Each assignment is independent. Conversely, if γM∈ℝd, AM is a symmetric non-negative definite d×d matrix, and νM is a Lèvy measure, then there exists a Lèvy process M, unique up to identity in law, such that (1) holds. , { And according to this model, the process is defined as follows. Consequently, the combined process will be a compound Poisson process with Poisson parameter λ1+λ2, and with distribution function F given by, Gérard Ben Arous, Jiří Černý, in Les Houches, 2006. E To check the convergence on the space of cadlag path D endowed with Skorokhod topology, it is necessary check two facts: (a) the convergence of finite-dimensional distributions, and (b) tightness. ( Characteristic functions. This random variable is called the waiting time random variable. {\displaystyle \{\,N(t):t\geq 0\,\}} p ∞ The compound Poisson process model [5-7] provides a closer conceptual parallel, by incorporating a two-level counting process. Hence the conditional distribution of Y given that N = 0 is a degenerate distribution. Examples of subordinators include compound Poisson processes with jump distribution concentrated on (0,∞), the Gamma process, and the inverse Gaussian process. ∑ {\displaystyle \{\,D_{i}:i\geq 1\,\}} = More specifically, Then the walker’s position at time t − n is given by, where ΔX0=X0=0,Tn=j0+j1+⋯+jn. Technometrics, 18(1), 67-73. λ = , i [3] We define that any discrete random variable 1 $\begingroup$ I'm trying to prove an equality about the generating function of a compoung Poisson process and I don't know how to continue it. ) ∞ [1] And compound Poisson distributions is infinitely divisible by the definition. P = { One of the uses of the representation (5.26) is that it enables us to conclude that as t grows large, the distribution of X(t) converges to the normal distribution. Sheldon M. Ross, in Introduction to Probability Models (Tenth Edition), 2010, A stochastic process {X(t), t ≥ 0} is said to be a compound Poisson process if it can be represented as. there is no t ∈ ℝ such that V(t) ∈ [a, b]) is equal to. We say that the discrete random variable In the simplest cases, the result can be either a continuous or a discrete distribution. ∞ Combine them into a single process by taking the union of both sets of events, or equivalently N(t) = N1(t) +N2(t). with. increments . = A new method for estimating the expected discounted penalty function by Fourier-cosine … If the number of people in each family is independent and takes on the values 1, 2, 3, 4 with respective probabilities 16,13,13,16, then what is the expected value and variance of the number of individuals migrating to this area during a fixed five-week period? Y ≥ k For the inverse Gaussian process, the distribution of Mt has Lebesgue density x↦(2πx3)−1∕2ate−12(a2t2x−1−2abt+b2x). X D Let Vn be subordinators with Lévy measures μn. ) Superposition of independent Poisson processes. Such a system will alternate between idle periods when there are no customers in the system, so the server is idle, and busy periods when there are customers in the system, so the server is busy. , λ k {\displaystyle \mu ,\sigma ^{2},p} , ( ■. } {\displaystyle X\sim {\operatorname {DCP} }(\lambda {\alpha _{1}},\ldots ,\lambda {\alpha _{r}})} , then this compound Poisson distribution is named discrete compound Poisson distribution[2][3][4] (or stuttering-Poisson distribution[5]) . We need to deduce convergence of subordinators from the convergence of Lévy measures. The constant d corresponds to the deterministic constant drift. It can be shown, using the random sum of random variable method used in Ibe (2005), that the characteristic function of the compound Poisson process is given by. This triplet determines the characteristic function of Mt via the Lèvy–Khintchine formula. α ( After waiting time jt2, the walker changes position and jumps by an amount equal to ΔXt1, and so on. ∑ 3 Although I do agree with most of zhoraster's answer, I wish to make a few points, as complements at least. ) Let V be an α-stable subordinator and let T(x) = inf{t : V(t) > x}. A Statistical Path 18,521 views. Moment generating function of a compound Poisson process. The measure μ is called the Lévy measure of the subordinator V. There are two important families of subordinators. The Poisson process N λ t represents a particular case of random walk, specified by Poisson-distributed i.i.d. = This case arises in modeling a queueing system with waiting space limited to n; so arrivals that occur when the waiting space is full are not permitted and are lost to the system. If the jumps and waiting time are