Critical Multitype Branching Processes with Random Migration
Miguel González Pedro Martín-Chávez Inés del Puerto

Classifying Human-Generated and AI-Generated Election Claims in Social Media
    Alphaeus Dmontesup1, Marcos Zampierisup1, Kevin Lybargersup1, Massimiliano Albanesesup1 and Genya Coultersup2
sup1Department of Information Sciences and Technology, George Mason University, USA 

sup2OSET Institute
{admonte, mzampier, klybarge, malbanes}@gmu.edu, genya@osetinstitute.org

    
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The aim of this paper is to introduce a multitype branching process with random migration following the research initiated with the Galton–Watson process with migration introduced in <cit.>. We focus our attention in what we call the critical case. Sufficient conditions are provided for the process to have unlimited growth or not. Furthermore, using suitable normalizing sequences, we study the asymptotic distribution of the process. Finally, we obtain a Feller-type diffusion approximation.

Keywords: multitype branching processes; random migration; unlimited growth; rate of convergence; diffusion approximation; critical case. 

MSC2020: 60J80


§ INTRODUCTION


The Galton–Watson branching processes with migration have been studied extensively in the literature.  The readers interested in the origin of these models are referred to the surveys <cit.>, <cit.> and <cit.>. These are  stochastic models used in population dynamic and other fields to describe the evolution of populations in which individuals produce offspring according to some probability distribution (often assumed to be independent and identically distributed),   and also have the possibility of migrating. Migration can  involve individuals leaving the population (emigrants) or individuals entering  the population from an external source (immigrants).


In this paper we generalized the  Galton–Watson process with migration introduced in <cit.>. Let ,  _+, ,  and _+ be the sets of positive integers, non-negative integers, integers, real numbers, and non-negative real numbers, respectively. Let 
{X_k,j: k∈_+, j∈} be  a sequence of _+-valued   independent and identically distributed (i.i.d.) random variables (offspring variables). On the same probability space, let  
{ I_k}_k∈_+ be defined as a set of _+-valued i.i.d. (immigration) random
variables, independent of the offspring variables. A discrete time homogeneous Markov chain (Y_k)_k∈_+ is called a Galton–Watson process with migration (GWPM) if Y_0=0 almost surely and, for k∈_+,

    Y_k+1= ∑_j=1^Y_k X_k,j+M_k+1,

(with ∑_1^0=0), where for p,q,r≥ 0 such that p+q+r=1,

    M_k+1={[               0              p,               ,;           I_k+1              q,               ,; -X_k,11_{Y_k>0}              r,               , ].

is the migration component. Here 1_A is the indicator function of a set A. As usual  Y_k represents the population size at generation k and X_k,j is the number of offspring of the j-th individual at the k-th generation. Notice that the model considers the possibility that only a single family can emigrate.

This model was studied in depth in the papers <cit.>, <cit.>, <cit.> and  <cit.>. Later, the model of GWPM introduced in <cit.> was extended in <cit.> and <cit.> by allowing two types of emigration (family and individual) as well as state-dependent immigration. 

The aim of this paper is to introduce a more general branching process with migration, by considering different types of individuals in the population, and a more general scheme of immigration and emigration components. In Section <ref>, we define the muti-type branching process with random  migration (MBPM), and obtain its first and second conditional moments. Focusing our attention in what we call critical case, we prove sufficient conditions for the unlimited growth or not of the process in Section <ref>. Finally, Section <ref> is dedicated to the study of the limiting behaviour of the process. First, we consider several normalizing sequences to obtain different limiting distributions, namely, gamma and normal distributions. Secondly, we consider a sequence of appropriately scaled random step
functions formed from a critical MBPM and prove its weak convergence to a Feller-type diffusion process. The proof of this result is relegated to the Appendix.



