#### Weak Stability of Centred Quadratic Stochastic Operators

Weak Stability of Centred Quadratic Stochastic Operators
Krzysztof Bartoszek 0 1 2 3 4
Małgorzata Pułka 0 1 2 3 4
Joachim Domsta 0 1 2 3 4
B Krzysztof Bartoszek 0 1 2 3 4
0 Present Address: Department of Computer and Information Science, Linköping University , 581 83 Linköping , Sweden
1 Department of Mathematics, Uppsala University , 751 06 Uppsala , Sweden
2 Communicated by Rosihan M. Ali
3 Department of Probability and Biomathematics, Gdan ́sk University of Technology , ul. Narutowicza 11/12, 80-233 Gdan ́sk , Poland
4 The Krzysztof Brzeski Institute of Applied Informatics, The State University of Applied Sciences in Elbla ̧g , ul. Wojska Polskiego 1, 82-300 Elbla ̨g , Poland
We consider the weak convergence of iterates of so-called centred quadratic stochastic operators. These iterations allow us to study the discrete time evolution of probability distributions of vector-valued traits in populations of inbreeding or hermaphroditic species, whenever the offspring's trait is equal to an additively perturbed arithmetic mean of the parents' traits. It is shown that for the existence of a weak limit, it is sufficient that the distributions of the trait and the perturbation have a finite variance or have tails controlled by a suitable power function. In particular, probability distributions from the domain of attraction of stable distributions have found an application, although in general the limit is not stable.
Mathematics Subject Classification 60E10 · 60E07 · 60F05 · 92D15
1 Introduction
The theory of quadratic stochastic operators (QSOs) is rooted in the work of Bernstein
[6]. Such operators are applied there to model the evolution of a discrete probability
distribution of a finite number of biotypes in a process of inheritance. The problem of a
description of their trajectories was stated in [20]. Since the seventies of the twentieth
century, the field is steadily evolving in many directions, for a detailed review of
mathematical results and open problems, see [15].
In the infinite dimensional case, QSOs were first considered on the 1 space,
containing the discrete probability distributions. Many interesting models were considered
in [16] which is particularly interesting due to the presented extensions and indicated
possibilities of studying limit behaviours of infinite dimensional quadratic
stochastic operators through finite dimensional ones. A comprehensive survey of the field
(including applications of quadratic operators to quantum dynamics) can be found in
[17]. Recently, in [5], different types of asymptotic behaviours of quadratic stochastic
operators in 1 were introduced and examined in detail.
Studies of QSOs on 1 are being generalized to more complex infinite dimensional
spaces (e.g. [1,12]). The results from [5] were also subsequently generalized in two
papers [3,4] to the L1 spaces of functions integrable with respect to a specified measure,
not necessarily the counting one. Also, an algorithm to simulate the behaviour of
iterates of quadratic stochastic operators acting on the 1 space was described [2].
The study of QSOs acting on L1 spaces is more complicated, in a sense because
Schur’s lemma does not hold. To obtain results, one needs more restrictive, appropriate
for L1 spaces, assumptions on the QSO, e.g. in [4] a kernel form (cf. Definition 1) was
assumed. But even in this subclass, it is not readily possible to prove convergence of
a trajectory of a QSO. Very recently in [18,21], a more restrictive subclass of kernel
QSOs corresponding to a model which “retains the mean” (according to Eq. (
9
) of [18])
was considered. The operators are built into models of continuous time evolution of the
trait’s distribution and the size of the population. With these (and additional technical
assumptions, like bounds on moment growth), they obtained a convergence slightly
stronger than weak convergence. Here, motivated by the model described in Example
2, Section 5.4 of [18], we consider a very special but biologically extremely relevant
type of “mean retention” where the kernel of the QSO corresponds to an additive
perturbation of the parents’ traits. Specific properties related to the considered class of
QSOs and basic assumptions of our models are presented in Sects. 2 and 3. Due to the
strong restrictions, our results presented in Sect. 4 are less general than consequences
of Theorems 3 and 4 in [18]. First, we consider discrete time evolution. Moreover,
we are concerned only with weak limits. But it is the price we pay for being allowed
to drop the assumptions of kernel continuity, moment growth, technical bounds on
elements of the birth-and-death process and other elements of the continuous time
process’ generator and kernel. Also, multidimensional traits are admitted. Our main
results only require the perturbing term to have a finite second moment or alternatively
to have tails of its distribution controlled by a power function, see Theorems 2, 3 and 4.
The model lacks uniqueness of the limit—it is seed specific. The family of all
possible limits is not yet characterized. However, a construction of a wide subfamily
of possible limits is obtained. This family is obtained from the fact that the model
of additive perturbation of the parents’ mean factorizes the problem into two parts—
the first one dependent on the initial distribution and the other one dependent on the
distribution of the perturbation. Some sufficient conditions for separate existence of
the limits are given in Sect. 5. It is an open problem whether a limit exists which is
not factorizable in this way. In Sect. 6, we introduce a very special class of dyadically
α-stable probability distributions which give further insight into the stability of the
studied operators.