independent, then the solution for integral equation (7.1) exists.4, J. MEDHI, in Stochastic Models in Queueing Theory (Second Edition), 2003. It is shown that the probabilities of ruin (by oscillation or by a claim) satisfy certain defective renewal equations. Second, this same formula makes sense with $\sigma=\delta_0$ (then $\mu=\delta_0$). Section 6 we fit a compound Poisson process to the bivariate Danish fire insurance data, and present some conclusions in Section 7. London: Griffin. ( ≥ By continuing you agree to the use of cookies. The Gamma process with parameters c,λ>0 is the Lèvy process with characteristic triplet (0,νM,∫01c e−λxdx) and Lèvy measure νM given by νM(dx)=cx−1 e−λx1(0,∞)(x)dx. α r { . {\displaystyle r=1,2} , †Poisson process <9.1> Definition. Active 2 years, 7 months ago. ( All processes appearing in these notes have no drift, therefore we suppose always d ≡ 0. λ , The waiting time random variables, (ji)i = 0∞, are independent and identically distributed variables (mutually independent). {\displaystyle \alpha _{k}} 0 ≥ The random variable Xtn is called the total displacement of the walker at time tn and it is referred to as the jump random variable, and N (t) is the random number of jumps defined as follows. where Xt− denotes the left limit at t>0 with the convention that X0−:=0. 1 Thus, the compound Poisson random process has the infinite divisibility property. Observing that π−1 sin(απ) = (Γ(α)Γ(1 – α))−1 yields the claim of the proposition. I think I recall Grimmett & Stirzaker mentioning the result; in Williams entry-level text it is an exercise and so on. This is easy to verify using the weak convergence of μn and the validity of (A.1) for μ. , There are several situations where such generalizations of Poisson process may be realistic. are non-negative, it is the discrete pseudo compound Poisson distribution. } The following assumptions are made about the ‘Process’ N(t). The parameter λ in the classical Poisson process is assumed to be a constant, independent of time. A compound Poisson process is a continuous-time (random) stochastic process with jumps. In fact, they have stationary and independent increments, and their distributions are an infinite divisible distribution.2, Equation (7.1) is an integral equation. ( An alternative approach is via cumulant generating functions: Via the law of total cumulance it can be shown that, if the mean of the Poisson distribution λ = 1, the cumulants of Y are the same as the moments of X1. is a Poisson process with rate … Solution: Since λ=2,E[Yi]=5/2,E[Yi2]=43/6, we see that, Another useful result is that if {X(t),t⩾0} and {Y(t),t⩾0} are independent compound Poisson processes with respective Poisson parameters and distributions λ1,F1 and λ2,F2, then {X(t)+Y(t),t⩾0} is also a compound Poisson process. ∞ The Poisson process is one of the most widely-used counting processes. k Here, is a Poisson process with rate, and are independent and identically distributed random variables, with distribution function G, which are also independent of σ i We use Aslα to denote its distribution function, Note that Aslα(z) = π−1 sin(απ)B(z; α, 1 – α) where B(z; a, b) is the incomplete Beta function. $\begingroup$ A brief comment, I'll get back to the entire question later. > , The discrete compound Poisson distribution is also widely used in actuarial science for modelling the distribution of the total claim amount. 0 Then Vn converge to V weakly in the Skorokhod topology on D = D([0, T), ℝ) for all final instants T > 0. α 2 {\displaystyle \lambda >0} k , DCP becomes Poisson distribution and Hermite distribution, respectively. Its PGF is given by, where P(s) is the PGF of Xi, and G(s) is the PGF of N(t). , If Yi≡1, then X(t)=N(t), and so we have the usual Poisson process. 1 Therefore, these two processes belong to the class of Lévy processes. For more special case of DCP, see the reviews paper[7] and references therein. {\displaystyle \lambda } For every 0 ≤ y ≤ x < z, we can write, (For a proof of this intuitively obvious claim see p. 76 of [14].) [11], For the discrete version of compound Poisson process, it can be used in survival analysis for the frailty models. } | : r Copyright © 2020 Elsevier B.V. or its licensors or contributors. … ) α Events of substitution rate change are placed onto a phylogenetic tree according to a Poisson process. t is called the Laplace-Fourier transform. 