Pliska Stud. Math. Bulgar. ….. (….),

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STUDIA MATHEMATICA
BULGARICA

MULTIVARIATE COMPOUNDS WITH EQUAL NUMBER OF SUMMANDS

Pavlina K. Jordanova 111The author is partially supported by Project
RD-08-69/02.02.2016 from the Scientific Research Fund in Konstantin Preslavsky
University of Shumen, Bulgaria.

###### Abstract

The paper considers multivariate discrete random sums with equal number of
summands. Such distributions describe the total claim amount received by a
company in a fixed time point. In Queuing theory they characterize cumulative
waiting times of customers up to time t. In Theory of branching processes they
model the number of heirs at a fixed point in time.

Here some general properties and formulae for numerical characteristics of
these distributions are derived and some particular cases are considered.

†† $2000$ Mathematics Subject Classification: 62E15 Key words: Compound
distributions; Multinomial distribution; Negative multinomial distribution

##  1\. Introduction

Univariate random sums, are considered intensively in the beginning of the
last century. Abraham Wald [17] found the relation between the numerical
characteristics of these random sums, their summands and their number of
summands. Nowadays his results are well known as ”Wald equalities”.
Investigation of the particular cases starts with Poisson distributed number
of summands and geometrically distributed on positive integers summands. This
case is related with the names of Alfred Aeppli (1924) and George
P$\acute{o}$lya (1930). An exploration of univariate compound Poisson
processes with their applications in insurance can be seen e.g. in the works
of O. Lundberg [10] (1940).

Here we consider the corresponding general multivariate distributions. Along
the paper ”$\stackrel{{\scriptstyle\rm d}}{{=}}$” stands for the equality in
distribution of two random variables (r.vs.), ”$\equiv$” for coincidence of
two classes of probability laws, $FI\,X:=\frac{Var\,X}{E\,X}$ is the Fisher
index of dispersion, $CV(X,Y)=\sqrt{\frac{cov(X,Y)}{EX\,EY}}$, and
$CV\,X=\frac{\sqrt{Var\,X}}{E\,X}$ is the coefficient of variation of the
distribution of the random variable (r.v.) $X$. The sign ”$\sim$” expresses
the fact, that the distribution of the corresponding r.v. belongs to a given
class of probability laws.

Let $N$ be a non-negative, integer valued r.v. with probability generating
function (p.g.f.) $G_{N}$. Suppose $(Y_{1i},Y_{2i},...,Y_{ki})$, $i=1,2,...$
are independent identically distributed (i.i.d.) random vectors, defined on
the same probability space $(\Omega,\mathcal{A},\mathrm{P})$. Denote their
p.g.f. by $G_{Y_{1i},Y_{2i},...,Y_{ki}}$, $i=1,2,...$. The random vector
$(X_{1N},X_{2N},...,X_{kN})$, which coordinates are defined by

(1) $X_{sN}=I_{N>0}\sum_{i=1}^{N}Y_{si},\quad s=1,2,...,k$

is called compound $N$($N$-stopped sum, random sum) with equal number of $Y$
summands. In any particular case, the letters $N$ and $Y$, will be replaced by
the name of the corresponding distribution. Briefly we will denote this in the
following way

$(X_{1N},X_{2N},...,X_{kN})\sim C\,N\,Y(\vec{a}_{N};\vec{b}_{Y}).$

Here the letters $C\,N\,Y(\vec{a}_{N};\vec{b}_{Y})$ have the following
meaning: $C$ comes from ”compound”, $N$ is the abbreviation of the
distribution of the number of summands, with parameters $\vec{a}_{N}$ and $Y$
is the abbreviation of the distribution of the vector of summands, with
parameters $\vec{b}_{Y}$.

The cases when $N$ is Poisson distributed and $(Y_{1i},Y_{2i},...,Y_{ki})$ are
one of the

  * •

Multinomial(Mn),

  * •

Negative Multinomial(NMn),

  * •

Multivariate Geometric(MGe),

  * •

Multivariate Poisson (MPo)

distributed are investigated in 1962, by G. Smith [14]. The particular case,
when the summands are MGe on positive integers and the number of summands are
Poisson, is investigated in series of papers of Minkova et. al, see e.g. [11]
and [12]. In 1971, Khatri [9] mentioned that the class of these distributions
is too wide and investigates some multivariate particular cases. In 1981
Charalambides [2], together with Papageorgiou [3] explored bivariate case.
Charalambides [2] has called all distributions with p.g.f.
$G_{N}(G_{Y_{11},Y_{21}}(z_{1},z_{2}))$ ”generalized”, and has mentioned that
many bivariate contagious and compound distributions belong to this class. In
1996 Wang [18], obtained multivariate compound Poisson distributions as the
limiting distributions of multivariate sums of independent random vectors. He
called the distribution, defined in (1), ”multivariate compound distribution
of type I”. The covariances formula for the general case of these
distributions

(2) $cov(X_{iN},X_{jN})=ENcov(Y_{i1},Y_{j1})+EY_{i1}EY_{j1}VarN$

can be found in Sundt et. al [16]. In series of papers they consider
recursions formulae for these distributions.

This paper develops the results in Jordanova [8]. Their proves are presented
and new explicit relations between the probability mass functions (p.m.fs.)
and the mean square regressions in particular cases are obtained. After the
characterisation, some simulations of realizations of random vectors with such
distributions are made, and the dependence between their coordinates is
visualised. These observations show that these distributions are appropriate
for modelling of both: linear dependence between the coordinates and
clustering in the observations. They can serve to overcome Simpson’s paradox,
described e.g. in Colin [4]. The main machinery of probability generating
functions (p.g.fs.) used here can be seen e.g. in the book of Johnson et. al
[7].

In the next section we investigate the general properties of these
distributions without specifying the probability laws of the summands in (1),
or those of the number of summands. Then, in Section 3, the particular cases
of Mn and NMn summands are considered. The paper finishes with some conclusive
remarks.

##  2\. General properties of multivariate compound distributions

In this section we investigate the main properties of the distribution of the
multivariate random sums, defined in (1), together with their numerical
characteristics and conditional distributions.

###### Theorem 1

Let $X_{1N},X_{2N},...,X_{kN}$ be a random vector, defined in (1).

Its distribution possesses the following properties.

  1. 1.

Characterisation of the joint probability distribution.

If the p.g.fs. exist in $(z_{1},z_{2},...,z_{k})$, then

(3)
$G_{X_{1N},X_{2N},...,X_{kN}}(z_{1},z_{2},...,z_{k})=G_{N}(G_{Y_{11},Y_{21},...,Y_{k1}}(z_{1},z_{2},...,z_{k})).$

Its probability mass function (p.m.f.) satisfies the following equalities

$P(X_{1N}=0,X_{2N}=0,...,X_{kN}=0)=G_{N}(G_{Y_{11},Y_{21},...,Y_{k1}}(0,0,...,0))=$

(4) $=G_{N}(P(Y_{11}=0,Y_{21}=0,...,Y_{k1}=0)).$

If the following derivatives exist, then for $s=1,2,..,k$, $x_{s}=0,1,...$:
$(x_{1},x_{2},...,x_{k})\not=(0,0,...,0)$,

$P(X_{1N}=x_{1},X_{2N}=x_{2},...,X_{kN}=x_{k})=$

(5)
$=\frac{\partial^{x_{1}+x_{2}+...+x_{k}}G_{N}(G_{Y_{11},Y_{21},...,Y_{k1}}(z_{1},z_{2},...,z_{k}))}{x_{1}!x_{2}!...x_{k}!\partial
z_{1}^{x_{1}}\partial z_{2}^{x_{2}}...\partial
z_{k}^{x_{k}}}|_{(z_{1},z_{2},...,z_{k})=(0,0,...,0)}=$ (6)
$=\sum_{n=1}^{\infty}P(N=n)\sum_{i_{11}+i_{12}+...+i_{1n}=x_{1}}\sum_{i_{21}+i_{22}+...+i_{2n}=x_{2}}...\sum_{i_{k1}+i_{k2}+...+i_{kn}=x_{k}}$
$\prod_{s=1}^{n}P(Y_{1s}=i_{1s},Y_{2s}=i_{2s},...,Y_{ks}=i_{ks}).$

  2. 2.

