Logarithmic Lévy process directed by Poisson subordinator
aa r X i v : . [ m a t h . P R ] D ec Modern Stochastics: Theory and Applications 6 (4) (2019) 419–441https://doi.org/10.15559/19-VMSTA142
Logarithmic Lévy process directed by Poissonsubordinator
Penka Mayster ∗ , Assen Tchorbadjieff Institute of Mathematics and Informatics, Bulgarian Academy of Sciences,Acad. G. Bonchev street, Bloc 8, 1113 Sofia, Bulgaria [email protected] (P. Mayster), [email protected] (A. Tchorbadjieff)Received: 5 April 2019, Revised: 13 September 2019, Accepted: 13 September 2019,Published online: 4 October 2019
Abstract
Let { L ( t ) , t ≥ } be a Lévy process with representative random variable L (1) defined by the infinitely divisible logarithmic series distribution. We study here the transitionprobability and Lévy measure of this process. We also define two subordinated processes.The first one, Y ( t ) , is a Negative-Binomial process X ( t ) directed by Gamma process. Thesecond process, Z ( t ) , is a Logarithmic Lévy process L ( t ) directed by Poisson process. Forthem, we prove that the Bernstein functions of the processes L ( t ) and Y ( t ) contain the iteratedlogarithmic function. In addition, the Lévy measure of the subordinated process Z ( t ) is ashifted Lévy measure of the Negative-Binomial process X ( t ) . We compare the properties ofthese processes, knowing that the total masses of corresponding Lévy measures are equal. Keywords
Infinitely divisible logarithmic series distribution, Bernstein function, Lévyprocess, change of time, compound Poisson process, Gauss hypergeometric function, Stirlingnumbers, harmonic numbers, partial Bell polynomials
Let { L ( t ) , t ≥ } be a Lévy process with representative random variable L (1) de-fined by the infinitely divisible logarithmic series distribution. The distribution of anyLévy process is completely determined by the distribution of its representative ran-dom variable, which is infinitely divisible [22]. The probability generating function ∗ Corresponding author.
Preprint submitted to VTeX / Modern Stochastics: Theory andApplications
P. Mayster, A. Tchorbadjieff (p.g.f.) of the random variable L (1) is expressed by the Gauss hypergeometric func-tion F (1 ,
1; 2; z ) [10]. This makes the usage of enumerative combinatorics methodsindispensable in this study [19]. Thus, using the partial Bell polynomials we obtain anexplicit representation of the transition probability and Lévy measure of this process.But, first of all, we distinguish two logarithmic distributions.The logarithmic series distribution supported by positive integers N = { , , . . . } was firstly introduced by R.A. Fisher, A.S. Corbet and C.B. Williams (1943) [9]. Itis not an infinitely divisible distribution. It is a Lévy measure for the well-knownNegative-Binomial process. The paper [9] represents an impressive combination ofempirical data and mathematical analysis, remaining a model for ecology today.The logarithmic series distribution supported by nonnegative integers Z + = { , ,. . . } is a particular case of the Kemp generalized hypergeometric probability distribu-tion (1956) [12]. Its infinite divisibility was proved by K. Katti (1967) [11]. Infinitelydivisible random variables with values in Z + were first studied by Feller (1968) [8],where it is shown that on Z + the infinitely divisible distributions coincide with thecompound Poisson distributions. A historical review on the origin of infinitely di-visible distributions “from de Finetti’s problem to Lévy–Khintchine formula” is pre-sented by F. Mainardi and S. Rogosin (2006) [15].At the present time, there are many works related to the topics of infinite di-visibility and discrete distributions. Some of them are monographs of F.W. Steuteland K. Van Harn [22], and N.L. Johnson, A.W. Kemp and S. Kotz [10]. Integer-valued Lévy processes and their application in financial econometrics are developedby O.E. Barndorff-Nielsen, D. Pollard and N. Shephard [1]. The compositions ofPoisson and Gamma processes are investigated by K. Buchak and L. Sakhno in [4, 5].The consecutive subordinations of Poisson and Gamma processes realized on two se-quences containing only four new processes are studied in [17]. It is shown therehow the additional randomness, caused by random time, is accumulated. The trans-formation of the Poisson random measure and the jump-structure of the subordinatedprocess is described in [16]. Other interesting integer-valued Markov processes arederived from Markov branching processes. In the model of branching particle systemwith a random initial condition, we obtained distributions describing the number ofparticles at time t , corresponding to the compound Poisson processes over the radiusof a flux of particles [18].The subordination by Bochner is also developed in many books and articles re-lated to applications in financial mathematics and functional analysis, see [20, 7].There is a special Chapter 3 in [3] devoted to the subordinators. The Lévy measureand potential kernel are also considered in [2]. The properties of the Bernstein func-tions are studied in [21]. The theory of subordinators and inverse subordinators isapplied to study risk processes in insurance models by N. Leonenko et al. in [13, 14].Our work in the topic is described in the following sections. In Section 2 we intro-duce the infinitely divisible logarithmic series distribution and its m -fold convolution.Our main tools of investigation