Counts of Bernoulli success strings in a multivariate framework

Statistics and Probability Letters 119 (2016) 1–10

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Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro

Counts of Bernoulli success strings in a multivariate framework Djilali Ait Aoudia a , Éric Marchand b,∗ , François Perron c a

École des HEC, Département de sciences de la décision, Montréal, Qc, Canada

b

Université de Sherbrooke, Département de mathématiques, Sherbrooke, Qc, Canada

c

Université de Montréal, Département de mathématiques et de statistique, Montréal, Qc, Canada

article

abstract

info

Article history: Received 5 November 2015 Received in revised form 8 July 2016 Accepted 8 July 2016 Available online 16 July 2016

Distributional results are obtained for runs of length 2 in Bernoulli arrays Xk,j with multinomial distributed rows. These include a multivariate Poisson mixture representation with Dirichlet mixing for the joint distribution of the number of runs in each column. © 2016 Published by Elsevier B.V.

MSC: 60C05 60E05 62E15 Keywords: Arrays Binomial moments Dirichlet Poisson distribution Poisson mixtures Runs

1. Introduction In many studies of data, one is concerned with the occurrence or non-occurrence of a special event over time. In the counting of species, one wishes to know whether a new catch is from a new species; in the study of records, one wishes to assess the likelihood of a new record or not; in classification, one wishes to know whether a current observation is from a different population than past observations. Distributional properties relative to such Bernoulli sequences of 0’s and 1’s, as well as to runs of consecutive 1’s, have generated much interest over the years. Multivariate versions are also of interest, namely for situations with r dependent sequences and where a motivating question is to describe the nature of the dependence. Bernoulli arrays were introduced by Ait Aoudia and Marchand (2010) and consist of entries {Xk,j , k > 1, j = 1, . . . , r + 1}, where the rows X k = (Xk,1 , . . . , Xk,r +1 ) are independently distributed as Multinomial(1, pk,1 , . . . , pk,r +1 ), k > 1. Such arrays arise in sampling one object at a time from an urn with r + 1 colors, with replacement and reinforcement (i.e., adding balls as in a Polyà urn). Quantities of interest include the number of runs n Sn,j = X X k , j k + 1,j of length 2 in the jth column, their limits Sj = limn→∞ Sn,j , and the random vectors k=1 S n = (Sn,1 , . . . , Sn,r ),

∗

and S = lim S n . n→∞

Corresponding author. E-mail address: [email protected] (É. Marchand).

http://dx.doi.org/10.1016/j.spl.2016.07.009 0167-7152/© 2016 Published by Elsevier B.V.

(1.1)

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D.A. Aoudia et al. / Statistics and Probability Letters 119 (2016) 1–10

For the univariate case (r = 1), an elegant result consists of the Poisson mixture representation S1 |U ∼ Poisson(aU ) with U ∼ Beta(a, b),

for pk,1 =

a a+b+k−1

, a > 0, b > 0,

(1.2)

as obtained by Mori (2001), as well as Holst (2008). This class of solutions contains degenerate cases with b = 0, i.e., P(U = 1) = 1, and where S1 is Poisson distributed with mean parameter a. Moreover, the a = 1, b = 0 Poisson(1) solution was recognized in the mid 1990s by Persi Diaconis, as well as Hahlin (1995), and earlier versions are due to Arratia et al. (1992), Kolchin (1971), and Goncharov (1944). This elegant result has inspired much interest and lead to various findings relative to the distributions of Sn,j and Sj for various other configurations of the {pk,j }’s, as well as relationships and implications for Pólya urns, records, matching problems, marked Poisson processes, etc., as witnessed by the work of Chern et al. (2000), Csörgó and Wu (2000), Holst (2007, 2008), Huffer and Sethuraman (2012), Huffer et al. (2009), Joffe et al. (2004, 2000), Mori (2001) and Sethuraman and Sethuraman (2004), among others. With such Poisson distributions and Poisson mixtures arising naturally in these univariate (r = 1) situations, it seems natural to investigate multivariate versions r of such models. r The findings of Ait Aoudia et al. (2014) are contributions in this direction with analysis for the totals j=1 Sn,j and T = j=1 Sj , and representations for the distributions of S n and S in the bivariate case (r = 2). For instance, they obtain the following multivariate extension of (1.2):

 T |U ∼ Poisson

aU



r

with U ∼ Beta(a, b),

a

for pk,l =

r (a + b + k − 1)