Characterisation of subsets of the coordinates.

For all $r=1,2,...,k$ and for all subset of coordinates
$X_{i_{1}N},X_{i_{2}N},...,X_{i_{r}N}$,

(7)
$G_{X_{i_{1}N},X_{i_{2}N},...,X_{i_{r}N}}(z_{i_{1}},z_{i_{2}},...,z_{i_{r}})=G_{N}(G_{Y_{11},Y_{21},...,Y_{k1}}(\vec{z}_{i\overline{i}})),$

where the vector $\overrightarrow{z}_{i\overline{i}}$ has coordinates equal to
$1$ on the places different from $i_{1},i_{2},...,i_{r}$ and correspondingly
coordinates $z_{i_{1}},z_{i_{2}},...,z_{i_{r}}$ on the other places.

$P(X_{i_{1}N}=x_{i_{1}},X_{i_{2}N}=x_{i_{2}},...,X_{i_{r}N}=x_{i_{r}})=$

(8)
$=\frac{\partial^{x_{i_{1}}+x_{i_{2}}+...+x_{i_{r}}}G_{N}(G_{Y_{11},Y_{21},...,Y_{k1}}(\vec{z}_{i\overline{i}}))}{x_{i_{1}}!x_{i_{2}}!...x_{i_{r}}!\partial
z_{i_{1}}^{x_{i_{1}}}\partial z_{i_{2}}^{x_{i_{2}}}...\partial
z_{i_{r}}^{x_{i_{r}}}}|_{(z_{i_{1}},z_{i_{2}},...,z_{i_{r}})=(0,0,...,0)}.$

  3. 3.

Marginal distributions and their dependence on $N$. For $i=1,2,...,k$,

(9)
$G_{X_{iN}}(z_{i})=G_{N}(G_{Y_{11},Y_{21},...,Y_{k1}}(1,1,...,1,z_{i},1,...,1)),$

$P(X_{iN}=0)=G_{N}(P(Y_{i1}=0)).$

For $x_{i}\not=0$, $P(X_{iN}=x_{i})=$

(10)
$=\sum_{n=1}^{\infty}P(N=n)\sum_{j_{1}+j_{2}+...+j_{n}=x_{i}}P(Y_{i1}=j_{1})P(Y_{i1}=j_{2})...P(Y_{i1}=j_{n}).$

If $E\,Y_{i_{1}}<\infty$ and $E\,N<\infty$, then

(11) $E\,X_{i_{N}}=E\,N\,E\,Y_{i_{1}},\quad
E\,(X_{i_{N}}|N=n)=n\,E\,Y_{i_{1}}.$

If $E\,(Y_{i1}^{2})<\infty$ and $E\,(N^{2})<\infty$, then

(12)
$Var\,X_{iN}=Var\,N\,(E\,Y_{i_{1}})^{2}+E\,NVar\,Y_{i_{1}},\,FI\,X_{iN}=FI\,N\,E\,Y_{i_{1}}+FI\,Y_{i_{1}}$
(13) $cov(X_{i_{N}},N)=EY_{i_{1}}\,Var\,N,$

$cor(X_{i_{N}},N)=$

(14)
$=E\,Y_{i_{1}}\sqrt{\frac{Var\,N}{Var\,N\,(E\,Y_{i_{1}})^{2}+E\,NVar\,Y_{i_{1}}}}=\sqrt{\frac{FI\,N}{FI\,N+(CV\,Y_{i_{1}})^{2}}}.$

  4. 4.

Conditional distributions.

(15)
$P(X_{iN}=x_{i}|X_{jN}=x_{j})=\frac{1}{x_{i}!}\frac{\frac{\partial^{x_{i}+x_{j}}G_{N}(G_{Y_{11},Y_{21},...,Y_{k1}}(\vec{z}_{ij\overline{ij}}))}{\partial
z_{i}^{x_{i}}\partial
z_{j}^{x_{j}}}|_{(z_{i},z_{j})=(0,0)}}{\frac{\partial^{x_{j}}G_{N}(G_{Y_{11},Y_{21},...,Y_{k1}}(1,...,1,z_{j},1,...,1))}{\partial
z_{i}^{x_{i}}}|_{z_{j}=0}},$

where the vector $\vec{z}_{ij\overline{ij}}$ has coordinates equal to $1$ on
the places different from $i$, $j$ and correspondingly coordinates
$z_{i},z_{j}$, on the places $i$ and $j$.

(16)
$G_{X_{iN}}(z_{i}|X_{jN}=x_{j})=\frac{\frac{\partial^{x_{j}}G_{N}(G_{Y_{11},Y_{21},...,Y_{k1}}(\vec{z}_{ij\overline{ij}}))}{\partial
z_{j}^{x_{j}}}}{\frac{\partial^{x_{j}}G_{N}(G_{Y_{11},Y_{21},...,Y_{k1}}(1,...,1,z_{j},1,...,1))}{\partial
z_{j}^{x_{j}}}}|_{z_{j}=0},$ (17)
$E(X_{iN}|X_{jN}=x_{j})=\frac{\frac{\partial}{\partial
z_{i}}\left[\frac{\partial^{x_{j}}G_{N}(G_{Y_{11},Y_{21},...,Y_{k1}}(\vec{z}_{ij\overline{ij}}))}{\partial
z_{j}^{x_{j}}}|_{z_{j}=0}\right]|_{z_{i}=1}}{\frac{\partial^{x_{j}}G_{N}(G_{Y_{11},Y_{21},...,Y_{k1}}(1,...,1,z_{j},1,...,1))}{\partial
z_{j}^{x_{j}}}|_{z_{j}=0}},$

  5. 5.

Dependence between the coordinates.

$cor(X_{iN},X_{jN})=$

(18)
$=\frac{EN\,cov(Y_{i1},Y_{i2})+EY_{i1}EY_{j1}Var\,N}{\sqrt{[Var\,N\,(E\,Y_{i_{1}})^{2}+E\,NVar\,Y_{i_{1}}][Var\,N\,(E\,Y_{j_{1}})^{2}+E\,NVar\,Y_{j_{1}}]}}=$
$=\frac{[CV\,(Y_{i1},Y_{i2})]^{2}+FI\,N}{\sqrt{[(CV\,Y_{i1})^{2}+FI\,N][(CV\,Y_{j1})^{2}+FI\,N]}}.$

  6. 6.

Distributions of the sum of the coordinates.

(19)
$G_{X_{1N}+X_{2N}+...+X_{kN}}(z)=G_{N}(G_{Y_{11},Y_{21},...,Y_{k1}}(z,z,...,z)),$
(20)
$P(X_{1N}+X_{2N}+...+X_{kN}=s)=\frac{\partial^{s}G_{N}(G_{Y_{11},Y_{21},...,Y_{k1}}(z,z,...,z))}{s!\partial
z^{s}}|_{z=0}.$

  7. 7.

Marginal distributions, given the sum of the coordinates. For $m=0,1,...,s$,

$P(X_{iN}=m|X_{1N}+X_{2N}+...+X_{kN}=s)=$

(21) $=\left(\begin{array}[]{c}s\\\ m\\\
\end{array}\right)\frac{\frac{\partial^{s}G_{N}(G_{Y_{11},...,Y_{i-11},Y_{i1},Y_{i+11},...,Y_{k1}}(z,...,z,z_{i},z,...,z))}{\partial
z^{s-m}\partial
z_{i}^{m}}|_{(z,z_{i})=(0,0)}}{\frac{\partial^{s}G_{N}(G_{Y_{11},...,Y_{k1}}(z,...,z))}{\partial
z^{s}}|_{z=0}},$

$G_{X_{iN}}(z_{i}|X_{1N}+X_{2N}+...+X_{kN}=s)=$

$=\left(\begin{array}[]{c}s\\\ m\\\
\end{array}\right)\frac{\sum_{m=0}^{s}z_{i}^{m}\frac{\partial^{s}G_{N}(G_{Y_{11},...,Y_{i-11},Y_{i1},Y_{i+11},...,Y_{k1}}(z,...,z,z_{i},z,...,z))}{\partial
z^{s-m}\partial
z_{i}^{m}}|_{(z,z_{i})=(0,0)}}{\frac{\partial^{s}G_{N}(G_{Y_{11},...,Y_{k1}}(z,...,z))}{\partial
z^{s}}|_{z=0}}.$

P r o o f.

