Weak Stability of Centred Quadratic Stochastic Operators
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Abstract
We consider the weak convergence of iterates of socalled centred quadratic stochastic operators. These iterations allow us to study the discrete time evolution of probability distributions of vectorvalued 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.
Keywords
Asymptotic stability Dyadic stability Infinite divisible distributions Quadratic stochastic operators Weak convergenceMathematics Subject Classification
60E10 60E07 60F05 92D151 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 \(\ell _{1}\) 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 \(\ell _{1}\) were introduced and examined in detail.
Studies of QSOs on \(\ell _{1}\) 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 \(L_{1}\) 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 \(\ell _{1}\) space was described [2].
The study of QSOs acting on \(L_{1}\) spaces is more complicated, in a sense because Schur’s lemma does not hold. To obtain results, one needs more restrictive, appropriate for \(L_{1}\) 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 birthanddeath 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\(\alpha \)stable probability distributions which give further insight into the stability of the studied operators.
2 Preliminaries
Definition 1
Different types of strong mixing properties of kernel quadratic stochastic operators were considered in [3, 5], on \({\mathscr {M}}_{0} \times {\mathscr {M}}_{0}\) and \(\ell _{1} \times \ell _{1}\), 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] nonergodicity of QSOs was studied.
However, very often the strong convergence of distributions is not appropriate and weak convergence for vectorvalued 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 longterm behaviour of quadratic stochastic operators based on the weak convergence of measures as described below.
Definition 2
Let \({\mathscr {M}}\) be the Banach lattice of all finite Borel measures on \(( X, {\mathscr {A}})\), where X is a complete separable metric space and \({\mathscr {A}}\) consists of all Borel sets. As before let \({\mathscr {P}}\) stand for the convex subspace of probability measures on \(( X, {\mathscr {A}})\). Then, a QSO \({\mathbf {Q}}\) on \({\mathscr {M}}\) is said to be weakly asymptotically stable at\(F\in {\mathscr {P}}\) if the weak limit of the sequence of values of the iterations of the diagonalized operators \({\mathbb {Q}}\) at F exists in \({\mathscr {P}}\) (we use the notation Open image in new window ).
3 The Centred QSO in \({\mathbb {R}}^{d}\)
Definition 3
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.
Proposition 1
Proposition 2
Proof
According to Definition 2 and the Lévy–Cramér continuity theorem (see e.g. Theorem 3.1 in [19], Chapter 13), we obtain
Corollary 1
By convolution theorems, we obtain the following Corollary.
Corollary 2
 (i)\(F^{(n)}\) is the probability distribution of the random ddimensional vector$$\begin{aligned} \xi ^{(n)} := \frac{\xi _{1}+\xi _{2}+\cdots +\xi _{2^{n}}}{2^{n}}. \end{aligned}$$
 (ii)\(G^{\{n\}}\) is the probability distribution of the random ddimensional vector$$\begin{aligned} \eta ^{\{n\}} := \sum \limits _{j=0}^{n1} \frac{\eta _{j;1}+\eta _{j;2}+\cdots +\eta _{j;2^{j}}}{2^{j}}. \end{aligned}$$
 (iii)\(H_n := ({\mathbb {Q}}_G)^n(F)\) is the probability distribution of the random ddimensional vector$$\begin{aligned} \zeta _{n} = \xi ^{(n)} + \eta ^{\{n\}}. \end{aligned}$$
For the onedimensional 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
Proof
Theorem 2
 (i)A probability distribution \(H\in {\mathscr {P}}^{(d)}\) satisfies the dyadically 1stable Eq. (11), if for some infinitely divisible \(H_{0}\in {\mathscr {P}}^{(d)}\) the logarithms of characteristic functions of H and \({H_{0}}\) are related as followsprovided that the series is convergent, almost uniformly with respect to \(s \in {\mathbb {R}}\). Then Open image in new window where,$$\begin{aligned} \ln (\varphi _H) (s) = \sum _{k\in {\mathbb {Z}}}\, 2^{k} \ln (\varphi _{H_{0}}) (2^{k} s) \end{aligned}$$$$\begin{aligned} \ln \left( \varphi _{F_{m}} (s) \right) := \sum _{k\in {\mathbb {N}}}\, 2^{k+m} \ln (\varphi _{H_0}) (2^{km} s), \; \text{ for } \; s\in {\mathbb {R}}, \; m \in {\mathbb {Z}}. \end{aligned}$$