are the Gauss hypergeometric function and partial Bellpolynomials, Stirling numbers and harmonic numbers. In Section 3 we present twomethods defining the transition probability of the Lévy process L ( t ) – starting withthe Lévy measure or starting with p.g.f. F ( t, s ) = E [ s L ( t ) ] and its Taylor expansion.Then, in the following two Sections 4 and 5 we consider the subordinated processes ogarithmic Lévy process directed by Poisson subordinator Y ( t ) and Z ( t ) . They are obtained respectively from the Negative-Binomial process X ( t ) directed by the Gamma one and the Logarithmic Lévy process L ( t ) directedby the Poisson one. In this study of subordinated processes, we proceed also by twomethods – either integrating the transition probability of the ground process, as it isshown in [20], or constructing the compound Poisson process with a prior definedLévy measure. The Bernstein functions of the processes L ( t ) and Y ( t ) contain theiterated logarithmic function. The Lévy measure of Z ( t ) is a shifted Lévy measureof X ( t ) . We compare the behaviour of all these processes in order to understand bet-ter the place of the Logarithmic Lévy process in the picture of compound Poissonprocesses. Several combinatorics identities arrive as auxiliary results. Finally, someapplications derived from studied processes are explained and demonstrated in Sec-tion 6. The Gauss hypergeometric function is defined in [10], page 20, for | z | < by F ( c, d ; g ; z ) = ∞ X k =0 [ c ] k ↑ [ d ] k ↑ [ g ] k ↑ z k k ! , where the increasing factorial, known as Pochhammer’s symbol, is denoted as [ c ] k ↑ = c ( c + 1) · · · ( c + k − , [ c ] ↑ = 1 . In particular, F (1 ,
1; 2; z ) = ∞ X k =0 k ! k !( k + 1)! z k k ! = − log(1 − z ) z . By definition, given in [22], Chapter 2, Example 11.7, the infinitely divisible randomvariable L (1) with logarithmic series distribution has the probability mass function(p.m.f.) supported by { , , , . . . } and given as follows: P ( L (1) = k ) = 1 A α k +1 ( k + 1) , < α < , A = − log(1 − α ) , k = 0 , , . . . . (1)The p.g.f. defined by F (1 , s ) = E [ s L (1) ] , | s | ≤ , is F (1 , s ) = ∞ X k =0 α k +1 k + 1 s k A = αA ∞ X k =0 ( αs ) k k + 1 = − log(1 − αs ) As and can be presented as follows: F (1 , s ) = αA (1 + G ( s )) , G ( s ) = ∞ X k =1 ( αs ) k k + 1 , G (1) = A − αα . (2)We remark that the following simple representation is a starting point of the Taylorexpansion, F (1 ,
1; 2; αs ) = 1 + G ( s ) . P. Mayster, A. Tchorbadjieff
Let us denote the finite sum of random variables L , L , . . . , L m , being independentcopies of L (1) , as L ( m ) = m X j =1 L j . (3)By convolution of p.m.f. we express the probability distribution of the random vari-able L (2) by means of the harmonic numbers as follows: P ( L (2) = n ) = (cid:16) αA (cid:17) α n n + 2 (cid:18) · · · + 1 n + 1 (cid:19) , n = 0 , , , . . . . Knowing the infinite divisibility of L (1) we write the p.g.f. of L ( m ) (3) by the power, F ( m, s ) = { F (1 , s ) } m , namely: (cid:18) − log(1 − αs ) As (cid:19) m = (cid:16) αA (1 + G ( s )) (cid:17) m . We present here two methods on expanding F ( m, s ) as power series of s , expandingonly ( − log(1 − αs )) m , or with the binomial expanding of ( αA (1 + G ( s ))) m . For thispurpose, the function G ( s ) is presented as an exponential generating function: G ( s ) = ∞ X k =1 g k s k k ! , g k = α k k ! k + 1 . (4)For convenience, we denote the sequences by bullet, as it is shown in [19], g • = ( g , g , . . . ) . Similarly, the sequences of the powers are expressed by a • = ( a, a , . . . ) , and inparticular: • = (1 , , . . . ) . In both cases of expanding we use the Faa di Brunoformula and the partial Bell polynomials B n,k [19], allowing to express the power [ G ( s )] k as follows: ( G ( s )) k k ! = 1 k ! ∞ X j =1 g j s j j ! k = ∞ X n = k B n,k ( g • ) s n n ! , (5)where B n,k ( g • ) = X ( k ,k , ··· ,k n ) n ! g k . . . g k n n k !(1!) k · · · k n !( n !) k n . The sum is over all partitions of n into k parts, that is over all nonnegative integersolutions ( k , k , . . . , k n ) of the equations: k + 2 k + · · · + nk n = n, k + k + · · · + k n = k. For example, B n, ( x • ) = x n , B n,n ( x • ) = ( x ) n and B n,k ( a • bx • ) = a n b k B n,k ( x • ) . (6) ogarithmic Lévy process directed by Poisson subordinator The falling factorials are defined as follows: [ x ] n ↓ = x ( x − · · · ( x − n + 1) = Γ( x + 1)Γ( x + 1 − n ) . Let us denote the Stirling numbers of the first kind by | s ( n, k ) | and s ( n, k ) , respec-tively, – unsigned | s ( n, k ) | := B n,k (( • − B n,k (0! , , , . . . . ) , and signed s ( n, k ) depending on the parity of n − k given by s ( n, k ) = ( − n − k | s ( n, k ) | . The Stirling numbers of the first kind transform the factorials into powers, [ x ] n ↑ = n X k =0 | s ( n, k ) | x k , [ x ] n ↓ = n X k =0 s ( n, k ) x k . (7)Thus, having these definitions and relations, the following useful lemma is formulatedand proved. Lemma 1.
The m-fold convolution of the infinitely divisible logarithmic series distri-bution (1) can be equivalently expressed in the following forms: P ( L ( m ) = n ) = (cid:16) αA (cid:17) m α n n ! | s ( m + n, m ) | m ! n !( m + n )! , n = 0 , , , . . . , (8) or P ( L ( m ) = n ) = (cid:16) αA (cid:17) m α n n ! m ∧ n X k =0 m !( m − k )! B n,k ( c • ) , B , = 1 , (9) where c • = (cid:18) k ! k + 1 , k = 1 , , . . . (cid:19) , B n, = B ,k = 0 , n > , k > . Proof.