  1. 1.

Equality (3) follows by the definition of p.g.f. and the Fubini’s theorem, in
particular also known as a ”Double expectations formula”. Double application
of equality (3) for $(z_{1},z_{2},...,z_{k})=(0,0,...,0)$, (see e.g. (34.2) in
[7]) leads us to (4). The main properties of p.g.f., see e.g. p.327 in [6],
together with (3) entails (5). The total probability formula leads us to (6).

  2. 2.

The property (7) follows by (3) and the definition of p.g.f. See e.g. p. 3 in
Johnson et. al [7]. The proof of the (8) is analogous to (5).

  3. 3.

(9) and $P(X_{iN}=0)=G_{N}(P(Y_{i1}=0))$ are particular cases, correspondingly
of (7) and (8), applied for $k=1$. Equality (10) is a consequence of the Total
probability formula. (11) and (12) are Wald’s equalities, applied for the
coordinates in our model. Formulae for double expectation and covariance
entail (13). (14) is a consequence of the formula for correlation, (13) and
(12).

  4. 4.

Definition about conditional probability and (8), entail (15). Replacement of
(8) and (9) in formula (34.48), Johnson et. al [7] gives (16). One of the main
properties of p.g.fs. is that its first derivative in $1$ is equal to the
expectation of the r.v. ((34.15), Johnson et. al [7]). Its application,
together with (16) gives (17).

  5. 5.

Formula for correlation, together with (2) and (12) prove (18).

  6. 6.

Following e.g. p.327 ([6]) we apply (3) for $z_{1}=z_{2}=...=z_{k}=z$ and
obtain (19). The main properties of p.g.f., see e.g. p.327 in [6], together
with (19) entails (20).

  7. 7.

Let $m=0,1,...,s$.

$P(X_{iN}=m,X_{1N}+...+X_{i-1N}+X_{i+1N}...+X_{kN}=j)=$

$=\frac{\partial^{m+j}G_{X_{iN},X_{1N}+...+X_{i-1N}+X_{i+1N}...+X_{kN}}(z_{i},z)}{m!j!\partial
z_{i}^{m}\partial z^{j}}|_{(z_{i},z)=(0,0)}=$
$=\frac{\partial^{m+j}G_{N}(G_{Y_{11},...,Y_{i-11},Y_{i1},Y_{i+11},...,Y_{k1}}(z,...,z,z_{i},z,...,z))}{m!j!\partial
z_{i}^{m}\partial z^{j}}|_{(z_{i},z)=(0,0)}.$

The last equality, together with the definition for conditional probability
and (20), entail

$P(X_{iN}=m|X_{1N}+X_{2N}+...+X_{kN}=s)=$

$=\frac{P(X_{iN}=m,X_{1N}+...+X_{i-1N}+X_{i+1N}...+X_{kN}=s-m)}{P(X_{1N}+...+X_{i-1N}+X_{i+1N}...+X_{kN}=s-m)}=$

$=\left(\begin{array}[]{c}s\\\ m\\\
\end{array}\right)\frac{\frac{\partial^{s}G_{N}(G_{Y_{11},...,Y_{i-11},Y_{i1},Y_{i+11},...,Y_{k1}}(z,...,z,z_{i},z,...,z))}{\partial
z^{s-m}\partial
z_{i}^{m}}|_{(z,z_{i})=(0,0)}}{\frac{\partial^{s}G_{N}(G_{Y_{11},...,Y_{k1}}(z,...,z))}{\partial
z^{s}}|_{z=0}}.$

□

Note: 1\. The second formula in (12) shows that, the bigger the $FI\,N$, or
$E\,Y_{i_{1}}$, or $FI\,Y_{i_{1}}$ are, the bigger the $FI\,X_{iN}$ is.

2\. The strength of the linear dependence between $X_{iN}$ and $N$, expressed
in (14), decreases with the increments of $FI\,N$ or $CV\,Y_{i1}$.

##  3\. Particular cases

Here we apply the results from Section 2 in some particular cases of compounds
with equal number of Mn or NMn distributed summands. We make a brief empirical
study of their mean square regressions and observe that in any of these
multivariate random sums it is close to linear.

###  3.1. Multivariate compounds with Multinomial summands.

Consider a partition $A_{1},A_{2},...,A_{k}$ of the sample space $\Omega$. Mn
distribution is the one of the numbers of outcomes $A_{i}$, $i=1,2,...,k$ in a
series of $n$ independent repetitions of a trial. More precisely, let
$n\in\mathbb{N}$, $0<p_{i}$, $i=1,2,...,k$ and $p_{1}+p_{2}+...+p_{k}=1$. A
vector $(\xi_{1},\xi_{2},...,\xi_{k})$ is called Multinomially distributed
with parameters $n,p_{1},p_{2},...,p_{k}$, if its probability mass function
(p.m.f.) is

$P(\xi_{1}=i_{1},\xi_{2}=i_{2},...,\xi_{k}=i_{k})=\left(\begin{array}[]{c}n\\\
i_{1},i_{2},...,i_{k}\end{array}\right)p_{1}^{i_{1}}p_{2}^{i_{2}}...p_{k}^{i_{k}},$

for $i_{1}+i_{2}+...+i_{k}=n,\,\,i_{s}=0,1,...,\,\,s=1,2,...,k.$ Briefly

$(\xi_{1},\xi_{2},...,\xi_{k})\sim Mn(n;p_{1},p_{2},...,p_{k}).$

The distribution of any subset $\xi_{i_{1}},\xi_{i_{2}},...,\xi_{i_{r}}$ of
its coordinates, can be easily described using relation that

(22)
$(\xi_{i_{1}},\xi_{i_{2}},...,\xi_{i_{r}},n-(\xi_{i_{1}}+\xi_{i_{2}}+...+\xi_{i_{r}}))\sim$

$\sim
Mn(n;p_{i_{1}},p_{i_{2}},...,p_{i_{r}},1-(p_{i_{1}}+p_{i_{2}}+...+p_{i_{r}})).$

A systematic investigation of this distribution, together with a very good
list of references, could be found e.g. in Johnson et al. [7]. Considering
this distribution for $k=1$, we obtain Binomial(Bi) distribution. For $n=1$
the Mn distribution is multivariate Bernoulli distribution. Some of the
following results, for the particular case when $N$ is Poisson distributed,
can be found in Smith [14]. This distribution is partially investigated on in
2002, Daley et. al [5], p. 113.

###### Theorem 2

Suppose $(Y_{1i},Y_{2i},...,Y_{ki})$, $i=1,2,...$ are independent Mn
distributed random vectors with parameters $(s,p_{1},p_{2},...,p_{k})$, where
$s\in\mathbb{N}$, $p_{1}>0,p_{2}>0,...,p_{k}>0$ and $p_{1}+p_{2}+...+p_{k}=1.$
Denote the resulting distribution, defined in (1) by
$(X_{1N},X_{2N},...,X_{kN})\sim
C\,N\,Mn(\vec{a}_{N};s,p_{1},p_{2},...,p_{k}),$ where the vector $\vec{a}_{N}$
contains the parameters of the distribution of the r.v. $N$.
$(X_{1N},X_{2N},...,X_{kN})$ possesses the following properties.

  1. 1.

$G_{X_{1N},X_{2N},...,X_{kN}}(z_{1},z_{2},...,z_{k})=G_{sN}(p_{1}z_{1}+p_{2}z_{2}+...+p_{k}z_{k}).$

  2. 2.