2 Preliminaries
We begin by putting our work in a more general context. We describe the theoretical
background of QSOs. Let (X, A ) be a separable metric space, where A stands for
the Borel σ -field. By M = M (X, A , · T V ), we denote the Banach lattice of all
signed measures on X with finite total variation where the norm is given by
F T V := sup {| F, f | : f is A − measurable, sup | f (x )| ≤ 1},
f ∈X x∈X
with F, f := X f (x ) d F (x ). By P := P (X, A ), we denote the convex set of all
probability measures on (X, A ). In our work, X can be the space of random values
of traits in inbreeding or hermaphroditic populations. Elements of P represent the
admitted single generation probability distribution of the trait. The model of heritability
is constructed with the use of quadratic stochastic operators, defined as below (we are
suitably extending the definitions given in [3]). Let M0 = M (μ) be the Banach
sublattice of M of all finite Borel measures on (X, A ) absolutely continuous with
respect to a fixed positive σ -finite measure μ and denote by P0 the set of probability
measures in M0, i.e. P0 = P ∩ M0. Clearly, M0 = L1(μ) and P0 is the convex
set of all probability densities with respect to μ.
Definition 1 A bilinear symmetric operator Q : M × M → M is called a quadratic
stochastic operator on M (briefly: QSO on M ) if for all F1, F2 ∈ M , F1, F2 ≥ 0
Q(F1, F2) ≥ 0 , and
Q(F1, F2) T V =
F1 T V
F2 T V .
The QSO Q on M is called a kernel quadratic stochastic operator if there exists a
A ⊗ A -measurable doubly-indexed family G = {G(·; x , y) : (x , y) ∈ X 2} ⊂ M of
probability measures on (X, A ), such that for F1, F2 ∈ M , G ∈ G we have
Q(F1, F2)( A) =
X×X
G( A; x , y) d F1×F2(x , y), for all A ∈ A .
Then, the family G is called the kernel of Q.
Notice that QSOs are bounded since Q(F1, F2) T V ≤ F1 T V F2 T V for all
F1, F2 ∈ M . Clearly, Q(P × P) ⊆ P and Q(P × P) ⊆ P0 if Q is a
kernel QSO with G(·, x , y) μ, for all x , y ∈ X . According to our interpretation
in terms of evolutionary biology, if F1, F2 ∈ P represent the trait distributions in
two different populations, then Q(F1, F2) ∈ P represents the distribution of this trait
in the next generation coming from the mating of independent individuals, one from
each of the two populations. Then, the nonlinear “diagonalized” mapping
P
F → Q(F ) := Q(F, F ) ∈ P.
describes the probability distribution of the offspring’s trait when F is the law of
the parents. In order to be more specific, the following Markov process is adequate
for the interpretation of the quadratic stochastic operators Q with kernel G , given by
Definition 1.
Let Ξ {n} =
Ξ1{n}, Ξ {n} , n = 0, 1, 2, . . . be a discrete time Markov process on
2
the product space X × X , with the product σ -field A ⊗ A of measurable sets. Assume
that the transition probability kernel is given by
P
Ξ {n+1} ∈ A × B | Ξ {n}
= P( A × B | Ξ {n} ), a.s.
where P( A × B | (x , y)) := G( A; x , y) · G(B; x , y), A, B ∈ A .
where F {n}( A) := P
Ξ {jn} ∈ A , j = 1, 2, satisfies the equations:
Note that the Markov operator P preserves product distributions of the form F × F
(with equal probability distribution on each coordinate). Thus, if Ξ {0} is a pair of i.i.d.
random elements of X , each following the probability distribution (p.d.) given by F {0},
then every pair Ξ {n} is also i.i.d. and follows the product distribution F {n} × F {n},
F {n+1}( A) =
X×X
G( A; x , y) d F {n} × F {n}(x , y) = Q F {n}, F {n} ( A)
= Q F {n} ( A),
for n = 0, 1, 2, . . . , A ∈ A . Therefore, the sequence of values of the iterates Qn(F ),
n = 0, 1, 2, . . ., can be seen as a model of the evolution of the probability distribution of
the X -valued trait of an inbreeding or hermaphroditic population, with F as the initial
distribution. A typical question when working with quadratic stochastic operators is
the long-term behaviour of the iterates Qn(F ), as n → ∞.
Different types of strong mixing properties of kernel quadratic stochastic operators
were considered in [3,5], on M0 ×M0 and 1 × 1, respectively. The distance between
measures is defined there by the total variation of the difference of the measures. In
particular, in these aforementioned works, equivalent conditions for uniform
asymptotic stability of such operators in terms of nonhomogeneous chains of linear Markov
operators are expressed. The study of the limit behaviour of quadratic stochastic
operators is becoming a more and more important topic, for instance, recently in [11,13,14]
non-ergodicity of QSOs was studied.