Expanding only the logarithmic function ( − log(1 − αs )) we obtain − log(1 − αs ) = ∞ X k =1 ( αs ) k k = ∞ X k =1 ( k − αs ) k k ! . Then, for the powers we have m ! ( − log(1 − αs )) m = ∞ X k = m B k,m ( α • ( • − s k k ! and m ! (cid:18) − log(1 − αs ) As (cid:19) m = 1 A m ∞ X k = m α k | s ( k, m ) | s k − m k ! . P. Mayster, A. Tchorbadjieff
The change of variable n = k − m leads to F ( m, s ) = (cid:16) αA (cid:17) m ∞ X n =0 α n s n n ! | s ( m + n, m ) | m ! n !( m + n )! . In the Taylor theorem on the binomial expansion of F ( m, s ) , there are only finitenumbers of terms: F ( m, s ) = (cid:16) αA (cid:17) m [1 + G ( s )] m = (cid:16) αA (cid:17) m m X k =0 m !( m − k )! [ G ( s )] k k ! . After replacing [ G ( s )] k k ! by the expansion (5) we change the summation order. Thenthe obtained result is F ( m, s ) = (cid:16) αA (cid:17) m ∞ X n =0 α n s n n ! m ∧ n X k =0 m !( m − k )! B n,k ( c • ) , where m ∧ n = min { m, n } . The p.m.f. P ( L ( m ) = n ) is given by the sequence ofcoefficients in front of s n in the p.g.f. F ( m, s ) .The comparison of two expressions (8) and (9) leads to the following combina-torics identity: m ∧ n X k =1 m − k )! B n,k ( c • ) = | s ( m + n, m ) | n !( m + n )! . Remark 1.
The harmonic numbers take part in the expansion of the hypergeomet-ric functions. A complete review on summation formulas involving generalized har-monic numbers and Stirling numbers is given in [6]. The generalized harmonic num-bers are defined as follows: H ( k ) n := 1 + 12 k + 13 k + · · · + 1 n k , H (1) n = H n . For m = 2 , , , . . . , we use the following relations between Stirling numbers ofthe first kind and generalized harmonic numbers to calculate directly the convolutionprobability of L ( m ) and to confirm the previous combinatorics identity. For example, | s (2 + n, | = ( n + 1)! (cid:18) · · · + 1 n + 1 (cid:19) (10)and | s ( n, | = ( n − { ( H n − ) − H (2) n − } , and | s ( n, | = ( n − { ( H n − ) − H n − H (2) n − + 2 H (3) n − } . The general recurrence formula on this relation is given in [6]. ogarithmic Lévy process directed by Poisson subordinator
The principal information on the behaviour of any Lévy process is given by the rep-resentative random variable and it is expressed by the canonical representation ofthe Bernstein function and the Lévy measure, [20, 21]. The Laplace transform of theprocess L ( t ) is given by E [ e − λL ( t ) ] = exp( − tψ L ( λ )) , λ ≥ , where the Laplace exponent is a Bernstein function defined by the random variable L (1) as follows: ψ L ( λ ) = − log (cid:16) E [ e − λL (1) ] (cid:17) , λ ≥ . For integer-valued Lévy processes it is also convenient to work with probability gen-erating functions, see [22], Chapter 2.Let us denote the Lévy measure of the process L ( t ) by Π L ( n ) , n = 1 , , . . . , itstotal mass by θ L = P ∞ n =1 Π L ( n ) , and its generating function by Q L ( s ) = ∞ X n =1 s n Π L ( n ) , | s | ≤ . The normalised Lévy measure is denoted by e Π L ( n ) and respectively, its p.g.f. as e Q L ( s ) = Q L ( s ) /θ L . Then in these notations, the p.g.f. F L ( t, s ) = E [ s L ( t ) ] is givenby F L ( t, s ) = exp {− tθ L [1 − e Q L ( s )] } = exp {− tθ L + tQ L ( s ) } , | s | ≤ , and the Bernstein function is in the form ψ L ( λ ) = ∞ X k =1 (cid:0) − e − λk (cid:1) Π L ( k ) , λ ≥ . All these characteristics of the Logarithmic Lévy process L ( t ) are specified in thefollowing lemma. Lemma 2.
The Lévy measure of the process L ( t ) generated by the infinitely divisiblelogarithmic series distribution L (1) (1) is given for n = 1 , , . . . by the partial Bellpolynomials as follows: Π L ( n ) = α n n ! n X k =1 ( − k − ( k − B n,k ( c • ) , c k = k ! k + 1 . (11) The generating function of the Lévy measure is Q L ( s ) = log(1 + G ( s )) , F (1 ,
1; 2; αs ) = 1 + G ( s ) , | s | ≤ . The Bernstein function of the Logarithmic Lévy process L ( t ) is ψ L ( λ ) = θ L (cid:26) − θ L log(1 + G ( e − λ )) (cid:27) , λ ≥ , P. Mayster, A. Tchorbadjieff where θ L = ψ L ( ∞ ) = − log (cid:16) αA (cid:17) . Proof.
Following representation (2) of p.g.f. F (1 , s ) , it is enough to write log( F (1 , s )) = log (cid:16) αA [1 + G ( s )] (cid:17) = log (cid:16) αA (cid:17) + log (1 + G ( s )) in order to get the generating function of the Lévy measure. The total mass of theLévy measure θ L = − log (cid:0) αA (cid:1) because G (0) = 0 .The logarithmic function log(1 + x ) is expanding by the signed Stirling numbersof the first kind and the expansion of G ( s ) is given previously in (5). Then log(1 + G ( s )) = ∞ X k =1 ( − k − ( G ( s )) k k = ∞ X k =1 ( − k − ( k − G ( s )) k k ! . (12)Exchanging the order of summation and in view of B n,k = 0 , k ≥ n + 1 , we write ∞ X k =1 ( − k − ( k − ∞ X n = k B n,k ( g • ) s n n ! = ∞ X n =1 n X k =1 ( − k − ( k − B n,k ( c • ) α n s n n ! . The Lévy measure is given by the sequence of coefficients in front of s n in Q L ( s ) .As a direct result of (11), the computations of several terms of the Lévy measureare simplified, such as Π L (1) = α , Π L (2) = α
2! 512 , Π L (3) = α
3! 34 , Π L (4) = α
4! 251120 , Π L (5) = α
5! 9512 . It is well known from the [22] (see Theorem 4.4, Chapter 2), that the p.m.f. P ( L (1) = n ) , n = 0 , , . . . , (1) is related to the sequence of the (canonical) Lévymeasure Π L ( n ) , n = 1 , , . . . , (11) by the following recurrence equation: ( n + 1) P ( L (1) = n + 1) = n X k =0 P ( L (1) = k )( n − k + 1)Π L ( n − k + 1) . It is equivalent to the next combinatorical identity: n + 1 n + 2 = n X k =0 k + 1)( n − k )! n − k +1 X j =1 ( − j − ( j − B n − k +1 ,j ( c • ) . There are two ways to define the transition probability P ( L ( t ) = n ) , n =0 , , . . . , of the Lévy process L ( t ) . We could proceed either by starting with p.g.f. F L ( t, s ) and its Taylor expansion or by using the Lévy measure to define the com-pound Poisson process L ( t ) . We present these methods separately in two independentproofs. ogarithmic Lévy process directed by Poisson subordinator Theorem 1.