, a > 0, b > 0,

(1.3)

which yields T ∼ Poisson(λ) for the degenerate case a = r λ, b = 0. The main findings here are presented in Section 3, are based on the assumption 1

pk,1 = · · · = pk,r =

b+k

,

with b > r − 1,

(1.4)

and include: (i) explicit expressions (Theorems 3.3 and 3.4) for the mixed binomial moments of Sn,1 , Sn,2 , . . . , Sn,r ; for all n, r; and of S1 , S2 , . . . , Sr , (ii) an explicit expression for the distribution function of S (Theorem 3.4), and (iii) two mixture representations (Theorem 3.5) for the distribution of S. As recalled in Section 2, mixed binomial moments yield expressions for probability mass and generating functions. The most intricate and challenging technical achievement is the binomial moment result of Theorem 3.3, which solves a recursive system of equations involving binomial moments related to the Sn,j ’s and auxiliary variables Wn,j , defined below in Section 3.1. Solutions to such a system of equations were provided by Ait Aoudia et al. (2014) in the bivariate case r = 2, but our result is unified and general for r > 2. The mixture representations are quite interesting. In the first mixture representation, the components Si of S are distributed as independent Poisson(Vi ) conditional on V1 , . . . , Vr which follows a Dirichlet distribution. This neatly summarizes the dependence structure of the Sj ’s, as well as replicating the marginal Beta mixture representation (1.2). The second mixture representation is quite different as the mixing parameter is univariate Beta and the dependence is reflected otherwise through a multivariate discrete distribution with Poisson and non-independent marginals, which connects for instance to the Schur-constant distributions of Castañer et al. (2015). 2. Multivariate Poisson mixtures and mixed binomial moments For non-negative integer variables Z1 , Z2 , . . . , Zr and constants ki ∈ N, i = 1, . . . , r, the mixed binomial  valued  random 

r

moments are defined as E

Zi ki

i =1

, where

z u

is taken to be equal to 0 for u > z. Our distributional findings involve

the determination of mixed binomial moments, which imply associated expressions for the probability generating and mass functions. Indeed, the Taylor series expansion about 1 of the probability generating function ψZ1 having radius of convergence greater than 1 can be expressed as

ψZ1 (t ) = E t 

Z1



=

  Z1  E

k

k>0

(t − 1)k ,

t ∈ [0, 1],

(2.1)

and the probability mass function (pmf) of Z can be written as

    ∞  k Z1 k −i P (Z1 = i) = (−1) E .

(2.2)

k

i

k=i

For the multivariate case, corresponding expressions are given by

 E

r  i=1

 Z ti i

 =

 k1 >0,...,kr >0

E

  r  r  Zi i=1

ki

j =1

(tj − 1)ki ,

(t1 , t2 , . . . , tr ) ∈ [0, 1]r ,

(2.3)

D.A. Aoudia et al. / Statistics and Probability Letters 119 (2016) 1–10

3

and

 P

r 

r 

 

{Zj = xj } =

(−1)



(ki −xi )

i=1

E

x1 6k1 ,...,xr 6kr

j =1

   r  r   Zi ki i=1

ki

i =1

xi

,

(2.4)

as long as the Taylor series expansion at (t1 , t2 , . . . , tr ) = (1, 1, . . . , 1) of the probability generating function converges on an open set containing the origin. Multivariate Poisson mixtures, defined as distributions of U = (U1 , . . . , Ur ), such that Ui |V ∼indep. Poisson(Vi ) and V = (V1 , . . . , Vr ) is a random vector on [0, ∞)r will play a key role below. For such mixtures, it is easy to verify that mixed binomial moments (assuming they exist) are given by

 E

 r   Ui i=1



 =E

ki

k r  V i

 ,

i

i=1

ki !