However, very often the strong convergence of distributions is not appropriate and
weak convergence for vector-valued traits will suffice. This is in particular if the
sequences consist of discrete measures and the limit distribution is not discrete. In
this situation, the total variation distance will always equal 2 hence never going to
zero. This makes strong convergence useless in such cases. Therefore, we analyse the
long-term behaviour of quadratic stochastic operators based on the weak convergence
of measures as described below.
Definition 2 Let M be the Banach lattice of all finite Borel measures on (X, A ),
where X is a complete separable metric space and A consists of all Borel sets. As
before let P stand for the convex subspace of probability measures on (X, A ). Then,
a QSO Q on M is said to be weakly asymptotically stable at F ∈ P if the weak limit
of the sequence of values of the iterations of the diagonalized operators Q at F exists
in P (we use the notation w-limn→∞Qn(F ) ∈ P ).
3 The Centred QSO in Rd
We will focus on a very specific subclass of quadratic stochastic operators which
we call centred. For this, we assume X = Rd , d ∈ N+, for the trait value space.
Correspondingly, for the domain of the QSOs, we have chosen the lattice M (d) =
M (Rd , B(d)) of all Borel finite measures (i.e. with finite total variation) on Rd .
Hence, the conditions of Definition 2 are fulfilled. For F, G ∈ M (d), the convolution
is defined by
F
G ( A) :=
Rd
F ( A − y) dG(y),
A ∈ B(d),
and for n ∈ N we denote by F ∗n the nth convolutive power of F , where F ∗0 is the
probability measure δ0 concentrated at the origin (of R(d)). Moreover, let us denote
the density-type convolution of any f, g ∈ L(1d) := L1(Rd , B(d), λ(d)) by
f
h(z) :=
Rd
f (z − y) g(y) dy, for z ∈ Rd .
For any F ∈ M (d), we define its characteristic function by
ϕF (s) :=
Rd
exp (i s · x ) d F (x ), for s ∈ Rd ,
where · stands for the canonical scalar product in Rd . Moreover, the vector of moments
of order 1 and the covariance matrix are defined by
m(F1) :=
vF :=
Rd
Rd
Rd
x d F (x ) =
x j d F (x ) : j ∈ {1, 2, . . . , d} ,
x j xk d F (x ) : ( j, k) ∈ {1, 2, . . . , d}2 ,
whenever they exist in Rd and Rd×d , respectively.
Definition 3 Let G ∈ P(d) be a probability measure on Rd . The centred QSO with
perturbation G denoted by QG is the QSO with kernel
GG :=
G(·; x , y) = G
· −
: (x , y) ∈ Rd × R
d .
Remark 1 Our work has a biological motivation in the background and the centred
QSO of Definition 3 can be interpreted as modelling traits with different values of
heritability. Heritability (in the quantitative genetic sense) looks at (amongst other
things) how the expectation of the offspring relates to the arithmetic average of the
parental traits. If the expectation of G were 0—this would relate to a heritability of 1.
Other expectations represent different families of quantitative genetic relationships.
We omit the straightforward proof that the above defined QG is a QSO. In order to
give the operator another form, let us introduce the following notation for measures
F ∈ M (d) and for densities f ∈ L(1d)
F˜ ( A) := F (2 A), for A ∈ B(d), f˜(x ) := 2 f (2 x ), for x ∈ Rd .
Proposition 1 Let F1, F2 and G be probability measures on Rd . If ξ1, ξ2 and η are
independent Rd -valued random vectors distributed according to F1, F2 and G,
respectively, then QG (F1, F2) is the probability distribution of
ζ =
In particular, if the measures are absolutely continuous with respect to the Lebesgue
measure λ(d), then their densities f1 := ddλF(d1) , f2 := ddλF(d2) , g := dλ(d) are elements
dG
of L(d) and the value of the operator at (F1, F2) is absolutely continuous, too, with
1
density
d
dλ(d) QG (F1, F2) = f˜1
f˜2
g.
As before, we pay special attention to the corresponding “diagonalized” mapping
M (d)
F −→
QG (F ) := QG (F, F ) ≡ F˜ 2
G ∈ M (d).
Equivalently, in terms of the corresponding characteristic functions we have
ϕQG (F)(s) = ϕF ( 2s ) 2 ϕG (s), for s ∈ Rd .
The above propositions and equations justify the interpretation that QG describes
the model of heritability of vector-valued traits in populations of inbreeding or
hermaphroditic species, whenever the offspring trait is equal to the additively
randomly perturbed arithmetic mean of the parents’ traits when the mating individuals
are chosen independently. Moreover, the iterates (QG )n, n ∈ N0 are then interpreted as
discrete time evolution operators acting in the space P(d) of probability distributions
of the vector-valued trait. The identity operator in P(d) is denoted then by (QG )0.