Let { L ( t ) , t ≥ } be a Lévy process generated by the infinitely divisi-ble logarithmic series distribution (1) supported by { , , , . . . } . Then its transitionprobability is given for n = 0 , , , . . . by P ( L ( t ) = n ) = (cid:16) αA (cid:17) t α n n ! n X k =0 [ t ] k ↓ B n,k ( c • ) , (13) or equivalently: P ( L ( t ) = n ) = (cid:16) αA (cid:17) t α n n ! n X k =0 t k B n,k ( y • ) , B , = 1 , (14) where y n = n X k =1 ( − k − ( k − B n,k ( c • ) , c k = k ! k + 1 . Proof 1.
The transition probability P ( L ( t ) = n ) is the coefficient in front of s n inthe expansion of p.g.f. F L ( t, s ) = (cid:0) αA (cid:1) t (1 + G ( s )) t . The Taylor theorem for thebinomial expansion following (5) leads to F L ( t, s ) = (cid:16) αA (cid:17) t ∞ X k =0 [ t ] k ↓ ∞ X n = k B n,k ( g • ) s n n ! . Then after exchanging the order of summation we find F L ( t, s ) = (cid:16) αA (cid:17) t ∞ X n =0 α n s n n ! n X k =0 [ t ] k ↓ B n,k ( c • ) . Because the partial Bell polynomials B , = 1 and B ,k = 0 , k = 1 , , . . . , thefollowing result is valid: F L ( t, s ) = (cid:16) αA (cid:17) t ∞ X n =1 α n s n n ! n X k =1 [ t ] k ↓ B n,k ( c • ) ! , c k = k ! k + 1 . In particular, it is easy to calculate several terms of the transition probability,directly from (13): P ( L ( t ) = 1) = (cid:16) αA (cid:17) t αt , P ( L ( t ) = 2) = (cid:16) αA (cid:17) t α (cid:18) t t ( t − (cid:19) ,P ( L ( t ) = 3) = (cid:16) αA (cid:17) t α (cid:26) t t ( t − . . t ( t − t − (cid:27) ,P ( L ( t ) = 4) = (cid:16) αA (cid:17) t α (cid:26) t t ] ↓ + [ t ] ↓ + 116 [ t ] ↓ (cid:27) . P. Mayster, A. Tchorbadjieff
Proof 2.
Let the positive random variable ξ be defined by the normalised Lévy mea-sure (11), having p.m.f. P ( ξ = n ) = Π L ( n ) /θ L , n = 1 , , . . . , θ L = log (cid:16) Aα (cid:17) , and p.g.f. E [ s ξ ] = Q L ( s ) /θ L . Let ( ξ , ξ , . . . , ξ k , k = 1 , , . . . ) be independentcopies of the random variable ξ . Following definition of the compound Poisson pro-cess, the transition probability is represented as follows: P ( L ( t ) = n ) = ∞ X k =0 e − θt ( θt ) k k ! P ( ξ + ξ + · · · + ξ k = n ) , θ = log (cid:16) Aα (cid:17) . Taking into account (12) and (6) we can represent the function θ Q L ( s ) = 1 θ log(1 + G ( s )) as an exponential generating function θ Q L ( s ) = P ∞ n =1 x n s n n ! , where x n = 1 θ α n n X k =1 ( − k − ( k − B n,k ( c • ) . It means that the normalised probability convolution distribution P ( ξ + ξ + · · · + ξ k = n ) = B n,k ( x • ) k ! n ! . Then P ( L ( t ) = n ) = ∞ X k =0 e − θt ( θt ) k k ! B n,k ( x • ) k ! n ! , k ≤ n. The infinite sum is reduced to the finite one because B n,k ( x • ) = 0 when k > n . Weknow that θ = − log( αA ) and e − θt = (cid:0) αA (cid:1) t . Let us denote y n = n X k =1 ( − k − ( k − B n,k ( c • ) . Then following the formula (6), we obtain, for n = 0 , , , . . . , P ( L ( t ) = n ) = (cid:16) αA (cid:17) t n X k =0 t k α n n ! B n,k ( y • ) . The probability P ( L ( t ) = 0) = ( αA ) t corresponds to the B , ( y • ) = 1 .We remark that in the matrix representation of partial Bell polynomials for com-position function the numbers B n,k ( x • ) , n ≥ k ≥ , are defined as product of ma-trices, [19], page 19. Let us denote by H ( s ) and G ( s ) respectively the exponentialgenerating functions of both sequences ( h • ) and ( g • ) . Likewise, by ( x • ) is denoted ogarithmic Lévy process directed by Poisson subordinator the sequence whose exponential generating function is the composition H ( G ( s )) ,such as H ( s ) = log(1 + s ) , H ( G ( s )) = log(1 + G ( s )) . Then the matrix associated with the sequence ( x • ) is the product of the triangularmatrices associated with ( g • ) and ( h • ) respectively: B n,k ( x • ) = n X j = k B n,j ( g • ) B j,k ( h • ) , k ≤ j ≤ n. The sequence ( h • ) defined by the function H ( s ) = log(1 + s ) is exactly the sequence h k = ( − k − ( k − and B n,k ( h • ) = s ( n, k ) , i.e. signed Stirling numbers ofthe first kind. Then, after applying formulas (6) and (7) and changing the order ofsummation, where k ≤ j , we confirm the equivalence of (13) and (14) as follows: n X k =1 t k B n,k ( x • ) = α n n X j =1 B n,j ( c • ) j X k =1 t k B j,k ( h • ) = α n n X j =1 B n,j ( c • )[ t ] j ↓ . The Lévy measure is the infinitesimal generator of the convolution semi-groupgiven by the transition probability P ( L ( t ) = n ) , n = 0 , , , . . . , see [2], page 172. Itis a limit in vague convergence, see [3], page 39, as follows: Π L ( n ) = lim t ↓ P L ( t, n ) t , n = 1 , , . . . . Then Π L ( n ) = lim t ↓ (cid:16) αA (cid:17) t α n n ! n X k =1 [ t ] k ↓ t B n,k ( c • ) . Finally, we know that [ t ] k ↓ = [ − t ] k ↑ ( − k . In this way, lim t ↓ [ t ] k ↓ t = ( − k − ( k − . The concept of subordination was introduced by S. Bochner in 1955 for the Markovprocesses, Lévy processes, and corresponding semigroups, as randomization of thetime parameter: Y ( t ) = X ( T ( t )) . There are two sources of randomness – the un-derlying process X ( t ) and a random time process T ( t ) , under the assumption oftheir independence. The time-change process T ( t ) is supposed to be a subordinator –the Lévy process with nonnegative increments, [3], Chapter 3. The independence ofthe ground process and the random time process ensures the preservation of Markovproperty and Lévy property for the subordinated process. The transformation of themain probabilistic characteristics, such as transition probability, Lévy measure andLaplace exponent, is stated and proved in [20], Chapter 6, Theorem 30.1. See also[7], Chapter 7, Theorem 6.2 and Theorem 6.18. They are our principal references. P. Mayster, A. Tchorbadjieff
In this paragraph, we study the effect of a random time-change for the Negative-Binomial process { X ( t ) , t ≥ } . The Lévy measure of a Negative-Binomial processis defined by a logarithmic series distribution supported by positive integers N = { , , . . . } with the same parameter < α < as for the Logarithmic Lévy process L ( t ) . The Gamma subordinator { T β ( t ) , t ≥ } with selective parameter β > canbe considered as a random observation time, where the mathematical expectation E [ T β ( t )] = βt . The obtained results are formulated and proved in the followingtheorem. Theorem 2.
Let { X ( t ) , t ≥ } be a Negative-Binomial process with the Bernsteinfunction ψ X ( λ ) = log (cid:18) − αe − λ − α (cid:19) , λ ≥ , ψ X ( ∞ ) = − log(1 − α ) = A. Let { T β ( t ) , t ≥ } be a Gamma subordinator with the Bernstein function ψ T ( λ ) = log(1 + βλ ) , λ ≥ , ψ T ( ∞ ) = ∞ . Suppose the processes X ( t ) and T β ( t ) are independent.Then for the subordinated process Y ( t ) = X ( T β ( t )) the following results arevalid.The Bernstein function of Y ( t ) is given by ψ Y ( λ ) = log (cid:18) β log (cid:18) − αe − λ − α (cid:19)(cid:19) , λ ≥ , ψ Y ( ∞ ) = log(1 + Aβ ) . The Lévy measure of the subordinated process is given by Π Y ( n ) = α n n ! n X k =1 | s ( n, k ) | ( k − (cid:18) β Aβ (cid:19) k , n = 1 , , . . . . The transition probability P ( Y ( t ) = n ) , n = 0 , , , . . . , is given by P ( Y ( t ) = n ) = α n n ! 1(1 + Aβ ) t n X k =0 | s ( n, k ) | [ t ] k ↑ (cid:18) β Aβ (cid:19) k , or equivalently: P ( Y ( t ) = n ) = α n n ! 1(1 + Aβ ) t n X k =0 t k B n,k ( w • ) , where the sequence ( w • ) is defined by w n = n X k =1 | s ( n, k ) || s ( k, | (cid:18) β Aβ (cid:19) k , | s ( k, | = ( k − . ogarithmic Lévy process directed by Poisson subordinator Proof.