(2.5)

linking binomial moments of the Ui ’s with mixed moments of the Vi ’s. The specific mixing densities for V that arise below with our findings are Dirichlet(a1 , a2 , . . . , ar +1 ); ai > 0; densities given by

r +1   Γ ai  i=1 r +1



1−

Γ (ai )

r 

ar +1 −1 vi

r 

i=1

a −1

vj j

ID (v),

j =1

i=1

with D the r dimensional simplex defined by D = {(v1 , . . . , vr ) : v1 > 0, . . . , vr > 0, v1 + · · · + vr < 1}, and their limiting versions as ar +1 → 0 which we denote Dirichlet(a1 , . . . , ar , 0). For multivariate Poisson mixtures with such mixing densities, we will make use of the lemma below. Hereafter, we denote (γ )k and (γ )k as the descending and ascending k−1 k−1 factorial with (γ )0 = (γ )0 = 1, (γ )k = j=0 (γ − j), and (γ )k = j=0 (γ + j) for k = 1, 2, . . . . Lemma 2.1. Consider a multivariate Poisson mixture distribution Ui |V ∼indep. Poisson(Vi ) with mixing variable V ∼ Dirichlet (a1 , . . . , ar +1 ), and ai > 0, i = 1, . . . , r. Then, (a) U = (U1 , . . . , Ur ) has mixed binomial moments

 E

r   Ui i=1

Γ



ai

r  ( ai ) k

i=1

=

ki

 r +1  

Γ

 ar + 1 +

r 

i

(ai + ki )

 i=1

ki !

,

i =1

and, for the particular case with a1 = 1, . . . , ar = 1, ar +1 = b − r + 1, we have

 E

 r   Ui i=1

ki

 =

1

(1 + b) r

.

(2.6)

ki

i=1

(b) For cases where a = · · · = ar = 1 the distribution of U1 + · · · + Ur is: (i) a Poisson mixture with U1 + · · · + Ur |W ∼ Poisson(W ), W ∼ Beta(r , ar +1 ) for ar +1 > 0; and (ii) Poisson(1) for ar +1 = 0. Proof. (a) Expression (2.6) follows immediately from the general result, while the latter is obtained by integrating the rhs of (2.5) with respect to the Dirichlet density. (b) From the definition of the Ui ’s, we have that U1 + · · · + Ur |V ∼ Poisson(W ) with W = V1 + · · · + Vr . Result (i) follows as one obtains readily that W ∼ Beta(r , ar +1 ) for ar +1 > 0, while result (ii) follows since W is degenerate at 1 for ar +1 = 0.  3. Distributions of S n and S 3.1. Mixed binomial moments of S n We set S0,j = 0 for j = 1, . . . , r, and consider the auxiliary random vectors

(Sn−1,1 , Sn−1,2 , . . . , Sn−1,j−1 , Wn,j , Sn−1,j+1 , . . . , Sn−1,r ), where Wn,1 , . . . , Wn,r are such that Wn,j = Sn−1,j + Xn,j , n > 1,

j ∈ {1, . . . , r }.

(3.1)

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D.A. Aoudia et al. / Statistics and Probability Letters 119 (2016) 1–10

In Theorem 3.3, we determine explicitly, for 1 6 r 6 1 + b, n > 0, and k = (k1 , . . . , kr ) ∈ Nr , the mixed binomial moments

 g (n, r : k) = E

 r   Sn,j

h(n, r : i, k) = E

,

kj

j =1





Wn+1,i



ki

(3.2)





Sn,j



kj

j∈{1,...,r }\{i}

,

i ∈ {1, . . . , r }.

(3.3)

The method of proof is highlighted by a two-stage induction argument (on r and n). Before proceeding, we require the two following lemmas whose proofs are relegated to the Appendix. Lemma 3.1. Let 1 6 r 6 1 + b, n > 0 and k ∈ Nr . Let Pr = {π : π is a permutation on {1, . . . , r }}. For k ∈ Nr , π ∈ Pr , let k′ ∈ Nr and, for r > 1, k′′ ∈ Nr −1 be defined as k′j = kπ(j) , kj = kj , ′′

′

j = 1, . . . , r , j = 1, . . . , r − 1.

Then, we have (a) g (n, r : k) = g (n, r : k′ ), h(n, r : i, k) = h(n, r : j, k′ ),

when π (j) = i, i ∈ {1, . . . , r };

(b) If i ∈ {1, . . . , r } and ki = 0 then h(n, r : i, k) = g (n, r : k)

= g (n, r − 1 : k′′ ),

when r > 1, π (r ) = i.

(c) If {i, ℓ} ⊂ {1, . . . , r }, r > 1, kℓ = 0 and π (r ) = ℓ then h(n, r : i, k) = h(n, r − 1 : j, k′′ ),

when π (j) = i.

