Some Examples of Multidimensional Shintani Zeta Distributions

JSIAM Letters Vol.6 (2014) pp.41–44 ⃝c 2014 Japan Society for Industrial and Applied Mathematics J S I A M Letters

Some examples of multidimensional Shintani zeta distributions

Takahiro Aoyama1 and Kazuhiro Yoshikawa2

1 Faculty of Culture and Education, Saga University, 1 Honjo-machi, Saga 840-8502, Japan 2 Graduate School of Science and Engineering, Ritsumeikan University, 1-1-1 Noji-higashi, Kusatsu, Shiga 525-8577, Japan E-mail 1aoyama cc.saga-u.ac.jp 2ra009059 ed.ritsumei.ac.jp Received January 22, 2014, Accepted January 29, 2014 Abstract In the studies of mathematical statistics, we often consider discrete distributions and their corresponding stochastic processes. Especially, probabilistic limit theorems of them may give us some progress in mathematical finance. There exist not so many properties of discrete distributions on Rd. In this paper, we treat multiple zeta functions as to define several forms of discrete distributions on Rd including those with infinitely many mass points. Our purpose is to obtain new methods in the relations between multiple infinite series and high dimensional integral calculus, which can provide us more opportunities to handle high dimensional phenomenon. Keywords Lévymeasure, multinomial distribution, zeta function Research Activity Group Mathematical Finance

∫ ( ) ) 1. Infinitely divisible distributions ⟨⃗ ⟩ i⟨⃗t, x⟩ + ei t,x − 1 − ν(dx) , ⃗t ∈ Rd, | |2 Infinitely divisible distributions are known as one of Rd 1 + x the most important class of distributions in probability (1.1) theory. They are the marginal distributions of stochas- where A is a symmetric nonnegative-definite d × d ma- tic processes having independent and stationary incre- trix, ν is a measure on Rd satisfying ∫ ments such as Brownian motion and Poisson processes. ( ) In 1930’s, such stochastic processes were well-studied by ν({0}) = 0 and |x|2 ∧ 1 ν(dx) < ∞, (1.2) P. Lévyand now we usually call them Lévyprocesses. Rd They often appear in mathematical finance as standard and γ ∈ Rd. stochastic processes. We can find the detail of Lévypro- (ii) The representation of µb in (i) by A, ν, and γ is cesses in Sato [1]. unique. In this section, we mention some known properties of (iii) Conversely, if A is a symmetric nonnegative- infinitely divisible distributions. definite d×d matrix, ν is a measure satisfying (1.2), and Definition 1 (Infinitely divisible distribution) A γ ∈ Rd, then there exists an infinitely divisible distribu- probability measure µ on Rd is infinitely divisible if, for tion µ whose characteristic function is given by (1.1). any positive integer n, there is a probability measure µn The measure ν is called the Lévymeasure and it gen- on Rd such that erates a jump type Lévyprocess. The following is also µ = µn∗, known as one of the most important classes of infinitely n divisible distributions. where µn∗ is the n-fold convolution of µ . n n Definition 4 (Compound Poisson distribution) Example 2 Normal, degenerate and Poisson distribu- A distribution µ on Rd is called compound Poisson if, tions are infinitely divisible. for some c > 0 and some probability measure ρ on Rd Rd { } Denote by I( ) the class of∫ all infinitely divisible with ρ( 0 ) = 0, Rd b ⃗ i⟨⃗t,x⟩ ⃗ ∈ Rd ( ) distributions on . Let µ(t) := Rd e µ(dx), t , µb(⃗t) = exp c[ρb(⃗t ) − 1] , ⃗t ∈ Rd. be the characteristic function of a distribution µ, where ⟨·, ·⟩ is the standard inner product in Rd. We also write Here the measure ρ is the Lévymeasure of the com- a ∧ b = min{a, b}. pound Poisson distribution µ and is finite. The Poisson The following is well-known. distribution is a special case where d = 1 and ρ = δ1, where δ is a delta measure at x. Proposition 3 (Lévy–Khintchine representation x (see, e.g. Sato [1])) (i) If µ ∈ I(Rd), then Remark 5 We have to note that any infinitely divis- ( ible distribution can be expressed as the weak limit of a 1 µb(⃗t) = exp − ⟨⃗t, A⃗t⟩ + i⟨γ,⃗t⟩ certain sequence of compound Poisson distributions. 2