For further analysis let us introduce probability distributions F (n), G{n} ∈ P(d)
defined by their characteristic functions as follows, where s ∈ R, n ∈ N+
s
ϕF(n) (s) := ϕF 2n
and their limits, whenever they exist, are denoted by
Obviously, the operations given by Eq. (
5
) can be expressed as
F (∞) = w-lim F (n)
n→∞
G{∞} = w-lim G{n}.
n→∞
F (n) = (Qδ0 )n(F ),
G{n} = (QG )n(δ0).
Proposition 2 The characteristic function of the nth iterate of QG at F equals
ϕ(QG )n(F)(s) = ϕF(n) (s) ϕG{n} (s), for s ∈ R, n ∈ N+.
Consequently, for the nth iterate, we have
(QG )n(F ) = F (n)
G{n}, for n ∈ N+.
Proof First let us notice that according to Eqs. (
4
)–(
5
), Eq. (
8
) will hold for n = 1.
Hence, for m := n + 1 ≥ 2 by the additional use of the nth equation, we get
ϕ(QG )m(F)(s) = ϕQG ((QG )n(F))(s) = ϕ(QG )n(F) 2s 2 ϕG (s)
(
4
)
(
6
)
(
7
)
(
8
)
(
9
)
=
=
ϕF
s/2
2n
s
ϕF 2n+1
2n·2 n−1
j=0
2n+1 n
j=0
ϕG
ϕG
s/2
2 j
s
2 j
= ϕF(m) (s) ϕG{m} (s).
2 j ·2
2 j
ϕG (s)
By induction, Eq. (
8
) holds for all natural n ∈ N+.
According to Definition 2 and the Lévy–Cramér continuity theorem (see e.g.
Theorem 3.1 in [19], Chapter 13), we obtain
Corollary 1 For G ∈ P(d), the centred QSO QG is weakly asymptotically stable at
F ∈ P(d) if and only if there exists the weak limit H∞ = w-limn→∞ F (n) G{n} ∈
P(d), i.e.
ϕ(QG )n(F)(s) ≡ ϕF(n) (s) ϕG{n} (s) → ϕH∞ (s) as n → ∞, for s ∈ R.
In particular, such a limit exists whenever the limits of F (∞) and G{∞} exist in P(d).
If this holds, then w-limn→∞ QnG (F ) = H∞ = F (∞) G{∞}.
By convolution theorems, we obtain the following Corollary.
Corollary 2 (cf. Example 2 in [18], Section 5.4) Let ξ1, ξ2, ξ3, . . . and η0;1, η1;1, η1;2,
η2;1, . . . , η2;4, . . . , η j;1, . . . , η j;2 j , . . . be independent sequences of random vectors
such that all ξ -s are i.i.d. according to F and all η-s are i.i.d. according to G. Then,
for every n ∈ N+, we have
(i) F (n) is the probability distribution of the random d-dimensional vector
(ii) G{n} is the probability distribution of the random d-dimensional vector
ξ (n) :=
ξ1 + ξ2 +2n· · · + ξ2n .
η{n} :=
j=0
n−1 η j;1 + η j;2 + · · · + η j 2 j
; .
2 j
(iii) Hn := (QG )n(F ) is the probability distribution of the random d-dimensional
vector
ζn = ξ (n) + η{n}.
For the one-dimensional case (d = 1), the model of heritability determined by Eq.
(
1
) has been previously discussed in Example 2 of [18]. In their case, the perturbation
distribution G is absolutely continuous, with mean value equal to 0 and finite variance
(amongst other assumptions). Although the whole model considered there is much
more complicated (a continuous time process, with random distance between the
mating instants etc.), the limit distribution of the trait values equals the limit of the discrete
time evolution (with instants counted by the number of consecutive generations). In
what follows here, we restrict ourselves to the discrete time model and extend the
class of possible weak limits of the iterations. It is possible that the obtained class of
limits is applicable to the continuous time model of the above cited Example 2 of [18].
We leave this question as well as the study of the convergence rate open for further
investigation.
4 Main Results
Theorem 1 Let the centred QSO QG be weakly asymptotically stable at F , where
F, G ∈ P(d). Then, the limit distribution H = w-limn→∞(QG )n(F ) ∈ P(d) is
a fixed point of QG , i.e. H = QG (H ), and the characteristic functions satisfy the
equation
ϕH (s) = ϕH ( 2s ) 2 ϕG (s), s ∈ Rd .
Conversely, if the characteristic function of H ∈ P(d) satisfies the above equation
for some probability measure G ∈ P(d), then H is a fixed point of QG , and so H is
the weak limit of (QG )n(H ), as n → ∞.
In particular H ∈ P(d) is a weak limit of the sequence F (n) = (Qδ0 )n(F ) defined
by Eq. (
5
) for some F ∈ P(d), if and only if its characteristic function satisfies the
following dyadically 1-stable equation
ϕH (2s) = (ϕH (s))2 , for all s ∈ Rd .
If this holds, then starting with F = H one gets F (n) = H for every n ∈ N+ (and
F (∞) := w-limn→∞ F (n) = H , as well).
