Ergodicity of non-homogeneous $$\mathbf {p}$$ p -majorizing quadratic stochastic operators
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Positivity
Ergodicity of non-homogeneous p-majorizing quadratic stochastic operators Mansoor Saburov1 Received: 31 January 2019 / Accepted: 4 December 2019 © Springer Nature Switzerland AG 2019
Abstract In this paper, we study the strong ergodicity of non-homogeneous p-majorizing quadratic stochastic operators acting on the finite dimension simplex. We first provide a criterion for the strong ergodicity of such operators. We then establish the strong ergodicity of non-homogeneous p-majorizing quadratic stochastic operators associated with scrambling, Sarymsakov, and Wolfowitz cubic p-stochastic matrices. Keywords Quadratic stochastic operator · p-Majorizing operator · Scrambling cubic stochastic matrix · Sarymsakov cubic stochastic matrix · Wolfowitz cubic stochastic matrix Mathematics Subject Classification 47H25 · 47H60 · 60J10 · 60K37
1 Introduction The classical vector majorization is a preorder of dispersion for vectors with the same length and the same sum of components (see [29]). A statement that x ≺ y can be regarded as saying that the elements of x are more equal than those of y. In other words, x is closer than y to the vector e = (e, . . . , e) with all components equal. Hence, the classical vector majorization can be viewed as a preorder of distance from the uniform vector e = (e, . . . , e). A preorder of distance from any fixed vector p = ( p1 , . . . , pm ) with positive components, the so-called p-majorization, is a generalization of the classical vector majorization. In the literature, these kinds of majorizations were called by different names such as d-majorization in [29,60], rmajorization in [21], q-majorization in [8,22], and p-majorization in [11] by refereeing to the reference vector in each case. In this paper, we choose p as the reference vector and stick to the name p-majorization. Several equivalent definitions of p-majorizations
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Mansoor Saburov [email protected]; [email protected] College of Engineering and Technology, American University of the Middle East, Egaila, Kuwait
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M. Saburov
and related concepts are discussed in [21,29]. Let us first provide some necessary notions and notations related to p-majorizations. m |x Let x1 = m k=1 k | be a norm of a vector x = (x 1 , . . . , x m ) ∈ R . We say that m−1 = {x ∈ Rm : x ≥ 0 (resp. x > 0) if xk ≥ 0 (resp. xk > 0) for all 1 ≤ k ≤ m. Let S x1 = 1, x ≥ 0} be the (m − 1)-dimensional standard simplex. An element of the simplex Sm−1 is called a stochastic vector. We setup intSm−1 = {x ∈ Sm−1 : x > 0} and ∂Sm−1 = Sm−1 \ intSm−1 . m Recall that a square matrix P = pi j i, j=1 is called stochastic if every row is a m stochastic vector. A square matrix P = pi j i, j=1 is called doubly stochastic if every row and column are stochastic vectors. For a given vector x = (x1 , . . . , xm ) ∈ Rm , let x[1] ≥ · · · ≥ x[m] denote the components of x in a non-increasing order. We say that x is majorized by y, written x ≺ y, if one has that k i=1
x[i] ≤
k i=1
y[i] , 1 ≤ k ≤ m,
m
xi =
i=1
m
yi .
i=1
The
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