– 41 – JSIAM Letters Vol. 6 (2014) pp.41–44 Takahiro Aoyama et al. 2. Zeta distributions Hu and Lin [7]. In one dimensional case, there exists a class of dis- Other cases are given by using multivariable zeta func- Rd crete distribution generated by the Riemann zeta functions and their corresponding distributions are on . tion. Our research is focused on this class and expanded Aoyama and Nakamura [8] introduced multidimensional to obtain several exact expressions of discrete multidi- Shintani zeta functions which are generalized to be mul- mensional distributions with Lévymeasures if they have. tivariable and multiple infinite series as in the following. We rarely see that both of the discrete distributions and Definition 9 (Multidimensional Shintani zeta Lévymeasures on Rd are computable in mathematical function (Aoyama and Nakamura [8])) Let ∈ N ∈ Cd ∈ Zr statistics as well as such relations between multiple se- d, m, r , ⃗s and (n1, . . . , nr) ≥0. For λlj, d ries and high dimensional measure theories even in pure uj > 0, ⃗cl ∈ R , where 1 ≤ j ≤ r and 1 ≤ l ≤ m, and mathematics. a function θ(n1, . . . , nr) ∈ C satisfying |θ(n1, . . . , nr)| = ε First, we introduce the Riemann zeta function and O((n1 + ··· + nr) ), for any ε > 0, we define a multidi- distribution. We can find the basic properties of zeta mensional Shintani zeta function ZS(⃗s) given by functions in Apostol [2]. ∞ ∑ θ(n , . . . , n ) Definition 6 (Riemann zeta function) The Rie- ∏ ∑ 1 r . m r ⟨⃗c ,⃗s⟩ ( λlj(nj + uj)) l mann zeta function is a function of a complex variable n1,...,nr =0 l=1 j=1 s = σ + it ∈ C, for σ > 1, t ∈ R given by We call the function θ(n1, . . . , nr) a generalized ∞ ∑ 1 Dirichlet character of the multidimensional Shintani zeta ζ(s) := . d ns function and write ⟨⃗c,⃗s⟩ := ⟨⃗c,⃗σ⟩ + i⟨⃗c,⃗t⟩ for ⃗c ∈ R n=1 and ⃗s ∈ Cd, where ⃗σ,⃗t ∈ Rd and ⃗s = ⃗σ + i⃗t. The series The Riemann zeta function converges absolutely in ZS(⃗s) converges absolutely in the region min1≤l≤m⟨⃗cl, ⃗σ⟩ the region σ > 1. In this region of absolutely conver- > r/m (see, Aoyama and Nakamura [8]), which we de- R gence, we have the following well-known distribution on . note by DS. Suppose that θ(n1, . . . , nr) is non-negative Definition 7 (Riemann zeta distribution) For or non-positive definite, then we can define the following d each σ > 1, a probability measure µσ on R is called a class of distribution on R . Riemann zeta distribution, if Definition 10 (Multidimensional Shintani zeta n−σ distribution (Aoyama and Nakamura [8])) For µσ ({− log n}) = , n ∈ N. ∈ Rd ζ(σ) each ⃗σ DS, a probability measure µ⃗σ on is called a multidimensional Shintani zeta distribution, if for all Then its characteristic function fσ can be written as ∈ Zr (n1, . . . , nr) ≥0, follows: ({ ( ) ∫ ∑m ∑r itx ζ(σ + it) − fσ(t) = e µσ(dx) = , t ∈ R. µ⃗σ cl1 log λlk(nk + uk) , R ζ(σ) l=1 k=1 ( )}) This class of distribution is first introduced in Jessen ∑m ∑r and Wintner [3] without normalization for an example ..., − cld log λlk(nk + uk) in the studies of infinitely many times convolutions. As l=1 k=1 ( )−⟨ ⟩ a probability distribution, it is first appeared in Khint- ∏m ∑r ⃗cl,⃗σ chine [4]. θ(n1, . . . , nr) = λlk(nk + uk) . ZS(⃗σ) Proposition 8 (See, e.g. Gnedenko and Kol- l=1 k=1 mogorov [5]) The characteristic function f (t) is a σ Then its characteristic function f⃗σ is given by compound Poisson with a finite Lévymeasure Nσ on R: ∫ ∫ ⃗ ∞ i ⟨⃗t,x⟩ ZS(⃗σ + it) d f⃗σ(⃗t) := e µ⃗σ(dx) = , ⃗t ∈ R , log fσ(t) = (exp(−itx) − 1)Nσ(dx), Rd ZS(⃗σ) 0 which can be regarded as a generalization of the Rie- where mann zeta distribution. ∑ ∑∞ −rσ p Remark 11 This class contains both infinitely divisi- Nσ(dx) = δr log p(dx). r Rd p∈P r=1 ble and non infinitely divisible distributions on . By applying Euler products, some simple examples of com- This proposition shows that the Riemann zeta func- pound Poisson case on R2 and generalized cases on Rd tion is treatable in the theory of Lévyprocesses as some are given in [9] and [10], respectively. other well-known functions. Further properties of this class is also studied in Hu and Lin [6]. 3. Relation between distributions and The Riemann zeta function is variously extended such as Hurwitz or Barnes types. (See, e.g. Apostol [2] in characters detail.) Also, several generalized zeta distributions are Many kinds of discrete distributions can be repre- introduced but most of them are not infinitely divisible. sented in the sense of multidimensional Shintani zeta The cases having the infinite divisibility are the following. functions by choosing suitable characters, and so their A special case of Hurwitz zeta function generates a characteristic functions can be written by multiple infi- compound Poisson distribution on R which is given in nite series. In this section, we pick up the multinomial