For $x_{1},x_{2},...,x_{k}=0,1,...$, such that $x_{1}+x_{2}+...+x_{k}$ is a
multiple of $s$, $P(X_{1N}=x_{1},X_{2N}=x_{2},...,X_{kN}=x_{k})=$

(23) $=\left(\begin{array}[]{c}x_{1}+x_{2}+...+x_{k}\\\
x_{1},x_{2},...,x_{k}\\\
\end{array}\right)p_{1}^{x_{1}}p_{2}^{x_{2}}...p_{k}^{x_{k}}P(Ns=x_{1}+x_{2}+...+x_{k}).$

$P(X_{1N}=0,X_{2N}=0,...,X_{kN}=0)=P(N=0).$

  3. 3.

For all $r=1,2,...,k$ and for any subset of coordinates
$X_{i_{1}N},X_{i_{2}N},...,X_{i_{r}N}$,

$G_{X_{i_{1}N},X_{i_{2}N},...,X_{i_{r}N}}(z_{i_{1}},z_{i_{2}},...,z_{i_{r}})=$

(24)
$=G_{sN}(p_{i_{1}}z_{i_{1}}+p_{i_{2}}z_{i_{2}}+...+p_{i_{r}}z_{i_{r}}+1-(p_{i_{1}}+...+p_{i_{r}})).$

$P(X_{i_{1}N}=x_{i_{1}},X_{i_{2}N}=x_{i_{2}},...,X_{i_{r}N}=x_{i_{r}})=$

(25)
$=\frac{p_{i_{1}}^{x_{i_{1}}}...p_{i_{r}}^{x_{i_{r}}}}{x_{i_{1}}!x_{i_{2}}!...x_{i_{r}}!}\sum_{m=0}^{\infty}\frac{(m+x_{i_{1}}+x_{i_{2}}+...+x_{i_{r}})!}{m!}.$

$.(1-(p_{i_{1}}+...+p_{i_{r}}))^{m}P[(sN=m+x_{i_{1}}+x_{i_{2}}+...+x_{i_{r}}),$

$(X_{i_{1}N},X_{i_{2}N},...,X_{i_{r}N},Ns-(X_{i_{1}N}+X_{i_{2}N}+...+X_{i_{r}N}))\sim$

$\sim
C\,N\,Mn(\vec{a}_{N};s,p_{i_{1}},p_{i_{2}},...,p_{i_{r}},1-(p_{i_{1}}+...+p_{i_{r}})).$

  4. 4.

For all $i=1,2,...,k$, and $m=0,1,...$,

(26)
$P(X_{iN}=m)=\frac{p_{i}^{m}}{m!}\sum_{j=0}^{\infty}\frac{(j+m)!}{j!}(1-p_{i})^{j}P(sN=j+m)=$
(27) $=\left\\{\begin{array}[]{ccc}G_{sN}(1-p_{i})&,&m=0\\\
\sum_{n=1}^{\infty}\left(\begin{array}[]{c}ns\\\ m\\\
\end{array}\right)p_{i}^{m}(1-p_{i})^{ns-m}P(N=n)&,&m=1,2,...\end{array}\right.$

If $E\,N<\infty$, $E\,X_{i_{N}}=sp_{i}E\,N.$

If $E\,(N^{2})<\infty$, then $Var\,X_{iN}=sp_{i}[sp_{i}Var\,N+(1-p_{i})E\,N]$
and

(28) $FI\,X_{iN}=1+p_{i}(sFI\,N-1).$

  5. 5.

$P(X_{iN}=x_{i}|X_{jN}=x_{j})=$

(29)
$=\frac{p_{i}^{x_{i}}}{x_{i}!}\frac{\sum_{m=0}^{\infty}\frac{(m+x_{i}+x_{j})!}{m!}(1-(p_{i}+p_{j}))^{m}P(sN=m+x_{i}+x_{j})}{\sum_{m=0}^{\infty}\frac{(x_{j}+m)!}{m!}(1-p_{j})^{m}P(sN=x_{j}+m)},$

$G_{X_{iN}}(z_{i}|X_{jN}=x_{j})=$

(30)
$=\frac{\sum_{m=0}^{\infty}\frac{(x_{j}+m)!}{m!}(p_{i}z_{i}+1-p_{i}-p_{j})^{m}P(sN=x_{j}+m)}{\sum_{m=0}^{\infty}\frac{(x_{j}+m)!}{m!}(1-p_{j})^{m}P(sN=x_{j}+m)}.$
(31)
$E(X_{iN}|X_{jN}=x_{j})=p_{i}\frac{\sum_{m=0}^{\infty}\frac{(x_{j}+m+1)!}{m!}(1-p_{j})^{m}P(sN=x_{j}+m+1)}{\sum_{m=0}^{\infty}\frac{(x_{j}+m)!}{m!}(1-p_{j})^{m}P(sN=x_{j}+m)}$

$cov(X_{iN},X_{jN})=sp_{i}p_{j}[-E\,N+s\,Var\,N]$

$cor(X_{iN},X_{jN})=$

(32)
$=\frac{\sqrt{p_{i}p_{j}}(sVar\,N-EN)}{\sqrt{[sp_{i}Var\,N+EN(1-p_{i})][sp_{j}Var\,N+EN(1-p_{j})]}}=$
(33) $=\sqrt{\frac{(FI\,X_{iN}-1)(FI\,X_{jN}-1)}{FI\,X_{iN}FI\,X_{jN}}}.$

  6. 6.

If the following expectation exists, then

(34)
$E(X_{iN}|X_{jN}=x_{j})=\frac{p_{i}}{p_{j}}(x_{j}+1)\frac{P(X_{jN}=x_{j}+1)}{P(X_{jN}=x_{j})}.$

  7. 7.

$G_{X_{1N}+X_{2N}+...+X_{kN}}(z)=G_{sN}(z),$

$P(X_{1N}+X_{2N}+...+X_{kN}=m)=P(sN=m).$

  8. 8.

For $j=1,2,...$, $i=1,2,...,k$, $(X_{iN}|X_{1N}+X_{2N}+...+X_{kN}=j)\sim
Bi(j;p_{i}).$

P r o o f.

  1. 1.

The form of p.g.f. follows by (3) and the formula of the p.g.f. of Mn
distribution. See e.g. (35.4) in [7].

  2. 2.

This property is an immediate consequence of (4), (5), the fact that

$X_{1N}+X_{2N}+...+X_{kN}=sN,$

and relation between p.m.f. and p.g.f., see e.g. (34.2) in [7].

  3. 3.

Here we replace the redundant variables in Equality 1), Th. 2 with 1, use one
of the main properties of p.g.f. (see e.g. p.327 in [6] or Daley et. al [5],
p. 113) and obtain (24). Using (22) and the definition (1), we have that

$(X_{i_{1}N},X_{i_{2}N},...,X_{i_{r}N},Ns-(X_{i_{1}N}+X_{i_{2}N}+...+X_{i_{r}N}))\sim$

$\sim
C\,N\,Mn(\vec{a}_{N};s,p_{i_{1}},p_{i_{2}},...,p_{i_{r}},1-(p_{i_{1}}+...+p_{i_{r}})).$

Now we sum up over the possible values of the last coordinate of this vector,
use 2), and obtain

$P(X_{i_{1}N}=x_{i_{1}},X_{i_{2}N}=x_{i_{2}},...,X_{i_{r}N}=x_{i_{r}})=$

$=\sum_{m=0}^{\infty}P(X_{i_{1}N}=x_{i_{1}},X_{i_{2}N}=x_{i_{2}},...,X_{i_{r}N}=x_{i_{r}},Ns-(X_{i_{1}N}+X_{i_{2}N}+...$

$...+X_{i_{r}N})=m)=$

$=\sum_{m=0}^{\infty}\left(\begin{array}[]{c}x_{i_{1}}+x_{i_{2}}+...+x_{i_{r}}+m\\\
x_{i_{1}},x_{i_{2}},...,x_{i_{r}},m\\\
\end{array}\right)p_{i_{1}}^{x_{i_{1}}}...p_{i_{r}}^{x_{i_{r}}}[1-(p_{i_{1}}+...+p_{i_{r}})]^{m}.$

$.P(Ns=x_{i_{1}}+x_{i_{2}}+...+x_{i_{r}}+m)=$

$=\frac{p_{i_{1}}^{x_{i_{1}}}...p_{i_{r}}^{x_{i_{r}}}}{x_{i_{1}}!x_{i_{2}}!...x_{i_{r}}!}\sum_{m=0}^{\infty}\frac{(m+x_{i_{1}}+x_{i_{2}}+...+x_{i_{r}})!}{m!}.[1-(p_{i_{1}}+...+p_{i_{r}})]^{m}.$

$.P(sN=m+x_{i_{1}}+x_{i_{2}}+...+x_{i_{r}}).$

  4. 4.