The main assumption in the definition of subordination by Bochner is the in-dependence of the ground process and the random time-change process. The methodsof the Laplace transform and conditional probability for independent processes givethe following convenient representations of the main characteristics, see [20], page197. The Bernstein function of the subordinated process is the composition of thecorresponding Bernstein functions, as follows: ψ Y ( λ ) = ψ T β ( ψ X ( λ )) . The transition probability of the subordinated process is expressed by the conditionalprobability and is given as the integral of transition probability of the ground processby the transition probability of the Gamma subordinator: P ( Y ( t ) = n ) = Z ∞ P ( X ( u ) = n ) u t − e − u/β duβ t Γ( t )= Z ∞ (1 − α ) u [ u ] n ↑ α n n ! u t − e − u/β duβ t Γ( t ) . Replacing the increasing factorials (7) [ u ] n ↑ = P nk =0 | s ( n, k ) | u k and (1 − α ) u = e u log(1 − α ) = e − Au , we obtain P ( Y ( t ) = n ) = α n n ! n X k =0 | s ( n, k ) | Z ∞ e − Au e − u/β u k + t − duβ t Γ( t )= α n n ! n X k =0 | s ( n, k ) | Γ( t + k ) β k Γ( t )(1 + Aβ ) t + k = α n n ! 1(1 + Aβ ) t n X k =0 | s ( n, k ) | [ t ] k ↑ (cid:18) β Aβ (cid:19) k . Let us remark, that the Lévy measure of the Gamma subordinator in our parametrisa-tion is given by Π T β ( du ) = e − u/β du/u, see [3], page 73. Then, from the results proved in [20, 7], the Lévy measure of thesubordinated process can be calculated as the integral of transition probability of theground process by the Lévy measure of the Gamma subordinator: Π Y ( n ) = Z ∞ P ( X ( u ) = n ) e − u/β duu = Z ∞ (1 − α ) u [ u ] n ↑ α n n ! e − u/β duu = α n n ! n X k =1 | s ( n, k ) | Z ∞ e − Au e − u/β u k duu = α n n ! n X k =1 | s ( n, k ) | Γ( k ) (cid:18) β Aβ (cid:19) k . P. Mayster, A. Tchorbadjieff
From the Bernstein function ψ Y ( λ ) we derive the generating function of the Lévymeasure Π Y ( n ) in the following form: Q Y ( s ) = θ Y e Q Y ( s ) = − log (cid:18) − β Aβ {− log(1 − αs ) } (cid:19) . It can be presented as an exponential generating function as follows: Q Y ( s ) = ∞ X n =1 u n s n n ! , u n = n X k =1 B n,k ( v • ) B k, ( s • ) , where the sequences ( v • ) and ( s • ) are defined respectively by v k = k ! α k k , s k = k ! k (cid:18) β aβ (cid:19) k . Moreover, B n,k ( v • ) = α n | s ( n, k ) | , B k, ( s • ) = (cid:18) β Aβ (cid:19) k | s ( k, | and u n = α n n X k =1 | s ( n, k ) || s ( k, | (cid:18) β Aβ (cid:19) k . Let ( ξ , ξ , . . . , ξ k , k = 1 , , . . . ) be independent copies of the positive random variable ξ with p.m.f. P ( ξ = n ) = Π Y ( n ) /θ, n = 1 , , . . . , θ = θ Y = log(1 + Aβ ) . In a complete analogy with the Proof 2 of Theorem 1, we find the normalised proba-bility convolution distribution P ( ξ + ξ + · · · + ξ k = n ) = 1 θ k B n,k ( u • ) k ! n ! . Then for n = 0 , , , . . . , we have: P ( Y ( t ) = n ) = ∞ X k =0 e − θt ( θt ) k k ! 1 θ k B n,k ( u • ) k ! n ! , B , = 1 . Obviously, the exponential decay is e − θt = ( Aβ ) t . Additionally, from the formula(6) we derived that B n,k ( u • ) = α n B n,k ( w • ) , w n = n X k =1 | s ( n, k ) || s ( k, | (cid:18) β aβ (cid:19) k . So, taking into account, that B n,k ( u • ) = 0 for all k > n , we see that the infinite sumis reduced to the finite one P ( Y ( t ) = n ) = α n n ! 1(1 + Aβ ) t n X k =0 t k B n,k ( w • ) , B , = 1 . ogarithmic Lévy process directed by Poisson subordinator Finally, we remark that for n = 1 , , . . . , lim t ↓ P ( Y ( t ) = n ) t = α n n ! B n, ( w • ) = α n n ! w n = Π Y ( n ) . The next studied process { Z ( t ) , t ≥ } is constructed as a random time-change ofthe Logarithmic Lévy process L ( t ) with the Poisson one in the assumption of theirindependence. The selective parameter b > of the Poisson process { T b ( t ) , t ≥ } is introduced, such as mathematical expectation E [ T b ( t )] = bt . The results areformulated and proved in the following theorem. Theorem 3.
Let { L ( t ) , t ≥ } be a Logarithmic Lévy process with the Bernsteinfunction ψ L ( λ ) = log (cid:18) Ae − λ − log(1 − αe − λ ) (cid:19) , λ ≥ , ψ L ( ∞ ) = log (cid:18) Aα (cid:19) > . Let { T b ( t ) , t ≥ } be a Poisson subordinator with the Bernstein function ψ T ( λ ) = b (1 − e − λ ) , λ ≥ , ψ T ( ∞ ) = b > . Suppose the processes L ( t ) and T b ( t ) are independent.Then for the subordinated Lévy process Z ( t ) = L ( T b ( t )) the following resultsare valid.The Bernstein function of the subordinated process Z ( t ) is given by ψ Z ( λ ) = b (cid:18) − αe − λ ) Ae − λ (cid:19) , λ ≥ , ψ Z ( ∞ ) = b (cid:16) − αA (cid:17) > . The Lévy measure of Z ( t ) is given by Π Z ( n ) = bα n +1 A ( n + 1) , n = 1 , , . . . . The transition probability of the subordinated process Z ( t ) is, for n = 0 , , . . . , P ( Z ( t ) = n ) = e − θt α n n ! n X k =0 (cid:18) αbtA (cid:19) k B n,k ( c • ) , c k = k ! k + 1 , θ = b − αbA . (15) Proof.