– 42 – JSIAM Letters Vol. 6 (2014) pp.41–44 Takahiro Aoyama et al. distribution which is well-known as a multidimensional acter discrete one, and show the relation with the character. θT (n1, . . . , nm) 3.1 Main result 1 (A character corresponding to a ∑∞ ∑ θN (n1, . . . , nm) multinomial distribution) := P r(T = N) m −⟨ ⟩ . ( ϕ(l)(j(l)) ⃗cl,⃗σ )N N=0 l=1 Fix N ∈ N. For each 1 ≤ l, k ≤ m, let ul = 1, λll = 1, ̸ ∈ d ∈ λlk = 0, (l = k). We also take ⃗σ DS,⃗cl = (clj)j=1 Then the characteristic function F⃗σ,T of a multidimen- d R , ϕ(l) ∈ R and j(1), . . . , j(m) ∈ N\{1} with relatively sional Shintani zeta distribution with a character θT has prime each other. Define a character by the form of F (⃗t) θN (n1, . . . , nm) ⃗σ,T  ∞ ∏m ∑  (ϕ(l))kl ∑ P r(T = N) N ! = m −⟨ ⟩  ( ϕ(l)(j(l)) ⃗cl,⃗σ )N  kl ! l=1  (l=1 ) N=0 ∑m ∑∞ := θN (n1, . . . , nm)  kl  nl + 1 = (j(l)) , kl = N , × ∏ ∑  m m ⟨⃗c ,⃗σ+i⃗t⟩  ( (λlk(nk + uk)) l  l=1 n1,...,nm=0 l=1 k=1 0 (otherwise). ∑∞ ∑ P r(T = N) d d = m −⟨ ⟩ Then, for each ⃗s = ⃗σ + i⃗t ∈ C , ⃗t ∈ R , ( ϕ(l)(j(l)) ⃗cl,⃗σ )N N=0 l=1 ∑ ∏m −⟨ ⟩ ( ) (ϕ(l))kl (j(l) ⃗cl,⃗s )kl ∑m N −⟨ ⃗⟩ ZS(⃗s) = N ! × ⃗cl,⃗σ+it kl ! ϕ(l)(j(l)) k1+···+km=N l=1 ( ) l=1 N ( ) ∑m ∞ N −⟨ ⟩ ∑ ∑m ⃗cl,⃗s ⟨ ⃗⟩ = ϕ(l)(j(l)) , = P r(T = N) q(l) ei ⃗xl,t , ⃗t ∈ Rd. l=1 ( ) N=0 l=1 ∑m N ZS(⃗σ + i⃗t) ⟨ ⃗⟩ Especially, if T belongs to a Poisson distribution with ⃗ i ⃗xl,t f⃗σ(t) := = q(l)e , mean λ, then ZS(⃗σ) l=1 ( ) ∞ N ∑ λN ∑m where −λ i⟨⃗xl,⃗t⟩ F⃗σ(⃗t) = e q(l) e −⟨⃗cl,⃗σ⟩ N ! ∑ ϕ(l)(j(l)) N=0 l=1 q(l) := m , ( ( )) −⟨⃗cl,⃗σ⟩ l=1 ϕ(l)(j(l)) ∑m ⟨ ⃗⟩ i ⃗xl,t − ⃗ ∈ Rd d − = exp λ q(l) e 1 , t . ⃗xl := (xlk)k=1, xlk := clk log j(l). l=1 If ϕ(1), . . . , ϕ(m) have the same sign, then the char- This is the characteristic function of a compound Pois- acter θ is non-negative or non-positive definite, and so d N son distribution with a finite Lévymeasure N⃗σ on R that f⃗σ is a characteristic function of a Shintani zeta given by distribution. That is, it is the characteristic function of ∑m a random variable X⃗σ defined by ( ( )) N⃗σ(dx) = λ q(l)δ⃗xl (dx). ∑m ∑m l=1 Pr X = x n ,..., x n ⃗σ l1 l ld l Remark 12 In mathematical finance, models caused l=1 l=1 by some one-dimensional Lévyprocess are studied. Now ∏m ∑m (q(l))nl we have discrete infinitely divisible distributions on Rd = N ! , when nl = N. nl! with finite Lévymeasures, which make us possible to sim- l=1 l=1 ulate some models associating with Rd-valued Lévypro- Especially, if m = d and ⃗x1, . . . , ⃗xd are the standard d cesses. basis of R , then X⃗σ belongs to a multinomial distribution. 4. Conditions to be characteristic func- Since multinomial distributions are the distributions which have densities at most finitely many points, their tions characteristic functions are also multiple finite sum. Non-negative or non-positive definiteness of charac- However, multidimensional Shintani zeta distributions ters are not necessary conditions for distributions to whose characteristic functions defined by multiple infi- be defined by multidimensional Shintani zeta functions. nite series may have densities at countably many points. Therefore, now we consider the case when they are not In the following, we give an example of them and men- non-negative nor non-positive definite. We have the fol- tion whether it is infinitely divisible or not. lowing lemma which holds under the settings in Main result 1 and 2. 3.2 Main result 2 (A character corresponding to a com- d d pound Poisson distribution on R ) Lemma 13 Suppose that R -valued vectors ⃗c1, . . . ,⃗cm R ··· ̸ We use the settings in Main result 1. For any non- are linearly independent over or ⃗c1 = = ⃗cm (= 0). negative integer valued random variable T , define a char- If ϕ(1), . . . , ϕ(m) do not have the same sign, then there