Lemma 3.2. Let 1 6 r 6 1 + b, k ∈ Nr with k1 + · · · + kr > 0. Let

σ (k) =

dim(k)



kj

and

d+ (k) =

j =1

dim(k)



1(kj > 0).

j =1

Then, the following hold: (a) (b) (c) (d)

g (n, r : h ( n, r : h ( n, r : h ( n, r :

k) = 0 if 0 6 n < σ (k) + d+ (k) − 1; i, k) = 0 if ki = 0, 0 6 n < σ (k) + d+ (k) − 1; i, k) = 0 if ki > 0, 06 n < σ (k) + d+ (k) − 2, i = 1, . . . , r; i, k) = (d+ (k) − 1)! (1 + b)σ (k)+d+ (k)−1 if ki > 0, n = σ (k) + d+ (k) − 2, i = 1, . . . , r.

Theorem 3.3. For g and h as defined in (3.2) and (3.3), under the assumption and setting pj = pj,1 = · · · = pj,r = 1/(b + j), j > 1 we have for all b > 0, 1 6 r 6 1 + b, k ∈ Nr , g (n, r : k) =

+ (k)

1

(n + 1 − σ (k)) d

(1 + b)σ (k)

(n + b + 1)d

+ (k)

,

(3.4)

if n > σ (k) + d+ (k) − 1 (and zero otherwise); and h(n, r : i, k) =

1

+ (k)−1(k >0) i

(n + 1 − σ (k))d

(1 + b)σ (k) (n + b + 1)d+ (k)−1(ki >0)

,

if n > σ (k) + d+ (k) − 1 − 1(ki > 0) (and zero otherwise); where σ (k) =

(3.5)

dim(k) j =1

kj and d+ (k) =

dim(k) j=1

1(kj > 0).

Proof. We use induction on r starting with r = 1. Eqs. (3.2)–(3.5) yield g (n, 1 : 0) = 1,

h(n, 1 : 1, 0) = 1,

n > 0.

(3.6)

D.A. Aoudia et al. / Statistics and Probability Letters 119 (2016) 1–10

5

Suppose for now that k1 > 1. With Lemma 3.2, we infer that Theorem 3.3 holds for 0 6 n 6 k1 − 1, r = 1. For n > k1 > 1, r = 1, we use induction on n, n > k1 − 1. Suppose that Theorem 3.3 is valid for n = k1 − 1, . . . , m − 1, m > k1 > 1, r = 1. Conditioning on X m+1 , we obtain g (m, 1 : k) = E



Sm,1



k1

 = pm+1 E



Wm,1 k1

+ (1 − pm+1 )E



Sm−1,1



k1

= pm+1 h(m − 1, 1 : 1, k) + (1 − pm+1 )g (m − 1, 1 : k) 1 (m − k1 ) 1 (m + b) 1 = + (m + b + 1) (1 + b)k1 (m + b + 1) (1 + b)k1 (m + b) (m + 1 − k1 ) 1 . = (1 + b)k1 (m + 1 + b) Similarly, setting κ = (κ1 , . . . , κr ) such that κ1 = k1 − 1 and κj = kj for j = 2, . . . , r, we obtain h(m, 1 : 1, k) = E



Wm+1,1 k1



  Sm−1,1 + (1 − pm+1 )E k1 k1       Wm,1 Wm,1 Sm−1,1 pm+1 E + + (1 − pm+1 )E k1 − 1 k1 k1 pm+1 [h(m − 1, 1 : 1, κ) + h(m − 1, 1 : 1, k)] + (1 − pm+1 )g (m − 1, 1 : k) pm+1 h(m − 1, 1 : 1, κ) + g (m, 1 : k) 1 1 (m + 1 − k1 ) 1 + (m + b + 1) (1 + b)k1 −1 (1 + b)k1 (m + 1 + b)

= pm+1 E = = = = =



1

(1 + b)k1

Wm,1 + 1



.

In the above development, notice that if k1 > 1, then h(m − 1, 1 : 1, κ) = 1/(1 + b)κ1 by the induction hypothesis while h(m − 1, 1 : 1, 0) = 1 by Eq. (3.6). With 1/(1 + b)0 = 1, we have indeed h(m − 1, 1 : 1, κ1 ) = 1/(1 + b)k1 −1 for all k1 > 1. Assume now that Theorem 3.3 is valid for r = 1, . . . , r0 − 1, r0 > 1, for all n > 0. Suppose that there exists j ∈ {1, . . . , r0 } such that kj = 0. Let π be a permutation on {1, . . . , r0 } such that π (r0 ) = j and consider k′ ∈ Nr0 and k′′ ∈ Nr0 −1 the new vectors given by k′ℓ = kπ(ℓ) , k′′ℓ

=

k′ℓ ,

ℓ = 1, . . . , r0 , ℓ = 1, . . . , r0 − 1.