§ PROBABILITY MODEL

Let us fix a probability space (Ω, ℱ,P) where all the random vectors are defined and let  p∈ be the  dimension of the vectors. 
For =(z_1,…,z_p)^⊤∈^p, let 
   ^+ =(z_1^+,…,z_p^+)^⊤∈_+^p, 
 where  x^+ stands for the positive part of x∈ and  ^⊤ for the transpose.


Let  {X_k,j,i = (X_k,j,i,1,…,X_k,j,i,p)^⊤:i∈{1,…,p},  k∈_+, j∈} be
 _+^p-valued independent random vectors, i.d. for each i∈{1,…,p}. Let 
{D_k() = (D_k,1(),…,D_k,p())^⊤ : k∈_+} be  _+^p–valued  i.d. random vectors for each 
∈_+^p such that D_k,i()=0 if z_i=0, and it has range on [1,z_i] if z_i>0, i∈{1,…,p}. Let 
{I_k () = (I_k,1(),…,I_k,p())^⊤ : k∈_+} be
 also ^p–valued  identically distributed random vectors for each 
∈_+^p. For z∈_+^p, let  {M_k() = (M_k,1(),…,M_k,p())^⊤ : k∈_+} 
be identically distributed ^p-valued random vectors  with range contained in 
[-z_1,∞) ×⋯× [-z_p,∞), defined, for all k∈_+ and i∈{1,…,p}, as

    M_k,i() = 
           0     with probability  p_i()∈[0,1] ,
       I_k,i()     with probability  q_i()∈[0,1] ,
    
            -D_k,i()     with probability  r_i()∈[0,1] ,

with p_i()+q_i()+r_i()=1, and
 r_i() vanishing if z_i = 0. 

Let  _0 = (Z_0,1,…,Z_0,p)^⊤ be a _+^p–valued random vector (the initial generation) and let us assume that {Z_0 ,X_k,j,i , M_k(): k∈_+,  j∈,  i∈{1,…,p}, z∈_+^p} are
independent ^p–valued random vectors. 

We define a multitype branching process with random migration (MBPM) as a sequence of  p-dimensional random vectors, (Z_k)_k∈_+,  recursively as 

    Z_k+1  = 
        ∑_i=1^p∑_j=1^Z_k,i+M_k,i(_k)X_k,j,i, 
             k∈_+.

Notice that (<ref>) generalizes (<ref>) in several directions. Model (<ref>) considers that there exist p-types of individuals and each one can give birth to individuals of any type. Moreover, several individuals (not necessary from the same family) can leave the population. The immigrants  produce offspring in the same generation in which they arrive.  

Let us introduce notations for some moments. For i∈{1,…,p} and ∈_+^p,

    m_i    =  X_0,1,i∈_+^p,    𝗆   =( m_1,…,  m_p)∈_+^p× p
    Σ_i    = X_0,1,i∈^p× p,    Σ   = (Σ_1, …,Σ_p)∈ (^p× p)^p
    
        a_i()   =I_0,i()∈_+,    a()   =( a_1(),…, a_p())^⊤∈_+^p
    
        b_i()   =D_0,i()∈_+,    b()   =( b_1(),…, b_p())^⊤∈_+^p

Let us denote
the operators 

    ∘ :^p ×^p   →   ^p,
    
     (_1,_2)   →   _1∘_2=(z_1,1z_2,1,…,z_1,pz_2,p)^⊤

 and 
 
    ⊙ :^p × (^p× p)^p  
        →   ^p× p
    
     (z,𝖠)   →   z⊙𝖠 
        =  ∑_i=1^p z_i𝖠_i.
 
Notice that the operator ∘ is the Hadamard product of matrices (see <cit.>) applied to column vectors. 