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d d exist R -valued vectors ⃗t1, ⃗t2 such that Take ⃗t1 ∈ R such that T0 = ⟨⃗c1,⃗t1⟩. Then ( ) | ⃗ | | ⃗ | ∑m f⃗σ(t1) > 1, F⃗σ(t2) > 1. ⟨ ⃗ ⟩ Re q(l)ei ⃗xl,t1 b It is known that characteristic functions µ of any prob- l=1 ability measures µ on Rd satisﬁes |µb(⃗t)| ≤ 1, ⃗t ∈ Rd. ( ) ∑m Hence, if normalized functions f (⃗t) and F (⃗t) have vec- i2πT θ ⃗σ ⃗σ = Re q(l)e 0 l > L − ϵ > 1. tors ⃗t , ⃗t such that |f (⃗t )| > 1, |F (⃗t )| > 1 holds, then 1 2 ⃗σ 1 ⃗σ 2 l=1 they can not be characteristic functions. Now we have Hence the following result. ∑m N Theorem 14 (A necessary and suﬃcient condi- ⟨ ⃗ ⟩ |f (⃗t )| = q(l)ei ⃗xl,t1 > 1. tion to be characteristic functions) Suppose that ⃗σ 1 d l=1 R -valued vectors ⃗c1, . . . ,⃗cm are linearly independent over R or ⃗c1 = ··· = ⃗cm (≠ 0). Then, f⃗σ, F⃗σ are charac- (QED) teristic functions if and only if ϕ(1), . . . , ϕ(m) have the The rest of the proofs and further results are given in same sign. Aoyama and Yoshikawa [12].