Proof By the Lévy–Cramér continuity theorem and Eq. (
4
), for every G ∈ P(d) the
operator F → QG (F ) is a continuous self-mapping of P(d) with respect to the weak
convergence. Denoting Hn := (QG )n(F ), H := w-limn→∞ Hn, by continuity we
have
QG (H ) = QG (w- lim Hn) = w- lim QG (Hn) = w- lim Hn+1 = H.
n→∞ n→∞ n→∞
Equation (
10
) and the second part of the theorem is a consequence of the equivalence
of Eqs. (
3
) and (
4
). The last claim is a particular case of the first one by taking G = δ0
the point measure concentrated at the origin of R(d), cf. Eq. (
7
).
Theorem 2 (i) A probability distribution H ∈ P(d) satisfies the dyadically 1-stable
Eq. (
11
), if for some infinitely divisible H0 ∈ P(d) the logarithms of characteristic
functions of H and H0 are related as follows
(
10
)
(
11
)
provided that the series is convergent, almost uniformly with respect to s ∈ R.
Then H = w-limn→∞(Fm )(n) where,
k∈N
ln ϕFm (s) :=
2−k+m ln(ϕH0 )(2k−m s), for s ∈ R, m ∈ Z.
(ii) Let F ∈ P(d) satisfy ϕF ( ns ) n → ϕS(s), as n → ∞ for some S ∈ P(d). Then
F (∞) = S. Moreover, in the case of d = 1, the weak limit is an element of the
extended family of Cauchy distributions with characteristic function
ϕF(∞) (s) = exp {−c|s| + i ms} , for s ∈ R, where c ≥ 0, m ∈ R.
(iii) Conditions of part (ii) are satisfied in each of the following situations:
1. F ∈ P(d) is of finite mean m(
1
) ∈ Rd ; then F (∞) is concentrated at m(
1
).
2. F ∈ P(d) is the p.d. (probability distribution) of the random vector ξ =
A(ξ ) + B, where all coordinates of ξ ∈ Rd are independent random
variables, satisfying the condition of part (ii) transformed by a (deterministic)
linear map A : Rd → Rd and shifted by a (deterministic) vector B ∈ Rd .
Proof The infinite divisibility of H0 implies that all terms of the series in part (i)
are logarithms of characteristic functions (of probability distributions on R(d)), and
therefore, the partial sums are also. By almost uniform convergence, the infinite sum is
a logarithm of a characteristic function, as well. Equation (
11
) follows now by standard
properties of limits. The second part of (i) holds since (Fm )(n) = Fm+n, for m ∈ Z,
n ∈ N.
The first part of (ii) follows since ϕF(n) (s) n∈N is a subsequence of ϕF ( ns ) n n∈N+ .
For the one-dimensional case, by standard analysis, the limit S is stable with
characteristic exponent 1 (unless concentrated at a single point) and in every case its
characteristic functions can be obtained (see e.g. Eq. (
3
) in [19], Chapter 15, Section
3)
ln ϕS(s) = i m s − c |s| (1 + i θ sign(s) ln |s|),
s n
where c ≥ 0, m ∈ R, |θ | ≤ 1. Moreover, by properties of limits, we have ϕS( n )
ϕS(s), for all s ∈ R, n ∈ N+, implying that θ = 0.
=
Case 1 of (iii) follows from the strong law of large numbers combined with Corollary
2(i).
Case 2 of (iii) We denote by ◦ the composition of matrices and treat vectors as rows
unless transposed by T to columns. Using a random vector ξ with p.d. F ∈ P(d) and
the expectation functional E, we can write for s ∈ R,
ϕF (s) = ϕξ (s) = E{exp(i s ◦ ξ T )} = E{exp(i s ◦ ( A ◦ ξ T + BT ))}
= E{exp(i (s ◦ A) ◦ ξ T + i s ◦ BT )} = ϕξ (s ◦ A) exp(i s ◦ BT ).
Thus, the probability distribution of ξ is in the domain of attraction of a p.d. on Rd
given by the following limit characteristic function
lim
n→∞
ϕξ ( n1 s)
n
= exp(i s ◦ BT ) lim
n→∞
d
= exp(i s ◦ BT )
k=1
ϕξ ( n1 s ◦ A)
n
ϕSk (s · A,k ),
where A,k stands for the kth column of the coefficient matrix A and Sk is the attracting
p.d. of the extended Cauchy type for the kth coordinate of ξ , k = 1, 2, . . . , d .
Remark 2 It is possible to give an example of a dyadically 1-stable H law that will
not be 1-stable. Namely, such a “partially 1-stable” H example is provided by P. Lévy.
Take, ϕH (ns) = (ϕH (s))n, where the equality is valid for n = 2k , k = 0, 1, 2, . . .
only. This is a particular case of Theorem 2 (i) with d = 1, where H0 is the p.d. of the
difference of two i.i.d. Poisson random variables with ln(ϕH0 (s)) = −1 + cos s (cf.