It is easy to see that a MBPM is a homogeneous
multitype Markov chain, whose state space  is _+^p. It is interesting to notice that the state  0 is absorbing if and if q_i( 0)=0 for every i∈{1,…,p}. In addition, if 

    P(⋂_i=1^p({M_0,i()=-z_i}∪{M_0,i()>-z_i,  X_0,j,i=  0, j=1, …, z_i+M_0,i()}))>0,

for each non-null state , then 

    P(Z_n=  n>0|Z_0=)<1,

because P(Z_n = 0|Z_0=)>0 for all n>0. Consequently,  such state  is transient. Under the assumptions that 
 0 is an absorbing state and that each non null state is transient, making use of Markov chain theory, we have the extinction–explosion duality

    P(Z_n→ 0 )+P(Z_n→∞)=1.


In the following proposition we establish the first and second conditional moments of the process. Let us consider the canonical filtration of the process ℱ_k =σ(Z_0,…,Z_k), k∈_+. 
 For ∈_+^p, let

    h() = M_0()= a()∘q()-b()∘r(),

where q()=( q_1(),…, q_p())^⊤ and r()=( r_1(),…, r_p())^⊤.


Let (Z_k)_k∈_+ be a  MBPM. Then, for each k∈, 
      
    Z_kℱ_k-1   = 𝗆(Z_k+h(Z_k)), 
    Z_k|ℱ_k-1    =  (Z_k+h(Z_k))⊙Σ+𝗆M_0(Z_k)𝗆^⊤,

    almost surely.
   

  Using the Markov property and the independence between the random  vectors in the model, we have
 
    Z_k+1|Z_k=   =   ∑_i=1^p∑_j=1^z_i+M_k,i()X_k,j,i=∑_i=1^p∑_j=1^z_i+M_k,i()X_k,j,i| M_k,i()
       =   ∑_i=1^p (z_i+M_k,i())m_i= 𝗆 (+M_0())
       =   𝗆(+h()),


    Z_k+1|Z_k=   =   ∑_i=1^p∑_j=1^z_i+M_k,i()X_k,j,i| M_k,i()
          +∑_i=1^p∑_j=1^z_i+M_k,i()X_k,j,i| M_k,i()
       =   ∑_i=1^p
    (z_i+M_k,i())Σ_i+𝗆M_k()
       =    (+M_0())⊙Σ+𝗆M_0()𝗆^⊤
        =     (+h())⊙Σ+𝗆M_0()𝗆^⊤.

    






§ UNLIMITED GROWTH OF THE PROCESS

 In this section we focus our attention in studying P(Z_n→∞). Here Z_n is what we call a critical MBPM. It satisfies the following hypothesis:

Hypothesis A: The offspring mean matrix 𝗆 is primitive with Perron–Frobenius eigenvalue equals 1.


The Perron–Frobenius' theorem (see <cit.>) guarantees that there exist left and right eigenvectors, u=(u_1…,u_p)^⊤ and v=(v_1,…,v_p)^⊤, respectively, of the Perron-Frobenius eigenvalue such that all their coordinates are positive,  ∑_i=1^p u_i=1 and u^⊤v =1.

We also consider the following hypothesis for the expected value of the random migration component:

Hypothesis B: The vector h()=(h_1(),…, h_p())^⊤, ∈_+^p, satisfies h_i()=() as →∞, for every i∈{1,…,p}.

It is interesting to notice that under Hypotheses A and B the mean growth rates of the functional u^⊤Z_n converge to 1. Indeed, as →∞,

    1/u^⊤u^⊤Z_n+1|Z_n= = 1/u^⊤u^⊤𝗆 ( + h()) = 1/u^⊤u^⊤ ( + h()) = 1 + u^⊤h()/u^⊤→ 1.


Let us define σ^2()u^⊤Z_n+1|Z_n=, ∈_+^p. Taking into account Hypothesis A and Proposition <ref>, it is easy to obtain that

    σ^2()=  u^⊤((+h())⊙Σ+ M_0())u.


Next we establish sufficient conditions under which the process does not have unlimited growth.