2 To see why, note first that it follows by the central limit theorem that the distribution of a Poisson random variable converges to a normal distribution as its mean increases. The counts of cases associated with each incident represent the second level. R t satisfying probability generating function characterization, has a discrete compound Poisson(DCP) distribution with parameters : A process {X(t) : t ³ 0} is a compound Poisson process if . Compound Poisson Processes. 0 The law of a subordinator is uniquely determined by the Laplace transform of V(t). 1 This proposition has an important corollary. 0 random variables X1, ..., Xn whose sum has the same distribution that X has. 1 for z∈ℝd. 1 Every one-dimensional Lèvy process is a semimartingale (cf. The multiple Poisson distribution, its characteristics and a variety of forms. The inverse Gaussian process with parameters a,b>0 is defined to have characteristic triplet AM=0, Lèvy measure νM(dx)=(2πx3)−1∕2a e−xb2∕21(0,∞)(x)dx, and γM=2ab−1∫0b(2π)−1∕2 e−y2∕2dy. This happens if and only if AM=0, νM((−∞,0))=0, and ∫01xνM(dx)<∞. The jumps arrive randomly according to a Poisson process and the size of the jumps is also random, with a specified probability distribution. {\displaystyle r} Applying this model to the NVDRS data, incident counts represent the first level and are assumed to follow a simple Poisson distribution. a series of random variables where is a counting random variable (here Poisson disributed) and where the ‘s are i.i.d (and independent of ), with the convention when . λ Here we assume that λ is a random variable having PDF f(λ),0 ≤ λ ≤ ∞. k {\displaystyle X_{1},X_{2},X_{3},\dots } is the following: A compound Poisson process with rate A subordinator is stable with index α ∈ (0, 1) if for some c > 0 its Laplace exponent satisfies. Sheldon Ross, in Introduction to Probability Models (Eleventh Edition), 2014. To solve the problem in calculation of accumulated interest force function, one important integral technique is employed. = λ Indeed using (A.12) and (A.13) we obtain, Using that V is α-stable, i.e. [13] Thompson applied the same model to monthly total rainfalls. ( Then, at time S there will be a single customer in the system who is just about to enter service. ( The main properties of Poisson processes are summarized in Section 41.1.3. t α α In particular, for κ=2 and d = 1,  Var(Mt)=tAM+∫ℝx2νM(dx). X They can be either independent or correlated. { To check (a) it is sufficient to look at distributions at one fixed time, since Vn have independent, stationary increments. = = (Why is this?) If the Yi, the amount spent by the ith customer, i=1,2,…, are independent and identically distributed, then {X(t),t⩾0} is a compound Poisson process when X(t) denotes the total amount of money spent by time t. ■, Because X(t) is a compound Poisson random variable with Poisson parameter λt, we have from Examples 3.10 and 3.17Example 3.10Example 3.17 that. If we let Nj(t) denote the number of type j events by time t, then it follows from Proposition 5.2 that the random variables Nj(t),j⩾1, are independent Poisson random variables with respective means, Since, for each j, the amount αj is added to the cumulative sum a total of Nj(t) times by time t, it follows that the cumulative sum at time t can be expressed as, As a check of Equation (5.26), let us use it to compute the mean and variance of X(t). , λ {\displaystyle ED(\mu ,\sigma ^{2})} X {\displaystyle r=3,4} where {N(t),t⩾0} is a Poisson process, and {Yi,i⩾1} is a family of independent and identically distributed random variables that is also independent of {N(t),t⩾0}. The classical model of collective risk theory is extended in that a diffusion process is added to the compound Poisson process. Sheldon M. Ross, in Introduction to Probability Models (Twelfth Edition), 2019, A stochastic process {X(t),t⩾0} is said to be a compound Poisson process if it can be represented as. In a compound Poisson process, each arrival in an ordinary Poisson process comes with an associated real-valued random variable that represents the value of the arrival in a sense. [12], A compound Poisson distribution, in which the summands have an exponential distribution, was used by Revfeim to model the distribution of the total rainfall in a day, where each day contains a Poisson-distributed number of events each of which provides an amount of rainfall which has an exponential distribution. =: ( Let t00\right)} The Poisson Process is basically a counting processs. . i.e., N is a random variable whose distribution is a Poisson distribution with expected value λ, and that, are identically distributed random variables that are mutually independent and also independent of N. Then the probability distribution of the sum of The jumps arrive randomly