Using Lemma 3.1, we have for n > 0: g (n, r0 : k) = g (n, r0 − 1 : k′′ ), h(n, r0 : j, k) = g (n, r0 − 1 : k′′ ), h(n, r0 : i, k) = h(n, r0 − 1 : ℓ, k′′ ),

i ̸= j, ℓ = π (i).

We also have

σ (k) = σ (k′′ ) and d+ (k) = d+ (k′′ ). Hence, with the induction hypothesis, we have our result whenever there exists j ∈ {1, . . . , r0 } such that kj = 0. Suppose now that d+ (k) = r0 and assume that Theorem 3.3 holds for r = 1, . . . , r0 − 1, r0 > 1, for all n > 0. Lemma 3.2 tells us that Theorem 3.3 is valid for r = r0 and 0 6 n < σ (k) + r0 − 1. It remains to prove that Theorem 3.3 holds for n > σ (k) + r0 − 1, r = r0 . This is done by induction on n, n > σ (k) + r0 − 2. Suppose that Theorem 3.3 is valid for n = σ (k) + r0 − 2, . . . , m − 1, m > σ (k) + r0 − 1, r = r0 . Let R = {1, . . . , r0 } R (i) = R \ {i},

κj =



kj − 1 kj

i ∈ R, if if

j = i, i, j ∈ R . j ̸= i,

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D.A. Aoudia et al. / Statistics and Probability Letters 119 (2016) 1–10

Please note that the κj ’s have been redefined here. Conditioning on X m+1 we obtain that

 g (m, r0 : k) = E

  Sm,j  kj

j∈R

= pm+1







E

Wm,i ki

i∈R

= pm+1



    Sm−1,j kj

j∈R(i)

 + (1 − r0 pm+1 )E

   Sm−1,j  kj

j∈R

h(m − 1, r0 : i, k) + (1 − r0 pm+1 )g (m − 1, r0 : k)

i∈R

= =

1

r0

(m + b + 1) (1 + b)σ (k) 1

(1 + b)σ (k)

(m − σ (k))r0 −1 1 (m − σ (k))r0 (m + b) + (m + b + 1) (1 + b)σ (k) (m + b)r0 (m + b)r0 −1

(m + 1 − σ (k))r0 . (m + b + 1)r0

(3.7)

Similarly, h(m, r0 : i, k) = E



Wm+1,i ki

= pm+1 E 

    Wm,i + 1 Sm−1,ℓ ki

 E

Wm,j

  = pm+1 E

+

E

Wm,j kj

  = pm+1 E

+

Wm,i ki − 1

 E

j∈R

kℓ

ℓ∈R(j)



Wm,i

j∈R(i)



    Sm−1,ℓ

ki − 1



kℓ

ℓ∈R(i)

kj

j∈R(i)



kℓ

ℓ∈R(i)

 

+

    Sm,ℓ

 +

+ (1 − r0 pm+1 )E

  Sm,ℓ  ℓ∈R



kℓ

    Wm,i Sm−1,ℓ ki

ℓ∈R(i)

   Sm−1,ℓ



kℓ

ℓ∈R(i)

kℓ

 + (1 − r0 pm+1 )E

  Sm,ℓ  ℓ∈R



kℓ

    Sm−1,ℓ kℓ

ℓ∈R(i)

    Wm,j Sm−1,ℓ kj



ℓ∈R(j)



= pm+1 h(m − 1, r0 : i, κ) +

kℓ



 + (1 − r0 pm+1 )E

  Sm,ℓ  ℓ∈R



kℓ



h(m − 1, r0 : j, k) + (1 − r0 pm+1 )g (m − 1, r0 : k)

j∈R

= pm+1 h(m − 1, r0 : i, κ) + g (m, r0 : k) (using (3.7)) 1 (m + 1 − σ (k))r0 −1  (b + σ (k)) (m + 2 − σ (k) − r0 )  + = (1 + b)σ (k) (m + b + 1)r0 −1 ( m + b + 2 − r0 ) (m + b + 2 − r0 ) =

1

(1 + b)σ (k)

(m + 1 − σ (k))r0 −1 . (m + b + 1)r0 −1

Notice that ki > 1 implies κj > 0 for j = 1, . . . , r0 , and the hypothesis of induction on n implies that