The first equality in (26) is a particular case of (25), for $r=1$. The rest
part of (26) is a consequence of the substitution $j+m=ns$ and the definition
of p.g.f. The results about the numerical characteristics follow by the Wald’s
equalities and (22). More precisely $EY_{i1}=sp_{i}$ and
$VarY_{i1}=sp_{i}(1-p_{i})$.

  5. 5.

The definitions for conditional probability and (25) imply (29). By (16)

$G_{X_{iN}}(z_{i}|X_{jN}=x_{j})=$

$=\frac{\frac{\partial^{x_{j}}G_{sN}(p_{i}z_{i}+p_{j}z_{j}+1-p_{i}-p_{j})}{\partial
z_{j}^{x_{j}}}}{\frac{\partial^{x_{j}}G_{sN}(p_{j}z_{j}+1-p_{j})}{\partial
z_{j}^{x_{j}}}}|_{z_{j}=0}=\frac{p_{j}^{x_{j}}\left(\frac{\partial^{x_{j}}}{\partial
y^{x_{j}}}G_{sN}(y)\right)|_{y=p_{i}z_{i}+1-p_{i}-p_{j}}}{p_{j}^{x_{j}}\left(\frac{\partial^{x_{j}}}{\partial
y^{x_{j}}}G_{sN}(y)\right)|_{y=1-p_{j}}}=$
$=\frac{\sum_{n=0}^{\infty}\left(\frac{\partial^{x_{j}}}{\partial
y^{x_{j}}}y^{sn}P(N=n)\right)|_{y=p_{i}z_{i}+1-p_{i}-p_{j}}}{\sum_{n=0}^{\infty}\left(\frac{\partial^{x_{j}}}{\partial
y^{x_{j}}}y^{sn}P(N=n)\right)|_{y=1-p_{j}}}=$ $=\frac{\sum_{n:sn\geq
x_{j}}^{\infty}\frac{(sn)!}{(sn-x_{j})!}(p_{i}z_{i}+1-p_{i}-p_{j})^{sn-
x_{j}}P(sN=sn)}{\sum_{n:sn\geq x_{j}}^{\infty}\frac{(sn)!}{(sn-
x_{j})!}(1-p_{j})^{sn-x_{j}}P(sN=sn)}.$

After the Substitution $m=sn-x_{j}$ we obtain (30).

In order to calculate covariation we use (2) and the formula for covariance of
Mn random vector $cov(Y_{i1},Y_{j1})=-np_{i}p_{j}$. See e.g. Johnson et. al
[7]. Together with $Var\,X_{iN}=sp_{i}[sp_{i}Var\,N+(1-p_{i})E\,N]$ and
formula for the correlation they imply (32). Divide the numerator and
denominator in $cor(X_{iN},X_{jN})$ to $EN$ and come to

$\frac{\sqrt{p_{i}p_{j}}(sFI\,N-1)}{\sqrt{[sp_{i}FI\,N+1-p_{i}][sp_{j}FI\,N+1-p_{j}]}}.$

Compare this result with (28) we see that it is equal to (33).

  6. 6.

By (31) and (26),

$E(X_{iN}|X_{jN}=x_{j})=p_{i}\frac{\sum_{m=0}^{\infty}\frac{(x_{j}+m+1)!}{m!}(1-p_{j})^{m}P(sN=x_{j}+m+1)}{\sum_{m=0}^{\infty}\frac{(x_{j}+m)!}{m!}(1-p_{j})^{m}P(sN=x_{j}+m)}=$
$=p_{i}\frac{\frac{(x_{j}+1)!}{p_{j}^{x_{j}+1}}P(X_{jN}=x_{j}+1)}{\frac{x_{j}!}{p_{j}^{x_{j}}}P(X_{jN}=x_{j})}=\frac{p_{i}}{p_{j}}(x_{j}+1)\frac{P(X_{jN}=x_{j}+1)}{P(X_{jN}=x_{j})}.$

  7. 7.

These distributions are obtained by 1) Th. 2, (19), (20) and the properties of
p.g.fs.

  8. 8.

By (1), we have $(X_{iN},X_{1N}+X_{2N}+...+X_{i-1N}+X_{i+1N}+...+X_{kN})\sim$

$\sim C\,N\,Mn(\vec{a}_{N};s,p_{i},1-p_{i})$. This, together with the
definitions for conditional probability, (23) and $Mn$ (via its p.m.f.) entail

$P(X_{iN}=m|X_{1N}+X_{2N}+...+X_{kN}=j)=$

$=\frac{P(X_{iN}=m,X_{1N}+...+X_{i-1N}+X_{i+1N}+...+X_{kN}=j-m)}{P(X_{1N}+X_{2N}+...+X_{kN}=j)}=$
$=\frac{\left(\begin{array}[]{c}j\\\ m\\\
\end{array}\right)p_{i}^{m}(1-p_{i})^{j-m}P(Ns=j)}{P(Ns=j)}=\left(\begin{array}[]{c}j\\\
m\\\ \end{array}\right)p_{i}^{m}(1-p_{i})^{j-m},$

$m=0,1,...,j.$

Now, the definition of $Bi$ distribution entails the desired result.

□

Note: 1\. As corollaries of the above theorem, when we fix the distribution of
$N$, we can obtain many new and well known distributions and their numerical
characteristics and conditional distributions.

Fig. 1.

Fig. 2.

Fig. 3.

Fig. 4.

2\. Here we denote by $LS(p_{L})$ the Logarithmic series distribution with
parameter $p_{L}\in(0,1)$. In order to observe the form of the dependence
between the r.vs. $X_{1N}$, $X_{2N}$ and to model it via their mean square
regression (34), we simulate $100$ realizations of $(X_{1N},X_{2N})\sim
CLSMn(p_{L};s,p_{1},p_{2})$, for different values of the parameters $p_{L}$,
$s$, $p_{1}$ and $p_{2}$. Four of the scatter plots of the data are given on
Figures 1 - 4. These observations show that the dependence between these two
r.vs. is approximately linear, with slope almost $\frac{p_{i}}{p_{j}}$. The
relation (33) entails that the strength of the dependence between $X_{iN}$ and
$X_{jN}$ is governed by $FI\,X_{iN}$ and $FI\,X_{jN}$. When the last two
Fisher indexes increase, the correlation between $X_{iN}$ and $X_{jN}$ also
increases. Equation (28) entails that the bigger the $FI\,N$, or $s$ are, the
bigger the $FI\,X_{iN}$ is. The last increases the strength of the dependence
between $X_{iN}$ and $X_{jN}$. All these results were observed empirically.
More precisely Figures 1 - 4, show that for bigger $s$ or $p_{L}$ we obtain
stronger linear dependence between $X_{iN}$ and $X_{jN}$.