Section 17.3 in [10]). Regarding cases (ii) and (iii) of Theorem 2, it is worth noting
that many authors (e.g. Section 8.8 in [9]) provide a description of a wider class of
1-stable multidimensional distributions.
The following propositions present examples of conditions which are sufficient
for the existence of the weak limit of the sequences of probability distributions G{n}
defined by Eq. (
5
).
Theorem 3 Let the probability distribution G ∈ P(d) be of finite second moments.
Assume that m := m(G1) = 0 ∈ Rd . Then w-limn→∞ G{n} = G{∞}, where
ϕG{∞} (s) = nl→im∞ ϕG{n} (s) =
∞
j=0
ϕG (2− j s)
2 j
, s ∈ Rd .
(
12
)
Moreover m(
1
)
G{∞} = 0 and regarding the covariance we have vG{∞} = 2vG .
Proof According to Corollary 2, G{n} is the p.d. of the sum η{n} :=
independent averages U j := (η j;1 + η j;2 + · · · + η j;2 j )/2 j , j = 0, 1, 2 . . ., where all
η j;k are i.i.d. according to G. By the assumptions on G, we have mU(1j) = 0 ∈ Rd and
vU j = 2− j vG for every j = 0, 1, 2, . . . . Hence, the series η{∞} := limn→∞ η{n} =
∞
j=0 U j converges almost surely (as it converges coordinatewise, cf. Theorem 2.5.3 in
[8]) to a random vector with mean 0 and covariance matrix equalling ∞j=0 2− j vG =
2vG . In particular, the probability distributions G{n} of η{n} converge weakly to the
probability distribution of η{∞}, which we denote by G{∞}. Now Eq. (
12
) holds by
the continuity theorem.
nj=−01 U j , of
Theorem 4 Let the one-dimensional probability distribution G ∈ P(
1
) satisfy the
following condition
| ln ϕG (s)| ≤ C |s|1+ε for |s| ≤ s0, where s0 > 0, ε ∈ (0, 1], C > 0.
Then, the sequence of p.ds. G{n} ∈ P(
1
), n ∈ N, defined according by Eq. (
5
)
converges weakly to a p.d. G{∞} ∈ P(
1
) determined by the infinite product of
characteristic functions Eq. (
12
), convergent almost uniformly with respect to s ∈ R.
Proof Due to the bounds on ϕG , for any positive real number T > 0, there exists a
natural number J ∈ N such that
2 j
s
ln ϕG 2 j
Therefore, the infinite product in Eq. (
12
) with respect to j ∈ N is almost uniformly
convergent and defines a characteristic function of a probability measure G{∞} on R.
Then, by the continuity theorem, this measure is the weak limit of G{n}, as n → ∞.
Remark 3 The above assumption on ln ϕG implies m(G1) = 0 ∈ Rd . Moreover, the
assumed behaviour of the function ln ϕG near zero can be equivalently replaced by
estimates on ϕG − 1. This is a direct consequence of the following inequalities
|a|(1 − |a|) ≤ | ln(1 + a)| ≤ |a|(1 + |a|), for a ∈ C, |a| < 0.5.
Corollary 3 Let η ∈ Rd be a random vector with independent coordinates ηk
distributed according to Gk ∈ P(
1
), satisfying the condition of Theorem 4 with (possibly
different) parameters s0,k > 0, εk ∈ (0, 1] and Ak > 0, k = 1, 2, . . . , d , respectively.
Moreover, let G ∈ P(d) be the p.d. of the random vector η = A(η ), transformed
from η by a (deterministic) linear map A : Rd → Rd . Then the sequence of p.ds.
G{n}, n ∈ N, given by Eq. (
5
) converges weakly to a p.d. G{∞} ∈ P(d) determined by
the infinite product in Eq. (
12
).
Proof In terms of η we may write
ϕG (s) = ϕη(s) = ϕη (s ◦ A), for s ∈ Rd .
Therefore, the j th factor of the infinite product in Eq. (
12
) can be written as follows
2 j
ϕη 21j s
=
ϕη
21j s ◦ A
2 j
=
d
k=1
ϕGk 21j s · A,k
2 j
,
where A,k stands for the kth column of the coefficient matrix A. Now, taking into
account the assumptions and boundedness of the scalar product, according by
Theorem 4 the infinite product
ϕG{k∞} (s) := ϕGk{∞} (s · A,k ) =
∞
j=0
ϕGk 21j (s · A,k )
2 j
, for s ∈ Rd ,
converges almost uniformly with respect to s ∈ Rd to the characteristic function of
the probability distribution of the random vector ηk A,k , for every k = 1, 2, . . . , d .
Thus, the infinite product in Eq. (
12
) defines the weak limit of G{n}. This limit is the
convolution of the limits G{1∞} G{2∞} . . . G{d∞} ∈ P(d).
