Compound and Non-Stationary Poisson Processes

Lecture 05: Compound and Non-Stationary Poisson Processes

1 Compound Poisson process

Acompound Poisson processis a real-valued point process{Z_t,t≥0}having the following properties.

1. finite jumps: for allω∈Ω,t7−→Z_t(ω)has finitely many jumps in finite intervals.

2. independent increments: for allt,s≥0;Zt+s−Z_tis independent of past{Z_u,u≤t}.

3. stationary increments: for allt,s≥0, distribution ofZ_t+s−Z_tdepends only onsand not ont.

For eachω∈Ω, we can define time and size ofnth jump

S₀(ω) =0 S_n(ω) =inf{t>0 :Z_t(ω)>Z_Sn−1(ω)}, n∈N, X0(ω) =0 X_n(ω) =Z_n(ω)−Zn−1(ω).

LetNt,t>0 be the counting process associated with the number of jumps in[0,t). Then,Snare the arrival instants ofnth jumps.

Proposition 1.1. A stochastic process{Z_t,t>0}is a compound Poisson process iff its jump times form a Poisson process and the jump sizes form an iid random sequence independent of the jump times.

Proof. From independent increment property of compound Poisson processes, it follows thatZt+s−Z_t= 0 is independent of the pastZu,u6t. Further, it follows from the stationary increment property that the distribution ofZt+s−Zt=0 is independent oft. It follows thatNt is a Poisson process. Similarly, it follows thatX₁,X₂, . . .areiidrandom variables, independent ofS₁,S₂, . . ..

Conversely, letZ_t=∑^N_i=1^t X_i whereN_t is a Poisson process independent of the randomiidsequence

X₁,X₂, . . .. It is easy to check thatZ_t has finitely many jumps in finite intervals. Further, one can show

independent and stationary increment properties.

• Arrival of customers in a store is a Poison processN_t. Each customerispends aniidamount X_iindependent of the arrival process. Amount of money spentY_nby firstncustomers is

Y₀=0, Y_n=

n

∑

i=1

X_i,i∈[n].

Now defineZ_t=Y_Nt as the amount spent by the customers arriving in timet. Then{Z_t,t≥0}

is a compound Poisson Process.

• Let the time between successive failures of a machine be independent and exponentially dis- tributed. The cost of repair isiid random at each failure. Then the total cost of repair in a certain timetis a compound Poisson Process.

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2 Non-stationary Poisson process

From the characterization of Poisson process just stated, we can generalize to non-homogeneous Poisson Process. In this case, the rate of Poisson Processλis time varying.

An integer valued counting process {N(t), t >0} is said to be possibly non-stationary Poisson processif it has unit jumps and independent increments. That is,

1. for eachω∈Ω, the mapt7→Nt(ω)has jumps of unit size only,

2. for anyt,s≥0, the random variableNt+s−N_tis independent of the past{N_u,u6t}.

Letm(t) =EN_t for allt>0. From non-decreasing property of counting processes, it follows that the mean is also non-decreasing in timet. From right continuity of counting process and the monotone convergence theorem, it follows that mean function is also right continuous. Thetime inverseof mean is defined as

τ(t) =inf{s>0 :m(s)>t},t>0.

Since, inverse of a non-decreasing function is also non-decreasing, we conclude thatτ(t)is non-decreasing function of timet.

Theorem 2.1. Let Ntbe a non-stationary Poisson process, such that m(t) =ENtis continuous. Then, Mt(ω),N_τ(t)(ω), t>0,ω∈Ω,

is a stationary Poisson process with unit rate.

Proof. Fixt>s≥0 and lets⁰,τ(s)andt⁰,τ(t)−τ(s). Then, by definition ofMt,t⁰,s⁰and independent increment property of non-stationary Poisson processN_t, we have

E[M_t−Ms|M_u;u6s] =E[N_t0−N_s0|N_u;u6s⁰] =m(t⁰)−m(s⁰) =m(τ(t))−m(τ(s)) =t−s.

It follows that M_t is a simple counting process with independent and stationary increments and unit rate.

Corollary 2.2. Let m(t)be a continuous non-decreasing function. Then, S1,S2, . . .are the arrival instants in a non-stationary Poisson process N_twith mean function m(t) =EN_tiff m(S₁),m(S₂), . . .are the arrivals instants of a stationary Poisson process of unit rate.

Proof. We can write thenth arrival instantS⁰_nof unit-rate stationary Poisson processM_t, in terms of the nth arrival instantS_nof non-stationary Poisson processN_t as

S⁰_n=inf{t>0 :τ(t)>S_n}=inf{t>0 :m(S_n)>t}=m(S_n).

This corollary implies thatS_n∈[s,t)if and only ifm(S_n)∈[m(s),m(t)). Therefore, number of arrivals in[s,t)equals number of arrivals for unit-rate stationary Poisson process in [m(s),m(t)). Hence, we conclude that forb(s,t) =m(t)−m(s)

Pr{N_t−N_s=k}=e^−b(s,t)b(s,t)^k

k! , k∈N0.