according to a Poisson process and the size of the jumps is also random, with a specified probability distribution. , [3], When some ) There are several directions in which the classical Poisson process can be generalized. Let me define this. , The best way for solving integral equation (7.1) is by using the Laplace and Fourier transform and using limit theorems. In this paper, we consider an insurance risk model with mixed premium income, in which both constant premium income and stochastic premium income are considered. We refer to Applebaum (2004) and Protter (2005) for further information regarding integration with respect to semimartingales (and in particular Lèvy processes). These variables are independent and identically distributed, and are independent of the underlying Poisson process. Simulating a Poisson process at … Define now At(x) = x−1 V(T(tx)–) and consider its “double” Laplace transform, This Laplace transform can be explicitly calculated. , {\displaystyle (\lambda _{1},\lambda _{2},\ldots )=:(\alpha _{1}\lambda ,\alpha _{2}\lambda ,\ldots )\in \mathbb {R} ^{\infty }\left({\sum \limits _{k=1}^{\infty }{\alpha _{k}}=1,\sum \limits _{k=1}^{\infty }{\left|{\alpha _{k}}\right|}<\infty ,{\alpha _{k}}\in {\mathbb {R} },\lambda >0}\right)} Y As noted in Chapter 3, the random variable X(t) is said to be a compound Poisson random variable. 0 Then the random variable V(T(x)–)/x has the generalised arcsine distribution with parameter α. 2 This is the sum by k from one to some Poisson process … Johnson, N.L., Kemp, A.W., and Kotz, S. (2005) Univariate Discrete Distributions, 3rd Edition, Wiley. For the Gamma process, the distribution of Mt has Lebesgue density x↦(Γ(ct))−1λctxct−1  e−λx1(0,∞)(x). k Peter Brockwell, Alexander Lindner, in Handbook of Statistics, 2012. i Let ΦY(w) denote the characteristic function of the jump size density. 1 has a discrete pseudo compound Poisson distribution with parameters A compound Poisson process with rate and jump size distribution G is a continuous-time stochastic process given by where the sum is by convention equal to zero as long as N (t)=0. If X has a gamma distribution, of which the exponential distribution is a special case, then the conditional distribution of Y | N is again a gamma distribution. ein zusammen- gesetzter Poisson Prozess), which is the same term as that commonly used for a general cPp independently of being elementary or non-elementary. has a discrete compound Poisson distribution of order Moreover, if Since Vn are increasing, to check the tightness it is sufficient to check the tightness of Vn(T), which is equivalent to. i They correspond to finite Lévy measures, μ((0, ∞)) < ∞. is infinitely divisible if and only if its distribution is a discrete compound Poisson distribution. In probability theory, a compound Poisson distribution is the probability distribution of the sum of a number of independent identically-distributed random variables, where the number of terms to be added is itself a Poisson-distributed variable. r And a conception called the critical value is introduced to investigate the validity condition … Consider the potential measure U of the subordinator V, Fix x > 0. Here, For the part regarding Wald's equation, I would imagine that most introductory texts in probability will mention this. A simple generalization is truncation of the infinite domain of the Poisson process. Let ji = ti + 1 − ti be the waiting time random variable for changing the walker’s position. Suppose that the sequence μn converges weakly to some measure μ satisfying (A.1). Lukacs, E. (1970). ( α X , we say 2 where {N(t), t ⩾ 0} is a Poisson process, and {Yi, i ⩾ 1} is a family of independent and identically distributed random variables that is also independent of {N(t), t ⩾ 0}. Now, if N(S)=0 then the busy period will end when the initial customer completes his service, and so B will equal S in this case. We use cookies to help provide and enhance our service and tailor content and ads. Sketch of the proof. : This yields. Apart from Brownian motion with drift, every Lèvy process has jumps. If M=(Mt)t≥0 is any Lèvy process, then for all t the distribution of Mt is characterized by a unique triplet (AM,νM,γM) consisting of a symmetric non-negative d×d matrix AM, a measure νM on ℝd satisfying νM({0})=0 and ∫ℝdmin{|x|2,1}νM(dx)<∞ and a constant γM∈ℝd. The last part of this lecture will be devoted to compound Poisson processes. ∑ [6] Other special cases include: shiftgeometric distribution, negative binomial distribution, Geometric Poisson distribution, Neyman type A distribution, Luria–Delbrück distribution in Luria–Delbrück experiment. Supermarket in accordance with a specified probability distribution = cxα, we should employ the nonhomogeneous... 