Theorem 5 If the modulus of the characteristic function G ∈ P(
1
) is bounded as
|ϕG (s)| ≤ exp −C |s|1+ε for |s| ≤ s0, where s0 > 0, ε ∈ (0, 1], C > 0,
then
|ϕG{∞} (s)| ≤ exp −2−εC s0ε |s| , for |s| ≥ s0,
whenever G{∞} := w-limn→∞ G{n} exists in P(
1
). In particular, G{∞} is absolutely
continuous with respect to the Lebesgue measure λ(
1
).
Proof The modulus of each factor of the convergent infinite product in Eq. (
12
) is not
greater than 1, since they are all characteristic functions. Thus, it suffices to obtain the
desired estimates for at least one suitably chosen factor. Let us fix |s| ≥ s0 and assign
to it a natural n(s) := log2 |ss0| ∈ N. Obviously, if n = n(s), then
|s|
2n+1 < s0,
and
2−n
and therefore by our assumptions the following estimates hold
|ϕG{∞} (s)| ≤ ϕG
s
2n+1
2n+1
≤ exp
−C
1+ε
2n+1
s0
≥ s
| |
|s|
2n+1
≤ exp −2−εC |s|1+ε 2−n ε
≤ exp
−2−εC |s|1+ε
≤ exp −2−εC s0ε |s| ,
s0
|s|
ε
as required. Since this estimates hold for all |s| ≥ s0, the obtained bound implies that
ϕG{∞} is integrable over R. Hence, the absolute continuity of the limit p.d., G{∞},
follows from the inverse Fourier transformation theorem (see e.g. Theorem 3.3.5. in
[8]).
5 Examples and Problems
The conditions on the logarithm of the kernel’s characteristic function in Theorems
4 and 5 are not “uncommon” ones. Besides distributions with finite variance, there is
a large class of heavy tailed distributions (on R) with moments of order less than 2,
which cover interesting domains of attraction to stable p.ds. For illustrative purposes,
we consider some specific cases. We note for what follows that Markov’s inequality
implies the following estimate of the tails, for arbitrary H ∈ P(
1
),
H ((−∞, −x ]) + H ([x , ∞)) ≤ x − p
R |u| pd H (u), for x > 0, p > 0.
Proposition 3 If G ∈ P(
1
) is symmetric and satisfies
G(−∞, −x ] = G[x , ∞) ≤ C x −(1+ε), for x > 0, where ε ∈ (
0, 1
), C > 0,
then, possibly with another constant C > 0, we have
| ln ϕG (s)| < C |s|1+ε, for s ∈ R
(equivalently, |ϕG (s) − 1| < C |s|1+ε ). In particular, G satisfies conditions of
Theorem 4.
Proof Due to the symmetry of G it suffices to consider only positive s > 0. Moreover,
for every A > 0 we have
|1 − ϕG (s)| = 1 −
eisx dG(x ) =
(1 − cos(s x )) dG(x )
Integrating the first term by parts and taking A = π/s, we obtain
I := 2
For the second term, again with A = π/s we have
I I := 4
[A,∞)
dG(x ) = 4G([ A, ∞)) ≤ 4 C π −(1+ε)s1+ε.
Next, let us indicate some stronger results, based on stability theory, for the asymptotic
behaviour of the characteristic functions near the origin. Obviously, stronger
assumptions will be required. The following two propositions provide simple tests for the
existence and continuity of the limit distributions that may be easily exploited by the
applied user. Since they follow from well known facts about stable distributions, the
proofs are omitted. The interested reader is referred e.g. to Theorem 5 of Section 7.4
in [7], Proposition 2.2.13 in [9] or Chapter 17 in Feller’s book [10]. In order to make
the propositions more approachable we assume that, as x → ∞,
G(−∞, −x ] = C x −(1+ε) + o x −(1+ε) ,
G[x , ∞) = C x −(1+ε) + o x −(1+ε) .
(
13
)
Proposition 4 If G ∈ P (
1
) with mean value 0 has tails satisfying Eq. (
13
) for some
constants C > 0 and ε ∈ (
0, 1
) then
ϕG (s) − 1 = −2 C c(ε) |s|1+ε + o(|s|1+ε), as s → 0,
where c(ε) = ε−1 (1 + ε) Γ (1 − ε) sin( π2 ε). In particular,
– G is in the domain of attraction of a (1 + ε)-stable p.d. with characteristic function
ϕG(∞) (s) = exp(−c|s|1+ε), where c = 2C c(ε) is a positive constant;
– the assumptions of Theorems 4 and 5 are satisfied, implying that the limit G{∞}
given by Eq. (
12
) exists and is absolutely continuous.
The specific properties of 1-stable distributions allow us to make further statements
under only slightly stronger assumptions.
Proposition 5 If the tails of a symmetric p.d. F ∈ P (
1
) satisfy Eq. (
13
) with ε = 0
(and G replaced by F ), then the characteristic function ϕF , satisfies the following
ϕF (s) − 1 = −π C |s| + o(|s|), as s → 0.