We will see that the inter-arrival times for the non-stationary Poisson processN_t, defined as T₀=0, T_n=S_n−Sn−1, n∈N,

are not independent anymore.

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Proposition 2.3. For a non-stationary Poisson process with continuous mean function m(t), we have Pr{T_n+1>t|S₁,S₂, . . . ,S_n}=exp(−m(S_n+t) +m(S_n)).

Proof. We define eventsA={m(S_n+1)>m(S_n+t)} andB={m(S_n+1)≥m(S_n+t)}. Then, we have A⊆ {T_n+1>t} ⊆B. Hence, we can write

Pr{A|S₁,S₂, . . . ,S_n} ≤Pr{T_n+1>t|S₁,S₂, . . . ,S_n}.

The arrival instantsS₁, . . . ,S_ndeterminem(S₁), . . . ,m(S_n),m(S_n+t). Further, sincem(S_n+1)−m(S_n)is the inter-arrival time of the stationary Poisson processM(t), it is independent ofS₁,S₂, . . . ,S_n

Pr{A|S1, . . . ,S_n}=Pr{m(Sn+1)−m(S_n)>m(S_n+t)−m(S_n)|S1, . . . ,S_n}=exp(−m(S_n+t) +m(S_n)).

Result follows from the continuity of the exponential distribution.

A Laplace functional

The Laplace functionalLof a point processΦ and associated counting process N is defined for all non-negative function f:R^d→Ras

LΦ(f) =Eexp

− Z

R^d

f(x)dN(x)

.

For simple function f(x) =∑^k_i=1t_i1{x∈A_i}, we can write the Laplace functional LΦ(f) =Eexp(−

∑

i

t_iN(A_i)),

as a function of the vector(t₁,t₂, . . . ,t_k), a joint Laplace transform of the random vector(N(A₁), . . . ,N(A_k)).

This way, one can compute all finite dimensional distribution of the counting processN.

Proposition A.1. The Laplace functional of the Poisson process with intensity measureΛis

LΦ(f) =exp

− Z

R^d

(1−e^−f(x))Λ(dx)

.

Proof. For a bounded Borel measurable setA⊆R^d, considerg(x) = f(x)1{x∈A}. Then, LΦ(g) =Eexp(−

Z

R^d

g(x)dN(x)) =Eexp(−

Z

A

f(x)dN(x)).

ClearlydN(x) =δx1{x∈Φ}and hence we can write

LΦ(g) =Eexp −

∑

S_i∈Φ∩A Z

A

f(S_i)

! .

We know that the probability ofN(A) =|Φ(A)|=npoints in setAis given by

P{N(A) =n}=e^−Λ(A)Λ(A)ⁿ n! .

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Given there arenpoints in setA, the density ofnpoint locations are independent and given by

f_S1,...,Sn(x₁, . . . ,x_n) =

Λ(dx) Λ(A)

n

, x₁, . . . ,x_n∈A.

Hence, we can write the Laplace functional as LΦ(g) =e^−Λ(A)

∑

n∈N0

Λ(A)ⁿ n!

n

∏

i=1 Z

A

e^−f(xi)Λ(dx_i) Λ(A) =exp

− Z

R^d

(1−e^−g(x))Λ(dx)

.

Result follows from taking increasing sequences of setsA_k↑R^dand monotone convergence theorem.

A.1 Superposition of point processes

Theorem A.2. The superposition of independent Poisson point processes with intensitiesΛ_kis a Poisson point process with intensity measure∑_kΛ_kif and only if the latter is a locally finite measure.

A.2 Thinning of point processes

Consider a probabilityretention functionp:R^d→[0,1]and a point processΦ. Thethinningof point processΦ={S_n∈R^d:n∈N}with the retention functionpis a point process such that

Φ^p={S_n∈Φ:Y(S_n) =1},

whereY(S_n)is an independent indicator stochastic process at each pointS_nandEY(S_n) =p(S_n).

Theorem A.3. The thinning of the Poisson point process of intensity measureΛwith the retention prob- ability function p yields a Poisson point process of intensity measure pΛwith

(pΛ)(A) = Z

A

p(x)Λ(dx)

for all bounded Borel measurable A⊆R^d.

Proof. LetA⊆R^dbe a bounded Boreal measurable set, and let f:R^d→Rbe a non-negative function.

Consider the Laplace functional of the thinned point process Φ^p for a non-negative functiong(x) = f(x)1{x∈A}

LΦ^p(g) =e^−Λ(A)

∑

n∈N0

1 n!

Z

A

(p(x)e^−f(x)+ (1−p(x))Λ(dx) n

=exp

− Z

R^d

(1−e^−g(x))p(x)Λ(dx)

.

Result follows from taking increasing sequences of setsA_k↑R^dand monotone convergence theorem.

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