3\. The comparison of the coefficients before $P(N=n)$ in both expressions for
the p.m.f. of $X_{iN}$, $i=1,2,...,k$, (the first obtained using the Total
probability formula and the second is (27)) gives us another proof of the
equality

$\sum_{j_{1}+j_{2}+...+j_{n}=m}\left(\begin{array}[]{c}s\\\ j_{1}\\\
\end{array}\right)\left(\begin{array}[]{c}s\\\ j_{2}\\\
\end{array}\right)...\left(\begin{array}[]{c}s\\\ j_{n}\\\
\end{array}\right)=\left(\begin{array}[]{c}ns\\\ m\\\ \end{array}\right).$

In analogous way, working with the theoretical results, related with
multivariate compounds with equal number of summands we can obtain new similar
equalities.

###  3.2. Compounds with NMn summands

In this subsection we consider the case, when the summands in (1) are NMn. It
seems that in 1952, Bates et. al [1] introduced NMn distribution. They reached
it considering Poisson distribution with Gamma mixing variable. Their first
parameter could be a real number. Wishart [19] investigated the case, when the
first parameter could be only integer. He called this distribution Pascal
multinomial distribution. From practical point of view, the NMn distribution
is interpreted as the one of the numbers of outcomes $A_{i}$, $i=1,2,...,k$
before the $n$-th $B$, in series of independent repetitions, where $A_{i}$,
$i=1,2,...,k$ and $B$ form a partition of the sample space. Let us remind the
definition.

Let $n\in\mathbb{N}$, $0<p_{i}$, $i=1,2,...,k$ and $p_{1}+p_{2}+...+p_{k}<1$.
A vector $(\xi_{1},\xi_{2},...,\xi_{k})$ is called Negative multinomially
distributed with parameters $n,p_{1},p_{2},...,p_{k}$, if its probability mass
function (p.m.f.) is

$P(\xi_{1}=i_{1},\xi_{2}=i_{2},...,\xi_{k}=i_{k})=$

$=\left(\begin{array}[]{c}n+i_{1}+i_{2}+...i_{k}-1\\\
i_{1},i_{2},...,i_{k},n-1\end{array}\right)p_{1}^{i_{1}}p_{2}^{i_{2}}...p_{k}^{i_{k}}(1-p_{1}-p_{2}-...-p_{k})^{n},$

$i_{s}=0,1,...,s=1,2,...,k.$ Briefly $(\xi_{1},\xi_{2},...,\xi_{k})\sim
NMn(n;p_{1},p_{2},...,p_{k})$.

We will use the distribution of its coordinates. For $m=2,3,...,k-1$,

(35) $(\xi_{i_{1}},\xi_{i_{2}},...,\xi_{i_{r}})\sim
NMn(s;\rho_{i_{1}},\rho_{i_{2}},...,\rho_{i_{r}}),$

with
$\rho_{i_{m}}=\frac{p_{i_{m}}}{1-\sum_{j\not\in\\{i_{1},i_{2},...,i_{r}\\}}p_{j}}$,
$m=1,2,...,r.$ More properties of NMn distribution can be found in Johnson et
al. [7].

If $A_{1},A_{2},...,A_{k}$ describe all possible mutually exclusive
”successes” and the event
$\overline{A}_{1}\cap\overline{A}_{2}\cap...\cap\overline{A}_{k}$ presents the
”failure”, then the coordinates $\xi_{i}$ of the above vector, can be
interpreted as the number of ”successes” of type $A_{i}$ $i=1,2,...,k$ until
n-th ”failure”. Considering this distribution for $k=1$, we obtain $NBi$
distribution with parameters $s$ and $(1-p_{1})$. Let us note that if
$k=2,3,...$ the marginal distributions of NMn distributed random vector,
defined above are $NBi(n,1-\rho_{i})$,
$\rho_{i}=\frac{p_{i}}{1-\sum_{j\not=i}p_{j}}$. More precisely their p.g.f. is

$G_{\xi}(z)=Ez^{\xi_{i}}=\left(\frac{1-\rho_{i}}{1-\rho_{i}z}\right)^{n},\,\,|z|<\frac{1}{\rho_{i}}\,\,i=1,2,...,k.$

Having in mind the univariate distributions, the NMn distribution is sometimes
called Multivariate Negative Binomial distribution. For $n=1$ the NMn
distribution is Multivariate geometric distribution. Some properties of the
bivariate version of this distribution are considered e.g. in 1981, by Phatak
et al. [13]. A systematic investigation of multivariate version could be found
e.g. in Srivastava et al. [15]. Here we consider random sums with Mn
distributed summands. The case when $N$ is Poisson distributed is considered
in Smith [14]. The next theorem generalizes his results.

###### Theorem 3

Suppose $(Y_{1i},Y_{2i},...,Y_{ki})\sim NMn(s,p_{1},p_{2},...,p_{k})$,
$i=1,2,...$ are independent. Here $s\in\mathbb{N}$,
$p_{1}>0,p_{2}>0,...,p_{k}>0$ and

$p_{1}+p_{2}+...+p_{k}<1.$

Denote by $p_{0}=1-p_{1}-p_{2}-...-p_{k}$. Then the resulting distribution of

$(X_{1N},X_{2N},...,X_{kN})\sim
C\,N\,NMn(\vec{a}_{N};s,p_{1},p_{2},...,p_{k}),$

defined in (1), possesses the following properties.

  1. 1.

Its p.g.f. is

(36)
$G_{X_{1N},X_{2N},...,X_{kN}}(z_{1},z_{2},...,z_{k})=G_{sN}\left(\frac{p_{0}}{1-p_{1}z_{1}-p_{2}z_{2}-...-p_{k}z_{k}}\right).$

  2. 2.

The p.m.f. satisfies the equalities:

$P(X_{1N}=0,X_{2N}=0,,X_{kN}=0)=G_{sN}(p_{0}).$

For $s=1,2,..,k$, $x_{s}=0,1,...$: $(x_{1},x_{2},...,x_{k})\not=(0,0,...,0),$

$P(X_{1N}=x_{1},X_{2N}=x_{2},...,X_{kN}=x_{k})=$

(37)
$=p_{1}^{x_{1}}p_{2}^{x_{2}}...p_{k}^{x_{k}}\sum_{j=1}^{\infty}\left(\begin{array}[]{c}sj+x_{1}+x_{2}+...+x_{k}-1\\\
x_{1},x_{2},...,x_{k},sj-1\\\ \end{array}\right)p_{0}^{js}P(N=j).$

  3. 3.

For all $r=1,2,...,k$ and for all $X_{i_{1}N},X_{i_{2}N},...,X_{i_{r}N}$,

$G_{X_{i_{1}N},X_{i_{2}N},...,X_{i_{r}N}}(z_{i_{1}},z_{i_{2}},...,z_{i_{r}})=$

(38)
$=G_{sN}\left(\frac{\rho_{0r}}{1-\rho_{i_{1}}z_{i_{1}}-\rho_{i_{2}}z_{i_{2}}-...-\rho_{i_{r}}z_{i_{r}}}\right),$

where
$\rho_{i_{j}}=\frac{p_{i_{j}}}{p_{0}+\sum_{j=1}^{r}p_{i_{j}}},\quad\rho_{0r}=1-\rho_{i_{1}}-...-\rho_{i_{r}},$
i.e.

$(X_{i_{1}N},X_{i_{2}N},...,X_{i_{r}N})\sim
C\,N\,NMn(\vec{a}_{N};s,\rho_{i_{1}},\rho_{i_{2}},...,\rho_{i_{r}}).$

$P(X_{i_{1}N}=0,X_{i_{2}N}=0,,X_{i_{r}N}=0)=G_{sN}(\rho_{0r}).$

For $s=1,...,r$, $x_{i_{s}}=0,1,...$:
$(x_{i_{1}},x_{i_{2}},...,x_{i_{r}})\not=(0,0,...,0),$

$P(X_{i_{1}N}=x_{i_{1}},X_{i_{2}N}=x_{i_{2}},...,X_{i_{r}N}=x_{i_{r}})=$

(39)
$=\rho_{i_{1}}^{x_{i_{1}}}...\rho_{i_{r}}^{x_{i_{r}}}\sum_{j=1}^{\infty}\left(\begin{array}[]{c}sj+x_{i_{1}}+x_{i_{2}}+...+x_{i_{r}}-1\\\
x_{i_{1}},x_{i_{2}},...,x_{i_{r}}+sj-1\\\
\end{array}\right)\rho_{0r}^{js}P(N=j).$

  4. 4.