Let (Z_k)_k∈_+ be a  MBPM satisfying Hypotheses A and B. Then P(Z_n→∞)=0  if at least one of the following conditions holds:


 
  (a) For each i∈{1,…, p},  h_i()≤ 0 for all ∈_+^p with  large enough.
    
  (b) 
    lim sup_→∞2(u^⊤ )(u^⊤h()) /σ^2()<1,

     and, for every i∈{1,…, p},
     
    |z_i+M_0,i()|^1+δ/2=((u^⊤ )^1+δσ^2()),

  and 

    |M_0,i()-h_i()|^2+δ=((u^⊤ )^1+δσ^2()),

  as →∞, for some 0<δ≤ 1.

  
 If (a) is satisfied, Theorem 3 in <cit.> implies that P(Z_n→∞)=0. If (b) holds, the result is a consequence of Theorem 4 in <cit.>, provided that, as →∞,

    ξ()|u^⊤Z_n+1 -u^⊤Z_n+1|Z_n|^2+δ|Z_n==((u^⊤)^1+δσ^2()).

Indeed,    using the fact that for every a,b∈ and r>0 there exists a positive constant C_r such that |a+b|^r≤
C_r(|a|^r+|b|^r) (C_r-inequality) and the Marcinkiewicz-Zygmund's inequality (see Lemma <ref> in Appendix), it is easy to check that 

    ξ()   ≤    C_1 ∑_i=1^p ( |∑_j=1^z_i+M_0,i()u^⊤(X_0,j,i-m_i)|^2+δ+|M_0,i()u^⊤m_i-h_i() u^⊤m_i|^2+δ)
       ≤     C_2  ∑_i=1^p ((z_i+M_0,i())^1+δ/2 + |M_0,i()-h_i()|^2+δ(u^⊤m_i)^2+δ),

with C_1 and C_2 certain positive constants. Hence, using (<ref>) and (<ref>), we obtain (<ref>).






  

  i) In hypothesis (a) we assume that for each type the emigration dominates over the immigration given large enough population size. 

  ii) To establish Theorem <ref> is not necessary to require that the extinction-explosion duality   (<ref>) holds. It is also valid for migration models with immigration at 0, that is, with a reflecting barrier at 0. In the case in which 0 is an absorbing state and the non-null states are transient, the result provides sufficient conditions for the extinction of the population.



Next we study sufficient  conditions for a positive probability of unlimited growth of the process. 

   Let (Z_k)_k∈_+ be a  MBPM  satisfying Hypotheses A and B and such that, for every non-null ∈_+,  
  
    P(I_0,i()>0, 𝐗_0,j,i≠ 0, i∈ A(), j=1,… z_i+I_0,i()  )>0,

  where A()={i∈{1,…, p}: z_i>0}. Then P(Z_n→∞)>0 if 

    lim inf_→∞2(u^⊤ )(u^⊤h()) /σ^2()>1,

and, for every i∈{1,…, p},
     
    |z_i+M_0,i()|^1+δ/2=((u^⊤ )^1+δ (u^⊤h())/(log(u^⊤ ))^1+α)

     and 
      
    |M_0,i()-h_i()|^2+δ=((u^⊤ )^1+δ (u^⊤h())/(log(u^⊤ ))^1+α)

     as →∞, for some 0<δ≤ 1 and α>0.
  

Taking into account  Theorem 2 in <cit.> and (<ref>) holds, it is enough to show that
ξ()=((u^⊤ )^1+δ (u^⊤h())/(log(u^⊤ ))^1+α). This later follows from (<ref>), (<ref>) and (<ref>).



 Theorem <ref> also holds if (<ref>) and (<ref>) are replaced by
 
    |z_i+M_0,i()|^1+δ/2=((u^⊤ )^2+δu^⊤h())

  and 

    |M_0,i()-h_i()|^2+δ=((u^⊤ )^2+δu^⊤h()),

    respectively.
Indeed, (<ref>), (<ref>) and (<ref>) yield ξ()=((u^⊤ )^2+δu^⊤h()). Therefore,

    ξ()/σ^2()(u^⊤)^1+δ= ξ()/(u^⊤)^2+δ (u^⊤h())(u^⊤) (u^⊤h())/σ^2().

