps. In practice it can also be considered and the Xjs are i.i.d. random variables with logarithmic as the waiting time until the kth event: distribution with parameter α = λ/(λ + β). Pr(τ ≤ t) = Pr(N (t) ≥ k) (23) Remark 2 (A note on paths). The subordinated process k 2 ˜ ∞ N(k), has increasing paths with jumps. The jumps are e−βt(βt)j = of positive integer-valued size. It can thus possibly used j! j k to model branching phenomena in discrete-time, when X= k− an arbitrary number of offsprings is permitted. 1 e−βt(βt)j = 1 − . j! j=0 4 Poisson and Yule–Furry processes X subordinated to the inverse of an Clearly τ , k ≥ 1, is Erlang-distributed: k independent fractional Poisson k−1 β(βt) − Pr(τ ∈ dt)/dt = e βtdt. (24) process k (k − 1)! In this section we introduce the inverse process of a Our aim is to study the subordinated process N˜(k)= fractional Poisson process independent of N1(t), t> 0. N1(τk), k ≥ 0. For in-depth information on the latter we refer to [6] or, more recently [3]. In the following, the fractional Pr(N (τ )= r) (25) 1 k Poisson process will be indicated as N ν (t), t> 0, where ∞ −λs r k−1 e (λs) β(βs) − = e βsds ν ∈ (0, 1] is the index of fractionality. r! (k − 1)! Z0 The inverse process is r k ∞ λ β −(λ+β)s r+k−1 = e s ds ν ν r!(k − 1)! τk = inf(t: N (t)= k), k ≥ 0. (30) Z0 r k ∞ λ β −y r+k−1 1 ν = e y dy As in the classical case we can write that Pr(τk ≤ t)= r!(k − 1)! (λ + β)r+k ν Z0 Pr(N (t) ≥ k). From [7, formula (2.19)], the distribu- λrβkΓ(r + k) tion is = r+k r!(k − 1)!(λ + β) ν k νk−1 k ν Pr(τk ∈ dt)/dt = β t Eν,νk(−βt ), (31) (r + k − 1)! λrβk = r!(k − 1)! (λ + β)r+k where β > 0 is the rate of the fractional Poisson process and the function r + k − 1 λrβk = ∞ r (λ + β)r+k (δ) zr   Eδ (z)= r , α,γ,δ ∈ C, R(α) > 0, r k α,γ Γ(αr + γ)r! r + k − 1 λ λ r=0 = 1 − , X r λ + β λ + β (32)      Enzo Orsingher, Federico Polito 175

r is the so-called generalised Mittag–Leffler function [8, r − 1 − = βk (−1)j 1 page 91]. The distribution of the subordinated process j − 1 ˜ ν ν j=1   N (k)= N1(τk ) reads X∞ −λjs νk−1 k ν × e s Eν,νk(−βs )ds Pr N˜ ν (k)= r (33) 0 Zr ∞ −λs r k r − 1 j−1 1  e (λs) − = β (−1) . = βksνk 1Ek (−βsν )ds j − 1 ν k ν,νk j=1 [(λj) + β] 0 r! X   Z r k ∞ λ β νk+r−1 −λs k ν In the last step we exploited relation (2.3.24) of [8]. = s e Eν,νk(−βs )ds r! 0 Z− − λrβk λ νk r −β (k, 1), (νk + r, ν) = ψ , 5 Poisson process subordinated to the r! (k − 1)! 2 1 λν (νk,ν)   inverse of an independent fractional

Yule process where the function ∞ p m ν (ap,Ap) j=1 Γ(aj + mAj) z Let Y (t), t> 0, be a fractional Yule process with rate pψq z = q , (bq,Bq) Γ(b + mB ) m! β > 0 [4]. The probability density function of the kth   m=0 Qj=1 j j X birth W ν , is [9]: (34) k Q ν is the generalised Wright function. Notice that, in the fφk (t) (38) last step of (33), we used relation (2.3.23) of [8]. The k m m − 1 l−1 ν−1 ν distribution (33) can be further simplified by writing = (−1) βlt Eν,ν (−βlt ), l − 1 m=1 l explicitly the Wright function. X X=1   ∞ 1 k + m − 1 where t > 0 and ν ∈ (0, 1]. It is also the prob- Pr N˜ ν (k)= r = (−1)m (35) ability density function of the right-inverse process r! m m=0 ν ν     φk = inf(t: Y (t) = k), k ≥ 1. In order to derive the X k+m ν β Γ(ν(k + m)+ r) distribution of the subordinated process N1(φk), k ≥ 1, × . λν Γ(ν(k + m)) we proceed as in the preceding section.   ν From (35), when ν = 1, we retrieve formula (25): Pr(N1(φk)= r) (39) ∞ k m e−λs(λs)r m − 1 ˜ 1 = Pr N (k)= r (36) r! l − 1 Z0 m=1 l=1   ∞  k+m X X β (k + m + r − 1)! l−1 ν−1 ν = (−1)m × (−1) βls Eν,ν (−βls )ds λ r!m!(k − 1)! m=0   r k m X λ β m − 1 l− ∞ k+m = l(−1) 1 k + r − 1 β m k + m + r − 1 r! l − 1 = (−1) m=1 l=1   r λ m X∞ X m=0       −λs ν r− ν X ∞ × e s + 1E (−βls )ds k + r − 1 β k β m −(k + r) ν,ν = Z0 r λ λ m k m    m=0     1 m − 1 l X = (−1) k −(k+r) r! l − 1 k + r − 1 β β m=1 l = 1+ X X=1   r λ λ ∞ n+1      βl Γ(ν(n + 1) + r) r k × − . k + r − 1 λ λ λν Γ(ν(n + 1)) = 1 − . n=0   r λ + β λ + β X      References Following the above calculation, it is also possible to determine the distribution of a Yule–Furry process, in- [1] S. Bochner. Harmonic Analysis and the Theory of Prob- dicated here as Y (t), t > 0, with birth rate λ > 0, ability. University of California Press, Berkeley, 1955. ν subordinated to τk , k ≥ 0, as follows. [2] R. D DeBlassie. Iterated Brownian motion in a an open set. Ann. Appl. Probab., 14(3):1529–1558, 2004. ν Pr(Y (τk )= r) (37) [3] L. Beghin and E. Orsingher. Fractional Poisson processes ∞ − −λs −λs r 1 k νk−1 k ν and related planar random motions. Electron. J. Probab., = e 1 − e β s Eν,νk(−βs )ds 14(61):1790–1826, 2009. Z0
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