If $E\,N<\infty$, then $E\,X_{i_{N}}=s\frac{p_{i}}{p_{0}}E\,N.$

If $E\,(N^{2})<\infty$, then
$Var\,X_{iN}=s\frac{p_{i}^{2}}{p_{0}^{2}}[sVar\,N+(1+\frac{p_{0}}{p_{i}})E\,N],$

(40) $FI\,X_{iN}=1+\frac{p_{i}}{p_{0}}[sFI\,N+1].$

  5. 5.

Dependence between the coordinates.

(41)
$cor(X_{iN},X_{jN})=\sqrt{\frac{(FI\,X_{iN}-1)(FI\,X_{jN}-1)}{FI\,X_{iN}FI\,X_{jN}}}.$

  6. 6.

Its conditional distributions satisfy the next equalities.

For $x_{j}\not=0$, $x_{i}=0,1,...$,

(42)
$P(X_{iN}=x_{i}|X_{jN}=x_{j})=\frac{1}{x_{i}!}\left(\frac{p_{0}+p_{j}}{p_{0}+p_{i}+p_{j}}\right)^{x_{j}}\left(\frac{p_{i}}{p_{0}+p_{i}+p_{j}}\right)^{x_{i}}.$
$.\frac{\sum_{n=1}^{\infty}\frac{(sn+x_{i}+x_{j}-1)!}{(sn-1)!}\left(\frac{p_{0}}{p_{0}+p_{i}+p_{j}}\right)^{sn}P(N=n)}{\sum_{n=1}^{\infty}\frac{(sn+x_{j}-1)!}{(sn-1)!}\left(\frac{p_{0}}{p_{0}+p_{j}}\right)^{sn}P(N=n)}.$

For $x_{i}=1,2,...$, $P(X_{iN}=x_{i}|X_{jN}=0)=$

$=\frac{\left(\frac{p_{i}}{p_{0}+p_{i}+p_{j}}\right)^{x_{i}}}{G_{sN}\left(\frac{p_{0}}{p_{0}+p_{j}}\right)}\sum_{n=1}^{\infty}\frac{(sn+x_{i}-1)!}{x_{i}!(sn-1)!}\left(\frac{p_{0}}{p_{0}+p_{i}+p_{j}}\right)^{sn}P(N=n),$
$P(X_{iN}=0|X_{jN}=0)=\frac{G_{sN}\left(\frac{p_{0}}{p_{0}+p_{i}+p_{j}}\right)}{G_{sN}\left(\frac{p_{0}}{p_{0}+p_{j}}\right)}.$

  7. 7.

For $x_{j}=1,2,...$

(43)
$E(X_{iN}|X_{jN}=x_{j})=\frac{p_{i}}{p_{j}}(x_{j}+1)\frac{P(X_{jN}=x_{j}+1)}{P(X_{jN}=x_{j})}.$

  8. 8.

$X_{1N}+X_{2N}+...+X_{kN}\sim CNNMn(s;1-p_{0}).$

  9. 9.

For $j=1,2,...$, $i=1,2,...,k$,

$(X_{iN}|X_{1N}+X_{2N}+...+X_{kN}=j)\sim Bi(j;\frac{p_{i}}{1-p_{0}}).$

P r o o f.

  1. 1.

The equality (36) follows by (3) and the formula of the p.g.f. of NMn
distribution. See e.g. (36.1) in [7], who use different parametrisation.

  2. 2.

The formula of the p.m.f. at $(0,0,...,0)$ is an immediate consequence of (4),
the fact that $G_{N}(z^{s})=G_{sN}(z)$ and the form of p.m.f. of
$(Y_{11},Y_{21},...,Y_{k1})\sim NMn(s,p_{1},p_{2},...,p_{k})$ at
$(0,0,...,0)$.

To prove (37) we can use the Total probability formula (the partition of the
sample space describe the possible values of $N$), and the fact that $n$-th
fold convolution on $NMn(s,p_{1},p_{2},...,p_{k})$ is
$NMn(ns,p_{1},p_{2},...,p_{k})$. The same formula can be obtained also by (5)
and relation between p.m.f. and p.g.f.

  3. 3.

Equality 1), Th. 3 entails (38), when replace redundant variables with 1 and
use one of the main properties of p.g.f., see e.g. p.327 in [6]. Using (35)
and the definition (1), we have that

$(X_{i_{1}N},X_{i_{2}N},...,X_{i_{r}N})\sim
C\,N\,NMn(\vec{a}_{N};s,\rho_{i_{1}},\rho_{i_{2}},...,\rho_{i_{r}}).$

Now we use (39) and complete the proof.

  4. 4.

The substitution of the mean
$EY_{i1}=s\frac{\rho_{i}}{1-\rho_{i}}=s\frac{p_{i}}{p_{0}}$, and the variance
$VarY_{i1}=s\frac{p_{i}(p_{0}+p_{i})}{p_{0}^{2}}$, $i=1,2,...,k,$ of NMn
distribution in (11) and (12) gives us $E\,X_{i_{N}}$ and $Var\,X_{i_{N}}$,
$i=1,2,...,k$.

  5. 5.

Using the properties of $NMn$ distribution we have that

$[CV\,(Y_{i1},Y_{i2})]^{2}=\frac{1}{s},\,(CV\,Y_{i1})^{2}=\frac{1}{s}(1+\frac{p_{0}}{p_{i}}),\,(CV\,Y_{j1})^{2}=s^{-1}(1+\frac{p_{0}}{p_{j}}).$

Now the equation (18), together with (40) entail the desired result.

  6. 6.

The definitions for conditional probability and (39), applied for $r=1$ and
$r=2$, imply (42). In analogous way we prove the case when $X_{jN}=0$.

  7. 7.

Let $x_{j}$ be a fixed positive integer. In the next calculations we first use
the formula for the mean, then we change the order of summation, use the
derivatives of the geometric series, make some algebra and obtain.

$E(X_{iN}|X_{jN}=x_{j})=\sum_{x_{j}=1}^{\infty}\frac{1}{(x_{i}-1)!}\left(\frac{p_{0}+p_{j}}{p_{0}+p_{i}+p_{j}}\right)^{x_{j}}\left(\frac{p_{i}}{p_{0}+p_{i}+p_{j}}\right)^{x_{i}}.$
$.\frac{\sum_{n=1}^{\infty}\frac{(sn+x_{i}+x_{j}-1)!}{(sn-1)!}\left(\frac{p_{0}}{p_{0}+p_{i}+p_{j}}\right)^{sn}P(N=n)}{\sum_{n=1}^{\infty}\frac{(sn+x_{j}-1)!}{(sn-1)!}\left(\frac{p_{0}}{p_{0}+p_{j}}\right)^{sn}P(N=n)}=$
$=\left(\frac{p_{i}}{p_{0}+p_{i}+p_{j}}\right)\sum_{n=1}^{\infty}\frac{P(N=n)}{(sn-1)!}\left(\frac{p_{0}}{p_{0}+p_{i}+p_{j}}\right)^{sn}\left(\frac{p_{0}+p_{j}}{p_{0}+p_{i}+p_{j}}\right)^{x_{j}}.$
$.\frac{\sum_{x_{j}=1}^{\infty}\frac{(sn+x_{i}+x_{j}-1)!}{(x_{i}-1)!}\left(\frac{p_{i}}{p_{0}+p_{i}+p_{j}}\right)^{x_{i}-1}}{\sum_{n=1}^{\infty}\frac{(sn+x_{j}-1)!}{(sn-1)!}\left(\frac{p_{0}}{p_{0}+p_{j}}\right)^{sn}P(N=n)}=$
$=\left(\frac{p_{i}}{p_{0}+p_{i}+p_{j}}\right)\sum_{n=1}^{\infty}\frac{P(N=n)}{(sn-1)!}\left(\frac{p_{0}}{p_{0}+p_{i}+p_{j}}\right)^{sn}\left(\frac{p_{0}+p_{j}}{p_{0}+p_{i}+p_{j}}\right)^{x_{j}}.$
$.\frac{\frac{(sn+x_{j})!}{\left(1-\frac{p_{i}}{p_{0}+p_{i}+p_{j}}\right)^{sn+x_{j}+1}}}{\sum_{n=1}^{\infty}\frac{(sn+x_{j}-1)!}{(sn-1)!}\left(\frac{p_{0}}{p_{0}+p_{j}}\right)^{sn}P(N=n)}=$
$=(x_{j}+1)\frac{p_{i}}{p_{j}}\frac{\left(\frac{p_{j}}{p_{0}+p_{j}}\right)^{x_{j}+1}\sum_{n=1}^{\infty}\frac{P(N=n)(sn+x_{j})!}{(sn-1)!(x_{j}+1)!}\left(\frac{p_{0}}{p_{0}+p_{j}}\right)^{sn}}{\left(\frac{p_{j}}{p_{0}+p_{j}}\right)^{x_{j}}\sum_{n=1}^{\infty}\frac{(sn+x_{j}-1)!}{(sn-1)!x_{j}!}\left(\frac{p_{0}}{p_{0}+p_{j}}\right)^{sn}P(N=n)}=$
$=\frac{p_{i}}{p_{j}}(x_{j}+1)\frac{P(X_{jN}=x_{j}+1)}{P(X_{jN}=x_{j})}.$