In particular, the assumptions of Theorem 2 (ii) are fulfilled, implying that F is in the
domain of attraction of the Cauchy p.d.
ϕF(∞) (s) = nl→im∞
s
ϕF n
n
= exp(−C π |s|), for s ∈ R.
Remark 4 We now give some examples of assumptions on a symmetric distribution
G ∈ P (
1
), which are sufficient to satisfy Eq. (
13
).
– For negative x < 0, G(−∞, x ] = ((uv((xx))))βα , is a positive increasing function, where
u and v are polynomials of degree l and m, respectively, with ε := mβ − lα − 1,
ε ≥ 0;
– G{Z} = 1 and G{ j } = ((uv(( jj))))βα , j > 0, where u and v are positive on Z+
polynomials of degrees l and m, respectively, with ε := mβ − lα − 2 ≥ 0; for
1
instance, this holds if G{ j } = C | j |2+ε for k = 0;
2+ε
– ddλG(
1
) (x ) = C (1 + a|x − μ|α ) α , x ∈ R, where α > 0, ε ≥ 0.
Remark 5 According to Corollary 1, for G ∈ P (d) the centred QSO QG is weakly
stable at F ∈ P (d), whenever the weak limit H∞ of the convolutions F (n) G{n}
exists in P (d), as n → ∞. The main results supply some sufficient conditions for
the existence of the limits separately for F (n) and G{n}. It follows that the limit H∞
is independent of F only within such families of F which possess a common limit
F (∞) ∈ P (d). In particular, the class can consist of distributions with common mean
value m(
1
) ∈ Rd . In a more general class of initial probability distributions F which
possess the limit F (∞) ∈ P (d), the stability is equivalent to existence of the weak
limit G{∞} ∈ P (d). Can the limits exist separately? Indeed, by Theorem 1, it suffices
that writing H for F Eq. (
10
) is satisfied. Then F is a fixed point of QG and the limit
of (QG )n (F ) equals F as the sequence is constant. We leave it as an open problem
whether there are solutions of Eq. (
10
), for which G{n} is not weakly convergent to a
p.d. on Rd .
6 Dyadically Stable Distribution
We conclude our work by considering a special kind of stable p.ds.
Definition 4 A one-dimensional p.d. F ∈ P (
1
) is said to be dyadically α-stable if it
is infinitely divisible and its characteristic function satisfies, cf. Eq. (
11
),
ϕF (2 s) = (ϕF (s))2α , for s ∈ R, where α ∈ (0, 2].
(
14
)
Remark 6 It is worth reminding the reader that the assumed infinite divisibility of F
ensures that ϕF is strictly non-zero. Hence, the right-hand side of Eq. 14 is well defined
as the unique continuous at the origin branch of a power function of a nowhere equal
zero complex valued continuous function. Furthermore, the right-hand side of Eq. 14
as a positive power of characteristic function of an infinitely divisible distribution will
remain a characteristic function of an infinitely divisible distribution.
A wide family of dyadically 1-stable probability distributions is generated through
Theorem 2 (i). Following this, one can also prove the following.
Proposition 6 A probability distribution H ∈ P (d) satisfies Eq. (
14
), if for some
infinitely divisible H0 ∈ P (d) the logarithms of characteristic functions of H and H0
are related as follows,
k∈Z
provided that the series is convergent, almost uniformly with respect to s ∈ R.
Theorem 6 For α ∈ (1, 2], every one-dimensional dyadically α-stable p.d. H ∈ P (
1
)
is a weak limit of iterates of some centred QSO QG (δ0). For this relationship between
H and G to hold, it is sufficient that ϕG (s) = (ϕH (s)) 2α2α−2 . Furthermore, then H is a
fixed point of QG .
Proof By definition, G is also dyadically α-stable, and therefore we have (ϕG ( 2s )) =
(ϕG (s))2−α . By multiple application of the dyadic division, we obtain the following
limit for G{n} = (QG )n (δ0), as n → ∞,
n−1
j =0
ϕG
s
2 j
2 j
=
n−1
j =0
Thus by Theorem 1 we also obtain that H is a fixed point of QG .
1 2α
→ (ϕG (s)) 1−2(1−α) = (ϕG (s)) 2α −2 = ϕH (s).
We close by remarking that dyadically 1-stable distributions form the whole set of
weak limits of F (n), n ∈ N, with F running over P (
1
), cf. Theorem 1. On the other
hand, the union of the families of all dyadically α-stable distributions, α ∈ (1, 2], do
not cover the family of all weak limits of G{n}, n ∈ N. Take for example G equal
to a convolution of two dyadically stable distributions with different exponents, say
1 < α < β ≤ 2, both not concentrated at the origin (of R). Then, repeating the above
proof, one gets that the limit is again a convolution of two such distributions, which
is not dyadically stable with any exponent.
Acknowledgements We would like to acknowledge Wojciech Bartoszek for many helpful comments and
insights. We thank two anonymous reviewers whose comments significantly improved the work.
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