Hence, by assumption (<ref>)  and ξ()=((u^⊤ )^2+δu^⊤h()), we deduce   ξ()=((u^⊤)^1+δσ^2()). 

Analogously, in Theorem <ref> we can replace (<ref>) and (<ref>) with
 
    |z_i+M_0,i()|^1+δ/2=(σ^2()(u^⊤)^δ/(log(u^⊤))^1+α),

     and 
      
    |M_0,i()-h_i()|^2+δ=(σ^2()(u^⊤)^δ/(log(u^⊤))^1+α),

respectively.
Assumptions (<ref>) and  (<ref>)
 could be intuitively interpreted for a critical MBPM as speed conditions,  noticing that

    (u^⊤ ) (u^⊤h())/σ^2()=(u^⊤)^-1u^⊤Z_n+1|Z_n=-1/u^⊤Z_n+1|Z_n= (u^⊤)^-2.






§ LIMITING BEHAVIOUR OF THE PROCESS

Consider a positive and twice continuously differentiable real function h(x) such that u^⊤h()=h(u^⊤ ), for ∈_+^p.  We denote by {a_n}_n∈_+ the solution of the difference equation

    a_0=1,     a_n+1=a_n+h(a_n).
 

In the next result, we consider different normalizing sequence of constants that guarantee the convergence of the corresponding normalized process on the unlimited growth set.











 Let (Z_k)_k∈_+ be a  MBPM  satisfying Hypotheses A and B,  and (<ref>). Suppose that for every non-null vector ∈_+^p:


  (i) h_i()= c_i(u^⊤)^α+((u^⊤)^α) for each i=1,…, p and for some α<1 and c=(c_1,…,c_p)^⊤∈^p such that u^⊤c > 0.

  (ii) σ^2()=ν(u^⊤)^β+((u^⊤)^β) for some β≤ 1+α and ν>0.

  (iii) max_1≤ i≤ p{|z_i+M_0,i()|^1+δ/2, |M_0,i()-h_i()|^2+δ}=((σ^2())^1+δ/2) for some δ∈(0,1].

  (iv) max_1≤ i≤ p{|h_i()|}=((u^⊤)^δ_1) for some δ_1<1.

  (v)  For some α̃, 1<α̃≤ 2, max_1≤ i≤ p{z_i+h_i(), |M_0,i()-h_i()|^α̃}=((u^⊤)^δ_2α̃) for some δ_2<1.

Then 

  a) If β=1+α and ν<2(u^⊤c), then for every vector
 x∈ℝ^p

    lim_n→∞P(Z_n/n^1/(1-α)≤x|Z_n→∞)=F_uZ(x).
 Here F_uZ is the distribution function associated with the random vector uZ, and Z is a random variable such that Z^1-α follows a gamma distribution with parameters 2(u^⊤c)-να/ν(1-α)
and  ν(1-α)^2/2. Note that vector inequalities are evaluated component-wise.



  b) If 0<α<1 and β<α +1, then, on
{Z_n→∞}, n^1/(α-1)Z_n converges to ((u^⊤c)(1-α))^1/(1-α)u in L^1.

Moreover, if β≥ 3α-1 and
max{δ_1,δ_2}<(β-α+1)/2, then for every vector x∈ℝ^p, it is verified that

    lim_n→∞P(Z_n-ua_n/Λ_n≤x|Z_n→∞)=F_uU(x).
 Here F_uU(x) is the distribution function associated with the random vector uU and U is  a random variable with standard normal distribution. The normalization sequence is given by  

    Λ_n:=ν^1/2((u^⊤c)(1-α))^(3α-1)/(2(1-α))
    n^α/(1-α)(log n)^1/2   β=3α-1, (ν(β-3α+1)^-1(u^⊤c)^β/(1-α)
    ((1-α)n)^(β-α+1)/(1-α))^1/2   β>3α-1.