 E

Wm,i ki − 1

    Sm−1,ℓ ℓ∈R(i)

kℓ

= h(m − 1, r0 : i, κ) (m − σ (κ))r0 −1 (1 + b)σ (κ) (m + b)r0 −1 1 (m + 1 − σ (k))r0 −1 = . (1 + b)σ (k)−1 (m + b)r0 −1 =

1

D.A. Aoudia et al. / Statistics and Probability Letters 119 (2016) 1–10

7

Finally, when ki = 1, we have κi = 0, and a permutation π on R can be found such that π (r0 ) = i. With κ ′ ∈ Nr0 , κ ′′ ∈ Nr0 −1 such that

κℓ′ = κπ(ℓ) , κℓ′′ = κℓ′ ,

ℓ = 1, . . . , r0 , ℓ = 1, . . . , r0 − 1,

and the hypothesis of induction on r, we thus obtain

 E

Wm,i

    Sm−1,ℓ

ki − 1

ℓ∈R(i)

kℓ

= h(m − 1, r0 : i, k) = g (m − 1, r0 − 1 : κ ′′ ) 1 (m − σ (κ ′′ ))r0 −1 = (1 + b)σ (κ ′′ ) (m + b)r0 −1 (m + 1 − σ (k))r0 −1 1 = .  (1 + b)σ (k)−1 (m + b)r0 −1

3.2. Distribution of S and mixture representations As a consequence of Theorem 3.3, we obtain the following for the asymptotic case when n → ∞. Theorem 3.4. Under assumption (1.4), (a) The mixed binomial moments of S1 , S2 , . . . , Sr are given by

 E

 r   Sj j =1

kj

=

1

(1 + b) r

;

(3.8)

ki

i=1

(b) The probability mass function of S = (S1 , . . . , Sr ) is given by 1

P (S1 = s1 , . . . , Sr = sr ) = with t =

r

i=1 si

(1 + b)t

1 F1

(t + r ; t + b + 1; −1),

(3.9)

and 1 F1 the confluent hypergeometric function given by 1 F1 (γ1 ; γ2 ; z ) =

(c) For t ∈ N, the distribution of S | = t }.

r

i=1

(γ1 )j z j j=0 (γ2 )j j! ;

∞

Si = t is uniform on the simplex M (t ) = {v = (v1 , v2 , . . . , vr ) : vi > 0,

r

i =1

vi

Proof. Part (a) follows directly from Theorem 3.3 and Lebesgue’s monotone convergence theorem by taking n → ∞. Part (c) follows from part (b) since P (S1 = s1 , . . . , Sr = sr ) is a function of s1 + · · · + sr . Finally, for part (b), use (2.4) to obtain r 

P (S1 = s1 , . . . , Sr = sr ) = {(k1 ,...,kr

(ki −si )

(−1) (1 + b) r ) : k >s ,i=1,...,r } i=1

 i

i

ki

 r   ki i =1

si

i=1

∞  (−1)m−t = (1 + b)m m=t

 r   ki

  (k1 ,...,kr ) : ki >si ,i=1,...,r ,

r 



i=1

si

k i =m

i=1

  ∞  (−1)m−t m + r − 1 = (1 + b)m t + r − 1 m=t =

(−1)m (t + r )m , (1 + b)t (1 + b + t )m m! m=0 ∞ 

which is (3.9) and completes the proof except the of the next to last equality. This equality can be justified by justification r the following combinatorial argument. Let t = i=1 si and A = {1, 2, . . . , m + r − 1} with m > t. The number of ways to choose B ⊂ A with |B| = t + r − 1 may be re-expressed as follows. Order the elements of B from smallest to largest. Condition on the values of the (s1 + 1)th, (s1 + s2 + 2)th, . . . (s1 + s2 + · · · sr −1 + r − 1)th elements in B. These r − 1 values divide the other m values of A into r groups with sizes   k1 , k2 , . . . , kr satisfying ki > si for all i, and the remaining t elements of B may be selected from these r groups in

r

i =1

ki si

ways.



Our next result establishes two elegant mixture representations.