Once again, the composition of two Bernstein functions is obvious: ψ Z ( λ ) = b (1 − exp( − ψ L ( λ ))) . The Lévy measure of the subordinated process is given by the following infinite sum,as it is shown in [20, 7], Π Z ( n ) = ∞ X k =1 P ( L ( k ) = n )Π T ( k ) = bP ( L (1) = n ) = bA α n +1 ( n + 1) , n = 1 , , . . . , P. Mayster, A. Tchorbadjieff because the normalised Lévy measure e Π L ( n ) , n = 1 , , . . . , of the Poisson process isexactly the delta function in n = 1 . The total mass of the Lévy measure Π Z ( n ) , n =1 , , . . . , is calculated directly from (1) as θ Z = bA ( A − α ) . The exponential generatingfunction of the Lévy measure Π Z is given by (2) and (4) as follows: Q Z ( s ) = bαA G ( s ) = bαA ∞ X k =1 g k s k k ! , g k = α k k ! k + 1 . Let ( ξ , ξ , . . . , ξ k , k = 1 , , . . . ) be independent copies of the positive random variable ξ with p.m.f. P ( ξ = n ) = Π Z ( n ) /θ, n = 1 , , . . . , θ = θ Z = b − bαA . The normalised probability convolution distribution is given by P ( ξ + ξ + · · · + ξ k = n ) = (cid:18) αA − α (cid:19) k B n,k ( c • ) α n k ! n ! , αA − α = bαAθ . Then the elementary transformations lead to (15) as follows: P ( Z ( t ) = n ) = ∞ X k =0 e − θt ( θt ) k k ! (cid:18) αA − α (cid:19) k B n,k ( c • ) α n k ! n != e − θt α n n ! n X k =0 (cid:18) αθtA − α (cid:19) k B n,k ( c • ) = e − θt α n n ! n X k =0 (cid:18) αbtA (cid:19) k B n,k ( c • ) . In particular, P ( Z ( t ) = 0) = e − θt , P ( Z ( t ) = 1) = α bte − θt A .
Knowing that B , = c = and B , = ( c ) = we find P ( Z ( t ) = 2) = e − θt α ( αbt A + (cid:18) αbt A (cid:19) ) . In the same way, as B , = , B , = 1 and B , = we obtain P ( Z ( t ) = 3) = e − θt α ( αbtA (cid:18) αbtA (cid:19) + (cid:18) αbt A (cid:19) ) , (16)and so on. Remark 2.
In this situation, the range of the random time process T b ( t ) is a dis-crete integer-valued set Z + = { , , , . . . } . The subordination by Bochner gives the ogarithmic Lévy process directed by Poisson subordinator transition probability of the subordinated process Z ( t ) = L ( T b ( t )) as the followingconditional probability: P ( Z ( t ) = n ) = ∞ X k =0 P ( L ( k ) = n ) e − bt ( bt ) k k ! . (17)The transition probability of the ground process L ( t ) for integer-valued time t = k is given by the k -fold convolution of the representative random variable L (1) , as thetwo equivalent expressions (9) and (8).After replacing P ( L ( k ) = n ) in (17) by (9), it is enough to exchange the order ofsummation to prove (15), as follows: P ( Z ( t ) = n ) = ∞ X k =0 (cid:16) αA (cid:17) k α n n ! k ∧ n X j =0 k !( k − j )! B n,j ( c • ) ( bt ) k e − bt k != α n e − bt n ! n X j =0 (cid:18) αbtA (cid:19) j B n,j ( c • ) ∞ X k = j (cid:18) αbtA (cid:19) k − j k − j )! . But, if we take P ( L ( k ) = n ) from (8), then replacing it in (17) we obtain P ( Z ( t ) = n ) = e − bt α n n ! ∞ X k =0 (cid:18) αbtA (cid:19) k | s ( k + n, k ) | n !( n + k )! . (18)The relation of the Stirling numbers on the binomial coefficients explains the equiva-lence of (18) to (15) and the presence of e − bt = e − θt in (18). For n = 1 we have | s ( k + 1 , k ) | = ( k + 1)!( k − , | s (1 , | = 0 . Then P ( Z ( t ) = 1) = αe − bt ∞ X k =1 (cid:18) αbtA (cid:19) k | s ( k + 1 , k ) | k )!= αe − bt αbt A ∞ X k =1 k − (cid:18) αbtA (cid:19) k − = e − bt e αbtA α bt A .
For n = 2 we have | s ( k + 2 , k ) | = [3( k + 2) − k + 2)!( k − , | s (2 , | = 0 , | s (3 , | = 2 . We calculate P ( Z ( t ) = 2) = α e − bt ∞ X k =1 (cid:18) αbtA (cid:19) k (cid:18) k + 54 (cid:19) (cid:18) k − (cid:19) . Obviously, (cid:18) k + 54 (cid:19) (cid:18) k − (cid:19) = 14( k − k − , k = 2 , , . . . . P. Mayster, A. Tchorbadjieff
It means that the probability P ( Z ( t ) = 2) is equal to: α e − bt ( (cid:18) αbtA (cid:19) ∞ X k =2 (cid:18) αbtA (cid:19) k − k − αbtA ∞ X k =1 (cid:18) αbtA (cid:19) k − k − ) . Finally, P ( Z ( t ) = 2) = α e − bt e αbtA ( (cid:18) αbtA (cid:19) + 23 αbtA ) . For n = 3 we use the consecutive relations of the Stirling numbers on the binomialcoefficients: | s ( k + 3 , k ) | = ( k + 3)!( k + 1)!2! ( k + 3)!( k − , | s (3 , | = 0 , | s (3 , | = 3 . The equivalent representation of P ( Z ( t ) = 3) will be transformed as follows: P ( Z ( t ) = 3) = α e − bt ∞ X k =1 (cid:18) αbtA (cid:19) k ( k + 3)( k + 2)8( k − . Obviously, ( k + 3)( k + 2)8( k − k − k − k − , k = 3 , , . . . . In the same way we obtain (16), and so on. The Stirling numbers are very convenientin applications because they have recurrence relation and tables of their values.We confirm the expression of the Lévy measure by the following limit: lim t ↓ P ( Z ( t ) = n ) t = α n n ! αbA B n, ( c • ) = bα n +1 An ! n ! n + 1 = Π Z ( n ) , n = 1 , , . . . . An important problem in many applications is how to recognize the original processfrom the observation data when the registration is randomly perturbed. The problemis growing in cases when the process is composed of several different processes. Wesee that the probabilistic characteristics for the couples of processes Y ( t ) and L ( t ) as well as for Z ( t ) and X ( t ) are similar. The best way to demonstrate their differentproperties is the comparison between Bernstein functions and Lévy measures withdifferent parameters. The Bernstein functions ψ L ( λ ) and ψ Y ( λ ) , how they are definedin Theorems 2 and 3, contain the iterated logarithmic function and have the followingderivatives at zero: ψ ′ L ( λ ) = αe − λ (1 − αe − λ )( − log(1 − αe − λ )) − , ψ ′ L (0) = αA (1 − α ) − ogarithmic Lévy process directed by Poisson subordinator and ψ ′ Y ( λ ) = αβe − λ { Aβ + β log(1 − αe − λ ) } (1 − αe − λ ) ,ψ ′ Y (0) = αβ (1 + Aβ − Aβ )(1 − α ) = αβ − α . The first cumulants are equal to the corresponding mathematical expectations E [ L (1)] or E [ Y (1)] and for the subordinated processes we have E [ Y (1)] = E [ X (1)] E [ T β (1)] . Similarly, the Bernstein functions for processes X ( t ) and Z ( t ) are denoted by ψ X ( λ ) and ψ Z ( λ ) as in Theorems 2 and 3. Their derivatives at zero are as follows: ψ ′ X ( λ ) = αe − λ − αe − λ , ψ ′ X (0) = α − α and ψ ′ Z ( λ ) = bA αe − λ + (1 − αe − λ ) log(1 − αe − λ ) e − λ (1 − αe − λ ) ,ψ ′ Z (0) = bA α + (1 − α ) log(1 − α )1 − α = bA (cid:18) α − α − A (cid:19) . For the application, we constructed two different case tests, named as
Selection A and
Selection B . Selection A.