The proof follows from Corollary 4.1 and Remark 4.1 in <cit.>, observing that the MBPM is a multitype controlled branching process (MCBP) with ϕ_k()=+M_k(), k∈_+ and ∈_+^p.





  

  i) Notice that under the assumptions of Theorem <ref>, Theorem
<ref> holds, and therefore P(Z_n→∞)>0. In fact, from (i) and (ii), it is
derived that

    lim inf_z→∞2(u^⊤ )(u^⊤h()) /σ^2()= 2(u^⊤c)ν^-1    β=α+1     ∞
 Finally,  using again (ii) and (iii), we have that
ξ()=((u^⊤ )^1+δ (u^⊤h())/(log(u^⊤ ))^1+α). Note that,
if β>α +1 or if β=α+1 and 2(u^⊤c)ν^-1<1, then
P(Z_n→∞)=0 (see Theorem
<ref>).

  ii) It
is not hard to see that a_n ∼ ((u^⊤c)(1-α)n)^(1-α)^-1, as
n→∞.






 §.§ Scaling limits

We now address the problem of scaling limits of MBPMs. To this end, we consider our process as particular case of a MCBP and apply the results on scaling limits for MCBPs provided in  <cit.>.

Assume the following hypothesis holds.

Hypothesis C: There exist vectors a,b∈_+^p and q,r∈ [0,1]^p, such that, as →∞,

    a()→a,  b()→b,   q()→q  and  r()→r.



Notice that under Hypothesis C, we have that, as →∞,  
    h()→a∘q-b∘r.


Notice also that Hypothesis C implies Hypothesis B. Therefore, under Hypothesis C (replacing Hypothesis B) and the rest of assumptions of Theorem <ref>, we have that the process has a positive probability of unlimited growth. We will assume that P(Z_n→∞)=1.

 Furthermore, we suppose that in each generation the immigration and emigration components are independent, that is the random variables {I_k,i(),D_k,j():i∈{1,…,p}, j∈{1,…,p}, ∈_+^p} are independent, for every k ∈_+.
 
 Denote by ℒ⟶ the weak convergence on the space of ^p–valued càdlàg functions on _+ endowed with the Skorokhod metric.

Let (Z_k)_k∈_+ be a MBPM  satisfying Hypothesis A, Hypothesis C with a∘q-b∘r∈_+^p∖{0}, and the following assumptions:

    
  (i) _0^2,  I_0()^4 and X_0,1,i^4 are finite for each i=1,…,p, and ∈_+^p,
    
  (ii) D_0,i()^2=(z) and I_0,i()^2 =(z), as z→∞ for i=1,…,p,
    
  (iii) D_0,i()^4=(z^2) and I_0,i()^4=(z^2), as z→∞ for i=1,…,p.
 
    

Then 

    (n^-1_⌊ nt ⌋)_t∈_+ℒ⟶ 
        (𝒵_t v)_t∈_+    as  n→∞,

     where (𝒵_t)_t∈_+ is the pathwise unique strong solution of the SDE

    𝒵_t = u^⊤α t + √(u^⊤ (v⊙Σ)u𝒵_t^+)𝒲_t,      t∈_+,

with initial value 𝒵_0 = 0,
α = a∘q-b∘r,  
 Σ = (Σ_1,…, Σ_p) 
    and (𝒲_t)_t∈_+ a standard Wiener process.








  

  * If, besides the assumptions given in the definition of the model, we consider that, for each i ∈{1,…,p}, X_k,j,i,1, …,X_k,j,i,p are independent random variables (k ∈_+, j ∈),
 then the variance matrices are diagonal, i.e. Σ_i=diag(X_0,1,i,1,…,X_0,1,i,p), and it is easy to obtain that u^⊤ (v⊙Σ)u=∑_j=1^p u_j^2∑_i=1^p v_i X_0,1,i,j.