Finally we have just compared the previous expression with $P(X_{jN}=x_{j}+1)$
and $P(X_{jN}=x_{j})$, using (39), for $r=1$.

  8. 8.

These distributions are obtained by (19), 1) Th. 3 for
$z_{1}=z_{2}=...=z_{k}=z$ and (38), for $r=1$.

  9. 9.

By (1) and the definition of the NMn distribution, we have

$(X_{iN},X_{1N}+X_{2N}+...+X_{i-1N}+X_{i+1N}+...+X_{kN})\sim$

$\sim C\,N\,Mn(\vec{a}_{N};s,p_{i},1-p_{0}-p_{i})$.

This, together with the definitions for conditional probability, (23) and $Mn$
(via its p.m.f.) entail

$P(X_{iN}=m|X_{1N}+X_{2N}+...+X_{kN}=j)=$

$=\frac{P(X_{iN}=m,X_{1N}+...+X_{i-1N}+X_{i+1N}+...+X_{kN}=j-m)}{P(X_{1N}+X_{2N}+...+X_{kN}=j)}=$
$=\frac{p_{i}^{m}(1-p_{0}-p_{i})^{j-m}\sum_{n=1}^{\infty}\left(\begin{array}[]{c}sn+j-1\\\
m,j-m,sn-1\\\
\end{array}\right)p_{0}^{ns}P(N=n).}{(1-p_{0})^{j}\sum_{n=1}^{\infty}\left(\begin{array}[]{c}sn+j-1\\\
j\\\ \end{array}\right)p_{0}^{ns}P(N=n).}=$ $=\left(\begin{array}[]{c}j\\\
m\\\
\end{array}\right)\left(\frac{p_{i}}{1-p_{0}}\right)^{m}(1-\frac{p_{i}}{1-p_{0}})^{j},\quad
m=0,1,...,j.$

Now we use the definition of $Bi$ distribution and complete the proof.

□

Note: 1\. In order to observe the form of the dependence between the r.vs.
$X_{1N}$, $X_{2N}$, we have simulated $100$ realizations of
$(X_{1N},X_{2N})\sim$ $C\,LS\,NMn(p_{L};$ $s,p_{1},$ $p_{2})$ and have plotted
the scatter plots of the observations. Here $LS(p_{L})$ means the Logarithmic
series distribution with parameter $p_{L}\in(0,1)$. Four examples of such
scatter plots, for different values of the parameters $p_{L}$, $s$, $p_{1}$
and $p_{2}$, are given on Figures 5 - 8. They show that the dependence between
these two r.vs. is approximately linear, with slope, close to
$\frac{p_{i}}{p_{j}}$. The strength of the dependence increases with
parameters $s$ and $p_{L}$. These empirical observations are synchronised with
the theoretical relations, expressed in the equations (43), (41) and (40).

Fig. 5.

Fig. 6.

Fig. 7.

Fig. 8.

2\. Similarly to the previous case, the relation (41) shows that the strength
of the linear dependence between $X_{iN}$ and $X_{jN}$ is completely governed
by the values of $FI\,X_{iN}$ and $FI\,X_{jN}$. When the last two Fisher
indexes increase, the correlation between $X_{iN}$ and $X_{jN}$ also
increases. From the other side, the equation (40) entails that the bigger the
$FI\,N$, or $s$, or $p_{0}$, or $p_{i}$ are, the bigger the $FI\,X_{iN}$ is.
So, via $FI\,X_{iN}$, the last increases the strength of the straight-line
dependence between $X_{iN}$ and $X_{jN}$. On the examples on Figure 3 and
Figure 4 we observe that the bigger the s, the stronger the linear dependence
is.

##  4\. Conclusive remarks

The paper considers the properties of the distributions of multivariate random
sums with equal number of summands. Some simulations of realizations of
particular cases of random vectors with such distributions are made, and the
dependence between their coordinates is observed. These investigations show
that these distributions are appropriate for modelling of both: linear
dependence between the coordinates and clustering in the observations. From
theoretical point of view, in both particular cases, described here, these
results can be seen in equalities (34) and (43). For the general case, without
fixing neither the distribution of the number of summands, nor the
distribution of summands, the second formula in (12) shows that, the bigger
the Fisher index(FI) of the number of summands, or the expectation, or the FI
of the summands are, the bigger the FI of the corresponding random sum is. The
strength of the linear dependence between the coordinates $X_{iN}$ of the
random sums and the number of summands $N$, expressed in (14), decreases with
the increments of the FI of $N$ or the coefficient of variation of the
summands.

If $(X_{1N},X_{2N})\sim C\,LS\,Mn(p_{L};s,p_{1},p_{2})$ the dependence between
these two r.vs. is approximately linear, with slope almost
$\frac{p_{i}}{p_{j}}$. The relation (33) entails that the strength of the
dependence between $X_{iN}$ and $X_{jN}$ is governed by $FI\,X_{iN}$ and
$FI\,X_{jN}$. When the last two Fisher indexes increase, the correlation
between $X_{iN}$ and $X_{jN}$ also increases. Equation (28) entails that the
bigger the $FI\,N$, or $s$ are, the bigger the $FI\,X_{iN}$ is. Figures 1 - 4
show that for bigger parameters $s$ or $p_{L}$ we obtain stronger linear
dependence between $X_{iN}$ and $X_{jN}$.

It is interesting that in the other, considered particular case here, i.e. if
$(X_{1N},X_{2N})\sim C\,LS\,NMn(p_{L};s,p_{1},p_{2})$, the form of the
dependence between $X_{1N}$ and $X_{2N}$ is again approximately linear, with
slope almost $\frac{p_{i}}{p_{j}}$, for all possible discrete distributions of
$N$. This is synchronised with (43). The strength of the dependence increases
with $s$ and $p_{L}$. The equation (40) entails that the bigger the $FI\,N$,
or $s$, or $p_{0}$, or $p_{i}$ are, the bigger the $FI\,X_{iN}$ is. Similarly
to the previous case, the relation (41) shows that the strength of the linear
dependence between $X_{iN}$ and $X_{jN}$ is completely governed by the values
of $FI\,X_{iN}$ and $FI\,X_{jN}$. When the last two Fisher indexes increase,
the correlation between $X_{iN}$ and $X_{jN}$ also increases.

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Pavlina Kalcheva Jordanova

Shumen University

115 Universitetska str.,

9700 Shumen, Bulgaria

e-mail<EMAIL_ADDRESS>