We can choose the parameters β and b in a way that the correspondingtotal masses of the Lévy measures are equal, θ L = θ Y , and θ X = θ Z , as follows: β = A − αAα < , b = A A − α > . By this choice (selection) of parameters β and b the mathematical expectations are inthe following inequalities: ψ ′ L (0) < ψ ′ Y (0) < ψ ′ X (0) < ψ ′ Z (0) . Namely, αA (1 − α ) − < ( A − α ) A (1 − α ) < α − α < AA − α (cid:18) α − α − A (cid:19) . The values of the Lévy measures Π X and Π Z at n = 1 satisfy the following inequality Π Z (1) < Π X (1) , when P ∞ n =1 Π Z ( n ) = P ∞ n =1 Π X ( n ) = A , as it is demonstratedin Figure 1 and in Figure 2. Selection B.
If we choose β = 1 A − − αα > , b = Aαα − A (1 − α ) > , P. Mayster, A. Tchorbadjieff
Fig. 1.
Selection A. Comparison between the Lévy measure Π X and Π Z for α = 1 / (left)and α = 2 / (right), where P ∞ n =1 Π Z ( n ) = P ∞ n =1 Π X ( n ) = A Fig. 2.
Selection A. Comparative plot of main Bernstein functions after rescaling with α equalto / and / , where ψ ′ L (0) < ψ ′ Y (0) < ψ ′ X (0) < ψ ′ Z (0) , knowing that θ L = θ Y = log( Aα ) and θ X = θ Z = A = − log(1 − α ) ogarithmic Lévy process directed by Poisson subordinator Fig. 3.
Selection B: Comparative plot of main Bernstein functions after rescaling with α equalto / and / , where ψ Y ( ∞ ) < ψ L ( ∞ ) < ψ Z ( ∞ ) < ψ X ( ∞ ) , knowing that ψ ′ L (0) = ψ ′ Y (0) and ψ ′ Z (0) = ψ ′ X (0) then the mathematical expectations ψ ′ L (0) = ψ ′ Y (0) and ψ ′ Z (0) = ψ ′ X (0) , and thetotal masses of the Lévy measures are in the following inequalities: ψ Y ( ∞ ) < ψ L ( ∞ ) < ψ Z ( ∞ ) < ψ X ( ∞ ) . Specifically, log (cid:18) − A (1 − α ) α (cid:19) < log (cid:18) Aα (cid:19) < α ( A − α ) Aα − A + α < A. It is demonstrated in Figure 3.
Remark 3.
All these inequalities are related to the following: A − α < Aα < A − α ) and A < α − α , where we remark only that using (10) we have A = ( − log(1 − α )) = 2 ∞ X n =2 | s ( n, | α n n != 2 ∞ X n =2 α n H n − n = α + α + 1112 α + 1012 α + 137180 α · · · P. Mayster, A. Tchorbadjieff and α − α = α (1 + α + α + · · · ) . The Negative-Binomial process in consideration can be constructed by the subordi-nation of a Poisson process by a Gamma process. In this way, the process Y ( t ) isa Poisson process subordinated by an iterated Gamma process. In potential theory,a Gamma subordinator and an iterated Gamma subordinator are classified as slowsubordinators.In the selection A , the inter-arrival times of the processes L ( t ) and Y ( t ) are expo-nentially distributed with the same parameter θ L = θ Y , but the mathematical expec-tation of the Logarithmic Lévy process in the unit time interval E [ L (1)] is less than E [ Y (1)] . It is the same for the processes X ( t ) and Z ( t ) .In the selection B , the mathematical expectations of jumps altitude for the pro-cesses L ( t ) and Y ( t ) are equal ψ ′ L (0) = ψ ′ Y (0) and E [ L (1)] = E [ Y (1)] , but themean number of jumps in the unit time interval of the Logarithmic process is greaterthan that for the subordinated process Y ( t ) , θ L > θ Y . It is the same for the processes X ( t ) and Z ( t ) , θ X > θ Z .In the general setting: Y ( t ) = X ( T ( t )) , when the underlying process X ( t ) is acompound Poisson process without drift, any randomness of T ( t ) before it passes thelevel given by the first jump time of X ( t ) is not reflected by Y ( t ) = X ( T ( t )) , see[16]. Acknowledgement
The authors are very thankful to the anonymous referees for the valuable commentson the paper.
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