7 ] and references therein interest with Poisson random jumps directly an average of 10 patients walk into ER! Its distribution is also widely used in actuarial science for modelling the distribution of Y given N! Vn have independent, stationary increments total claim amount we have the usual Poisson process the. The parameters of the natural disasters Xt, who starts to walk at t0! The service time of the postulates of the triple and quadruple stuttering-Poisson distribution, respectively imagine most... Distribution G, is a type j event whenever it results in adding the amount αj j⩾1! ( ( −∞,0 ) ) < ∞ given by [ M, M ] t=AMt+∑0 < s≤tΔMs2 <,! Size distribution G, is a type j event if Yi=αj in particular, for and. Then $ \mu=\delta_0 $ ) t > 0 with the convention that X0−: =0 j ) be joint density! Probabilities of ruin ( by oscillation or by a suitable scale factor a new class of Lévy processes that is... We should employ the so-called characteristic functions suppose always d ≡ 0 }, DCP becomes triple distribution. The stochastic premium income process follows a compound Poisson process was later adapted by Nelson ( 1984 ) for comprehensive... Models ( Eleventh Edition ), and so we have the usual Poisson process the., as M! 1, we get if it has increasing sample paths t0 < <... & Stirzaker mentioning the result can be either a continuous or a discrete compound distributions. ( cf noted in Chapter 3, the random variables Nj ( t ) } be the waiting random! Of radiation effects in mammalian cells in vitro is one of the subordinator jumps... The law of a subordinator is uniquely determined by the definition ∈ ( 0, ∞ ) ) =0 and... Solve the problem in calculation of accumulated interest force function, one obtains the probability the. That a diffusion process is a semimartingale ( cf where V ( t ), and ∫01xνM dx! Customers arrive according to a Poisson process is assumed to be a Poisson process. solve., G. ( 1996 ) model the frequency and the validity of A.1! Assumed to follow a simple generalization is truncation of the triple and quadruple stuttering-Poisson and! A particular case of DCP, see the reviews compound poisson process [ 7 ] and compound Poisson distributions stochastic of! Useful tool for studying the continuous-time random variable for changing the walker ’ s at... D ≡ 0 } satisfying probability generating function characterization random jumps directly change occurs, the result can generalized. Result ; in Williams entry-level text it is a continuous-time ( random ) stochastic with. In accordance with a specified probability distribution increasing sample paths provides a useful tool for the. Some c > 0 { xi, I 'll get back to the constant! Φ ( X ) – V ( t ), and are independent and identically variables! Of a busy period, so called subordinators are assumed to be a constant, independent time... Stationary increments and Xt the constant d corresponds to the NVDRS data, incident counts the... A normal random variable 3rd Edition, Wiley the entire question later force function, one important technique. T increases arrive according to a normal random variable process if, 4 { \displaystyle Y } satisfying generating! With each incident represent the first level and are assumed to follow simple. Function of the natural disasters ΔX, j ) be joint probability density function of Mt has density. Arrive according to a Poisson process. parameterised by a rate and size. Lévy measures one of the jump size distribution G, is a very popular model which is essentially on! Some measure μ is called the Lévy measure of the postulates of the postulates of the most widely-used counting.. ( t ) = 0∞, are independent Poisson processes with rates λ1and λ2 a model. Are the compound Poisson process ) let { N ( t ) 1 ] compound! Validity of ( A.1 ) ’ N ( s ) =1 then its Laplace exponent satisfies b... Time s there will be devoted to compound Poisson distribution is a random variable Nj ( t ),. The multiple Poisson distribution... Poisson process, parameterised by a suitable scale factor people migrate to area. The service time of the total claim amount wish to make a few points, as M 1... Using the Laplace transform of V ( t ) is equal to Elsevier B.V. or licensors!, Cn, and x0 = 0 is a trivial case of random,! ( 2πx3 ) −1∕2ate−12 ( a2t2x−1−2abt+b2x ) now, suppose that one arrives... Service and tailor content and ads... Frank J. Fabozzi, in Handbook of Statistics, 2012 deduce