 (ii)Let \(F\in {\mathscr {P}}^{(d)} \) satisfy \(\left( \varphi _F(\tfrac{s}{n}) \right) ^n \rightarrow \varphi _S(s)\), as \(n \rightarrow \infty \) for some \(S\in {\mathscr {P}}^{(d)}\). Then \(F^{(\infty )}= S\). Moreover, in the case of \(d=1\), the weak limit is an element of the extended family of Cauchy distributions with characteristic function$$\begin{aligned} \varphi _{F^{(\infty )}}(s ) = \exp \left\{  c \vert s \vert + i m s \right\} , \; \text{ for } \; s \in {\mathbb {R}}, \; \text{ where } \; c \ge 0, \; m \in {\mathbb {R}}. \end{aligned}$$
 (iii)Conditions of part (ii) are satisfied in each of the following situations:
 1.
\(F\in {\mathscr {P}}^{(d)} \) is of finite mean \( m^{(1)} \in {\mathbb {R}}^d\); then \(F^{(\infty )}\) is concentrated at \(m^{(1)} \).
 2.
\(F\in {\mathscr {P}}^{(d)} \) is the p.d. (probability distribution) of the random vector \( \xi = A ( \xi ') + B\), where all coordinates of \(\xi ' \in {\mathbb {R}}^{d'}\) are independent random variables, satisfying the condition of part (ii) transformed by a (deterministic) linear map \(A: {\mathbb {R}}^{d'} \rightarrow {\mathbb {R}}^{d}\) and shifted by a (deterministic) vector \(B \in {\mathbb {R}}^{d}\).
 1.
Proof
The infinite divisibility of \(H_{0}\) implies that all terms of the series in part (i) are logarithms of characteristic functions (of probability distributions on \({\mathbb {R}}^{(d)}\)), 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 \((F_{m})^{(n)} = F_{m+n}\), for \( m \in {\mathbb {Z}}\), \( n \in {\mathbb {N}}\).
Case 1 of (iii) follows from the strong law of large numbers combined with Corollary 2(i).
Remark 2
It is possible to give an example of a dyadically 1stable H law that will not be 1stable. Namely, such a “partially 1stable” H example is provided by P. Lévy. Take, \(\varphi _{H}(n s ) = \left( \varphi _{H}(s )\right) ^{n}\), where the equality is valid for \(n= 2^{k}\), \(k=0,1,2, \ldots \) only. This is a particular case of Theorem 2 (i) with \(d=1\), where \(H_{0}\) is the p.d. of the difference of two i.i.d. Poisson random variables with \(\ln ( \varphi _{H_{0}} (s)) = 1 + \cos s \) (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 1stable 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
Proof
According to Corollary 2, \(G^{\{n\}}\) is the p.d. of the sum \(\eta ^{\{n\}} := \sum _{j=0}^{n1}U_{j}\), of independent averages \(U_{j} := (\eta _{j;1}+\eta _{j;2}+\cdots +\eta _{j;2^j})/2^{j}\), \(j = 0,1,2 \ldots \), where all \(\eta _{j;k}\) are i.i.d. according to G. By the assumptions on G, we have \(m_{U_{j}}^{(1)} =0 \in {\mathbb {R}}^d\) and \(v_{U_{j}}=2^{j}{v_{G}}\) for every \(j = 0,1,2,\ldots \) . Hence, the series \(\eta ^{\{\infty \}} := \lim _{n\rightarrow \infty } \eta ^{\{n\}} = \sum _{j=0}^{\infty }U_{j}\) 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 \( \sum _{j=0}^{\infty } \, 2^{j} v_G = 2 v_{G}\). In particular, the probability distributions \(G^{\{n\}}\) of \(\eta ^{\{n\}}\) converge weakly to the probability distribution of \(\eta ^{\{\infty \}}\), which we denote by \(G^{\{\infty \}}\). Now Eq. (12) holds by the continuity theorem.
Theorem 4
Proof
Remark 3
Corollary 3
Let \(\eta ' \in {\mathbb {R}}^{d'}\) be a random vector with independent coordinates \(\eta _k'\) distributed according to \(G_k' \in {\mathscr {P}}^{(1)}\), satisfying the condition of Theorem 4 with (possibly different) parameters \(s_{0,k} >0 \), \(\varepsilon _k\in (0,1]\) and \(A_{k}>0\), \(k = 1,2,\ldots , d'\), respectively. Moreover, let \(G\in {\mathscr {P}}^{(d)} \) be the p.d. of the random vector \( \eta = A ( \eta ')\), transformed from \(\eta '\) by a (deterministic) linear map \(A: {\mathbb {R}}^{d'} \rightarrow {\mathbb {R}}^{d}\). Then the sequence of p.ds. \(G^{\{n\}}\), \(n \in {\mathbb {N}}\), given by Eq. (5) converges weakly to a p.d. \(G^{\{\infty \}} \in {\mathscr {P}}^{(d)}\) determined by the infinite product in Eq. (12).
Proof
Theorem 5
Proof
5 Examples and Problems
Proposition 3
Proof
Proposition 4