8

D.A. Aoudia et al. / Statistics and Probability Letters 119 (2016) 1–10

Theorem 3.5. Under assumption (1.4), (a) The distribution of S is a multivariate Poisson mixture with Si |V ∼indep. Poisson(Vi ) for i = 1, . . . , r, and V = (V1 , . . . , Vr ) ∼ Dirichlet (a1 = 1, . . . , ar = 1, ar +1 = b − r + 1); (b) The distribution of S admits the representation S |α ∼ pα ,

α ∼ Beta(1, b),

(3.10)

with pα the multivariate probability mass function on Nr given by

  r −1  r −1 j pα (s1 , . . . , sr ) = (−1) j

j=0

e−α α s1 +···+sr +j

(s1 + · · · + sr + j)!

,

α ∈ (0, 1].

(3.11)

Proof. (a) This  follows as a direct consequence of Theorem 3.4 and Lemma 2.1. (b) Set t = i si . In view of expression (3.9), it suffices to show that

(1 + b)t



1

b(1 − α)b−1 gt ,r (α) dα = 1 F1 (t + r ; t + b + 1; −1),

(3.12)

0

for all r = 1, 2, . . . , b > r − 1, t = 0, 1, . . . , and with gt ,r (α) = pα (s1 , . . . , sr ). We prove by induction on r and we make use of the integral representation

Γ (γ1 ) Γ (γ2 − γ1 ) 1 F1 (γ1 ; γ2 ; −1), Γ (γ2 )

(3.13)

γ2 1 F1 (γ1 + 1; γ2 ; −1) = γ2 1 F1 (γ1 ; γ2 ; −1) − 1 F1 (γ1 + 1; γ2 + 1; −1),

(3.14)

1



α γ1 −1 (1 − α)γ2 −γ1 −1 e−α dα =

0

as well as the recurrence relation both valid for γ2 > γ1 > 0 (e.g., NIST Digital Library of Mathematical Functions, 2016). For r = 1, we have

gt ,1 (α) = e t ! α and (3.12) is readily verified by using (3.13). Now, assume (3.12) holds for a given r0 and observe that gt ,r0 +1 (α) = gt ,r0 (α) − gt +1,r0 (α) by making use of Pascal’s identity and by re-arranging terms. We now obtain by exploiting this along with (3.14) −α

(1 + b)t

t



1

b(1 − α)b−1 gt ,r0 +1 (α) dα = 1 F1 (t + r0 ; t + b + 1; −1)

0

1 1 F1 (t + r0 + 1; t + b + 2; −1) t +b+1 = 1 F1 (t + r0 + 1; t + b + 1; −1),

−

which indeed establishes (3.12).



As a consequence of the above, we recover a Poisson mixture representation for the total number of runs of length 2, and as expressed in (1.3). Corollary 3.6. Under assumption (1.4), the total number of runs T = Otherwise for b > r − 1, T admits the mixture representation T |L ∼ Poisson(L),

r

i=1

Si of length 2 is Poisson(1) distributed when b = r −1.

L ∼ Beta(r , b − r + 1).

Proof. The result is a direct application of Theorem 3.5 and Lemma 2.1.



Remark 3.7. The previous result can also be obtained by directly using Theorem 3.4. For instance with b = r − 1, we have using (2.3)



r 

 Si

E t i=1  = E

 r 



 t

Si

= {(k1 ,...,kr

i =1

=

(t − 1) i (r ) ki ) : k >0,i=1,...,r }

ki

 i

∞  k=0

i

  (k1 ,...,kr ) : ki >0,i=1,...,r ,

r 

(t − 1)k  (r )k ki =k

i=1

=

∞   k=0 t −1

=e

r +k−1 r −1



(t − 1)k (r )k

,

which establishes the Poisson(1) distributional result for T when b = r − 1.

D.A. Aoudia et al. / Statistics and Probability Letters 119 (2016) 1–10

9

Corollary 3.6 is already known and due to Ait Aoudia et al. (2014), but it is presented here for sake of completeness and since our derivation differs in how the joint distribution of S1 , S2 , . . . , Sr is used. The other findings above, which we now further discuss, generalize Ait Aoudia et al.’s (2014) bivariate case (r = 2) results. Theorem 3.5’s multivariate Poisson mixture representation of S extends in a most interesting way the known marginal distribution representation for Si , i = 1, . . . , r, which is a Beta mixture of Poisson distributions. Here the components Si are clearly dependent, but the representation tells us that they are conditionally independent and the dependence is reflected through the dependence of the mixing components of the Dirichlet. In contrast to the Dirichlet mixture, the dependence in representation (b) of Theorem 3.5 is reflected through the conditional distributions of S, and the mixing variable α is univariate. Furthermore, it is readily verified that the conditional marginal distributions of Si |α are Poisson (α), which is consistent with the univariate Beta mixture result. Remark 3.8. The joint probability mass functions (pmf) pα in (3.11), which has Poisson(α) marginals, are examples of the larger class of pmf’s of the form