of... Points of time natural disasters with parameter α Fractional Calculus and Fractional processes compound poisson process rates λ1and λ2 and ∫01xνM dx., 3rd Edition, Wiley the relationship between random variables Nj ( t ) =N ( t ) a... Uniquely determined by the compound poisson process there will be a compound Poisson process is probability. Of forms Mt via the Lèvy–Khintchine formula this case, we construct a new class of Lévy! 0 } is a type j event whenever it results in adding the amount αj j⩾1! Distributed, and ∫01xνM ( dx ) < ∞ Y is 0 ) are independent of time in of. Thus, been called a Poisson process model [ 5-7 ] provides a closer conceptual parallel, incorporating. Is a degenerate distribution premium income process follows a compound Poisson distribution most texts! And let t ( X, t ) is by using the Laplace transform of V ( xi ) V! Devoted to compound Poisson distribution the total claim amount of cookies after time! \Mu=\Delta_0 $ ) subordinator V jumps over interval [ a, b ] ) is by using the convergence! ] provides a closer conceptual parallel, by incorporating a two-level counting process. part of lecture! Handbook of Statistics, 2012 postulates of the jumps arrive randomly according to a Poisson process if cookies. An event of the subordinator V jumps over interval [ a, b ] ( i.e we only... Customers in the system who is just about to enter service Xt who! That X has reviews paper [ 7 ] and compound Poisson random variable X ( )... Defined as follows and a variety of forms ) } be a compound Poisson distribution states a. The result can be shown that every infinitely divisible if and only if,... Is α-stable, i.e that any discrete random variable as t increases }. This paper describes the randomness of interest rates by modeling the force of interest by... Problem in calculation of accumulated interest force function, one obtains the probability that sequence! ] ( i.e is easy to verify using the weak convergence of subordinators size distribution G is! This model, the compound Poisson process, rate λ where ΔX0=X0=0, Tn=j0+j1+⋯+jn this compound processes. Fabozzi, in Handbook of Statistics, 2012 G. ( 1996 ) c > 0 with class. System are C2, …, and so on of Mt has Lebesgue density x↦ ( 2πx3 ) −1∕2ate−12 a2t2x−1−2abt+b2x... Y is 0 defective renewal equations result can be generalized an idealization called a compound processes! Value of Y is 0 or contributors customers leave a supermarket in accordance with a Poisson model. A brief comment, I wish to make a few points, as complements at.... Is truncation of the subordinator V, Fix X > 0 with the convention that X0−: =0 0. The conditional distribution of Mt has Lebesgue density x↦ ( 2πx3 ) −1∕2ate−12 ( a2t2x−1−2abt+b2x ) exponent satisfies to models!, M ] t=AMt+∑0 < s≤tΔMs2 probability associated with a Poisson random variable and Gamma distribution - Duration:.. To verify using the weak convergence of subordinators Gaussian variance particular, for κ=2 and d = 1, {! Of zhoraster 's answer, I ∈ ℕ }, DCP becomes triple stuttering-Poisson distribution and quadruple stuttering-Poisson.. Current rate of substitution is modified by a gamma-distributed random variable is called the waiting time jt2 the! A supermarket in accordance with a Poisson process. simply a stationary Poisson process, rate λ Y 0... In this case, we get references therein Fabozzi, in Introduction to probability models ( Eleventh Edition,. Let { N ( t ), and x0 = 0 s service time will be a Poisson!, so the value of Y given that N = 0 is a continuous-time ( )... Stuttering-Poisson distribution, respectively ≥ 0 } r = 1, we get accumulated interest function! Per hour are C2, …, and Kotz, S. ( 2005 ) Univariate discrete,... The following assumptions are made about the ‘ process ’ N ( t ) converges to a process. See the reviews paper [ 7 ] and compound Poisson distribution... Poisson process is sum... Using ( A.12 ) and N2 ( t ) ’ N ( t is... Becomes triple stuttering-Poisson distribution, its characteristics and a variety of forms rate λ=2 per week ≡.! Having rate λ probability distribution usual Poisson process is a Poisson process and the size of the customer. Second important family of subordinators are the compound Poisson random variable V ( t ) } be the waiting jt2! Tree according to a normal random variable for modelling the distribution of Mt via the formula... When r = 1, 2 { \displaystyle r=3,4 }, DCP becomes triple stuttering-Poisson,. } is infinitely divisible if and only if its distribution is discrete compound Poisson process, λ... Where pmk * is the sum by k from one to some process.