We give the proof of Lemma 13 of the case of f⃗σ when By following our story, we can see that the characters ⃗c1 = ··· = ⃗cm (≠ 0) as in the same way as in Aoyama seem to be an important key of multidimensional Shin- and Nakamura [9,10]. The following proposition plays a tani zeta functions in view of defining distributions. We key role in its proof. still need new facts and methods of them as to make things in stochastic models clearer and more useful. Proposition 15 (Kronecker’s approximation theorem (see, e.g. [11])) If r1, . . . , rn are arbitrary real Acknowledgments numbers, if real numbers θ1, . . . , θn are linearly independent over the rationals, and if ϵ > 0 is arbitrary, then The authors would like to express sincere appre- there exist a real number t and integers h1, . . . , hn such ciations to Professor Jirô Akahori for his valuable that comments. This work was partially supported by JSPS KAKENHI Grant Numbers 23330190, 24340022, | − − | ≤ ≤ tθk hk rk < ϵ, 1 k n. 23654056 and 25285102. Proof of Lemma 13 Since ϕ(1), . . . , ϕ(m) do not have the same sign, there exists l0 such that q(l0) < 0. References Put [1] K. Sato, LévyProcesses and Infinitely Divisible Distributions, ∑ ∑m Cambridge Univ. Press, Cambridge, 1999. L := q(l) − q(l0) > q(l) = 1, [2] T. M. Apostol, Introduction to Analytic Number Theory,

l≠ l0 l=1 Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1976. and take n0 ∈ N and ϵ > 0 such that L − ϵ > 1. [3] B. Jessen and A. Wintner, Distribution functions and the Define Riemann zeta function, Trans. Amer. Math. Soc., 38 (1935), log j(l) 48–88. θl = (1 ≤ l ≤ m). [4] A. Ya. Khinchine, Limit Theorems for Sums of Independent 2π Random Variables (in Russian), GONTI, Moscow, 1938. Then, θ1, . . . , θm are linearly independent over the ra- [5] B. V. Gnedenko and A. N. Kolmogorov, Limit Distributions tionals. Therefore, the Kronecker’s approximation theo- for Sums of Independent Random Variables (Translated from ∈ R the Russian by Kai Lai Chung), Addison-Wesley, London, rem shows that there exists T0 such that 1968. i2πT θ ϵ [6] G. D. Lin and C.-Y. Hu, The Riemann zeta distribution, |e 0 l0 + 1| < ∑ , m | | Bernoulli, 7 (2001), 817–828. ( l=1 q(l) ) [7] C.-Y. Hu, A. M. Iksanov, G. D. Lin and O. K. Zakusylo, The ϵ |ei2πT0θl − 1| < ∑ (l ≠ l ). Hurwitz zeta distribution, Aust. N. Z. J. Stat., 48 (2006), m | | 0 ( l=1 q(l) ) 1–6. [8] T. Aoyama and T. Nakamura, Multidimensional Shintani Thus, we have Rd zeta functions and zeta distributions on , Tokyo J. Math., 36 (2013), 521–538. ∑m ei2πT0θl − L [9] T. Aoyama and T. Nakamura, Behaviors of multivariable finite Euler products in probabilistic view, Math. Nachr., 286 l=1 ∑ (2013), 1691–1700. i2πT0θl i2πT0θl ≤ |q(l)||e − 1| + |q(l0)||e 0 + 1| < ϵ [10] T. Aoyama and T. Nakamura, Multidimensional polynomial Euler products and infinitely divisible distributions on Rd, l≠ l0 submitted, http://arxiv.org/abs/1204.4041. and that is ( ) [11] T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Graduate Texts in Mathematics 41, ∑m Re q(l)ei2πT0θl − L Springer-Verlag, New York, 1990. [12] T. Aoyama and K. Yoshikawa, Multinomial distributions in l=1 ( ) Shintani zeta class, preprint.

∑m ≤ i2πT0θl − q(l)e L < ϵ. l=1

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