γ (x1 , . . . , xr ) =

 r −1   r −1 j =0

j

 (−1)j p

 

xi + j ,

x1 , . . . , xr ∈ N,

(3.15)

i

which were analyzed in the bivariate case (r = 2) and otherwise referred to by Ait Aoudia and Marchand (2014). In the above  form,p is a univariate pmf, γ has marginals given by p, and γ is a valid pmf as long as p is (r − 1)-monotone, i.e.,

r −1 j =0

r −1 j

(−1)j p(m + j) > 0 for all m > 0. These conditions indeed apply for p ∼ Poisson(α) with α 6 1 leading to γ

as in (3.11). For this and further properties of such distributions, we refer to Castañer et al. (2015). Concluding remarks Our findings involve the number of runs of length 2 in Bernoulli arrays with independent multinomial distributed rows and identically distributed row components. The multivariate Poisson–Dirichlet mixture representation for the distribution of S (Theorem 3.5) is a natural extension of a corresponding Poisson-Beta mixture representation (i.e., result (1.2) for a = 1). The key technique distinguishing the results here from previous work is related to the double-induction argument of Theorem 3.3 used in deriving the binomial moments. We believe the argument offers promise, subject to further technical challenges, to analyze the distributions of S n and S for other configurations of the probabilities pk,j . Finally, one wonders whether an alternate construction can be derived for the multivariate case based on other approaches that have been used in the univariate case. An example consists of the univariate Bernoulli randomly stopped sequence approach and related mixture of Poisson representation due to Huffer and Sethuraman (2012). Such a construction would avoid the Binomial moment computation and, more importantly, may provide an understanding of why the mixture representation arises. Acknowledgments We are indebted and thankful to two anonymous reviewers for quite useful and constructive comments. Éric Marchand and François Perron gratefully acknowledge the support for research provided by NSERC of Canada. Finally, we are grateful to Chris Jones for useful discussions and insight on the distributions represented by (3.15). Appendix Proof of Lemma 3.1. Part (a) follows because the columns of the matrix with entries {Xk,j : k > 1, j = 1, . . . , r } are exchangeable random variables. Parts (b) and (c) are readily established and follow from definitions. Proof of Lemma 3.2. Let Mr+ (k) = {j ∈ {1, . . . , r } : kj > 0}. (a) For g (n, r : k) to be positive, we require Sn,j > kj for all j ∈ Mr+ (k) and, hence, m=1 Xm,j > kj + 1 for all j ∈ Mr+ (k). With this last condition requiring n + 1 > σ (k) + d+ (k), the result follows. (b) Given part (a) above, this follows from part (b) of Lemma 3.1. n+1 n+1 (c) If ki > 0, Wn+1,i > ki , Sn,j > kj for all j ∈ Mr+ (k) \ {i}, then m=1 Xm,i > ki and m=1 Xm,j > kj + 1 for all j ∈ Mr+ (k) \ {i}. The property thus follows as in (a). (d) If ki > 0, n = σ (k) + d+ (k) − 2 then

n+1



Wn+1,i ki



> 0,



Sn,j kj



> 0,

j ∈ Mr+ \ {i},

if and only if Wn+1,i = ki and Sn,j = kj , j ∈ Mr+ \ {i}. Moreover Wn+1,i = ki and Sn,j = kj , j ∈ Mr+ (k) \ {i}, if and only if Xℓi = 1,

n + 2 − ki 6 ℓ 6 n + 1,

Xℓj = 1,

for kj + 1 consecutive values of ℓ ∈ {1, . . . , n + 1 − ki },

(A.16)

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D.A. Aoudia et al. / Statistics and Probability Letters 119 (2016) 1–10

j ∈ Mr+ (k) \ {i}. There are (d+ (k) − 1)! ways to select the values of (ℓ, j) ∈ {1, . . . , n + 1} × Mr+ (k) \ {i} such that (A.16) is satisfied. We thus obtain hn,r (i : k1 , . . . , kr ) = E [Y ],

Y ∼ Bernoulli (d+ (k) − 1)! (1 + b)σ (k)+d+ (k)−1 . 







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