G is in the domain of attraction of a \((1+ \varepsilon )\)stable p.d. with characteristic function \(\varphi _{G^{(\infty )}}(s) =\exp (  c \vert s \vert ^{1+ \varepsilon })\), where \(c = 2 C c(\varepsilon )\) is a positive constant;

the assumptions of Theorems 4 and 5 are satisfied, implying that the limit \(G^{\{\infty \}}\) given by Eq. (12) exists and is absolutely continuous.
The specific properties of 1stable distributions allow us to make further statements under only slightly stronger assumptions.
Proposition 5
Remark 4

For negative \(x<0\), \(G(\infty , x] = \frac{ ( u(x) )^{\alpha } }{ (v(x))^{\beta } }\), is a positive increasing function, where u and v are polynomials of degree l and m, respectively, with \(\varepsilon := m \beta  l \alpha 1\), \(\varepsilon \ge 0\);

\(G\{ {\mathbb {Z}}\} =1\) and \(G\{j\} = \frac{ ( u(j) )^{\alpha } }{ (v(j))^{\beta } }, \; j>0\), where u and v are positive on \({\mathbb {Z}}_{+}\) polynomials of degrees l and m, respectively, with \( \varepsilon := m \beta  l \alpha 2 \ge 0\); for instance, this holds if \(G\{j\} = C \frac{1}{\vert j \vert ^{2 + \varepsilon }}\) for \(k\ne 0\);

\( \frac{\mathrm {d}G}{\mathrm {d}\lambda ^{(1)}}(x) = C \left( 1 + a \vert x  \mu \vert ^\alpha \right) ^{\tfrac{2 + \varepsilon }{ \alpha }}, \; x \in {\mathbb {R}}, \) where \(\alpha >0, \varepsilon \ge 0\).
Remark 5
According to Corollary 1, for \(G \in {\mathscr {P}}^{(d)}\) the centred QSO \({\mathbf {Q}}_{G}\) is weakly stable at \(F\in {\mathscr {P}}^{(d)}\), whenever the weak limit \(H_{\infty }\) of the convolutions \( F^{(n)} \star G^{\{n\}}\) exists in \({\mathscr {P}}^{(d)}\), as \(n \rightarrow \infty \). 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 \(H_{\infty }\) is independent of F only within such families of F which possess a common limit \(F^{(\infty )} \in {\mathscr {P}}^{(d)}\). In particular, the class can consist of distributions with common mean value \(m^{(1)} \in {\mathbb {R}}^{d}\). In a more general class of initial probability distributions F which possess the limit \(F^{(\infty )} \in {\mathscr {P}}^{(d)}\), the stability is equivalent to existence of the weak limit \(G^{\{\infty \}} \in {\mathscr {P}}^{(d)}\). 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 \({\mathbb {Q}}_{G}\) and the limit of \(({\mathbb {Q}}_{G})^{n}(F)\) 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 \({\mathbb {R}}^{d}\).
6 Dyadically Stable Distribution
We conclude our work by considering a special kind of stable p.ds.
Definition 4
Remark 6
It is worth reminding the reader that the assumed infinite divisibility of F ensures that \(\varphi _{F}\) is strictly nonzero. Hence, the righthand 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 righthand 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 1stable probability distributions is generated through Theorem 2 (i). Following this, one can also prove the following.
Proposition 6
Theorem 6
For \(\alpha \in (1,2]\), every onedimensional dyadically \(\alpha \)stable p.d. \(H \in {\mathscr {P}}^{(1)}\) is a weak limit of iterates of some centred QSO \({\mathbb {Q}}_{G}(\delta _{0})\). For this relationship between H and G to hold, it is sufficient that \(\varphi _{G}(s) = \left( \varphi _H(s)\right) ^{\frac{2^\alpha 2}{2^\alpha }}\). Furthermore, then H is a fixed point of \({\mathbb {Q}}_{G}\).
Proof
We close by remarking that dyadically 1stable distributions form the whole set of weak limits of \(F^{(n)}\), \(n \in {\mathbb {N}}\), with F running over \({\mathscr {P}}^{(1)}\), cf. Theorem 1. On the other hand, the union of the families of all dyadically \(\alpha \)stable distributions, \(\alpha \in (1,2]\), do not cover the family of all weak limits of \(G^{\{n\}}\), \(n \in {\mathbb {N}}\). Take for example G equal to a convolution of two dyadically stable distributions with different exponents, say \(1< \alpha < \beta \le 2\), both not concentrated at the origin (of \({\mathbb {R}}\)). 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.
Notes
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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