  * Let us give one example of emigration distributions which satisfy assumptions (ii) and (iii). For all k∈_+, i=1,…,p and z∈_+^p, let D_k,i() be 0 if z_i=0, otherwise, we define the distribution of D_k,i() as P(D_k,i()=j) ∝1/j^3, j=1,…,z_i. It is easy to obtain that D_k,i()^2=(z_i) and D_k,i()^4=(z_i^2).




§ APPENDIX

(Marcinkiewicz-Zygmund's inequality)
If  {X_n}_n≥ 1 is a  sequence of  independent and
identically distributed random variables with E[X_1]=0 and
E[|X_1|^p]<∞, p≥ 2, then
E[|∑_j=1^nX_j|^p]=O(n^p/2).



We refer the reader    to  <cit.>, p. 387 for the proof.




 §.§ Proof of Theorem <ref>
 We rewrite the MBPM as a controlled multitype branching process (see <cit.>), with ϕ_k (z)  =  z + M_k() for  k∈_+ and ∈_+^p, and apply Theorem 3.3 from <cit.>. Therefore, we only need to check Hypotheses 1–6 of that theorem. 

Hypothesis 1 follows trivially from (i).

For each 
z∈_+^p, h() = M_0()= a()∘q()-b()∘r(), 
so ε(z) = ϕ_0 (z) = Λz + α + g(z) where Λ is the identity matrix of order p, α = a∘q-b∘r and g() = h() -(a∘q-b∘r).

Consequently, Hypothesis C implies that g()=(1) as →∞ and, therefore, Hypothesis 2 in <cit.> holds.

To verify Hypothesis 3, it is enough to show that ϕ_0,i() = () as →∞ for all i=1,…,p. Indeed, taking into account the independence of the immigration and emigration components, we have

    ϕ_0,i()   = M_0,i()^2 - M_0,i()^2 
       = q_i() I_0,i()^2 + r_i()D_0,i()^2 - (q_i() I_0,i() - r_i()D_0,i())^2,

and, by Hypothesis C and (ii), we obtain Hypothesis 3.

Lyapunov's inequality yields  |X|^3^1/3≤|X|^4^1/4. 
Furthermore,

    κ_i()    = (ϕ_0,i(z) - ϕ_0,i(z))^4
       = M_0,i(z)^4 - 3M_0,i(z)^4 - 4M_0,i(z)M_0,i(z)^3 + 6 M_0,i(z)^2M_0,i(z)^2
       = q_i() I_0,i()^4 + r_i()D_0,i()^4 - 3(q_i() I_0,i() - r_i()D_0,i())^4
          -4 (q_i() I_0,i() - r_i()D_0,i())(q_i() I_0,i()^3 - r_i()D_0,i()^3) 
          + 6 (q_i() I_0,i() - r_i()D_0,i())^2 (q_i() I_0,i()^2 + r_i()D_0,i()^2).

Hence, Hypothesis 4 in <cit.> holds as a consequence of Hypothesis C, (ii) and (iii).

Hypothesis A yields Hypothesis 5 because Λ is the identity matrix which implies 𝗆 = 𝗆Λ = 𝗆.

Finally, Hypothesis 6 follows from the fact that we are assuming that P(Z_n→∞)=1 and (iv) of Remark 3.1 in <cit.>. Hence, the result follows from the aforementioned Theorem 3.3 in <cit.>.









§ ACKNOWLEDGEMENTS

We would like to thank the referee for her/his comments that helped us improve the paper.



§ FUNDING

The authors  are supported by grant PID2019-108211GB-I00 funded by MICIU/AEI/10.13039/
501100011033. P. Martín-Chávez is also grateful to the Ministerio de Ciencias, Innovación y Universidades for support from a predoctoral fellowship Grant No. FPU20/06588.
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