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• W2570667799 abstract "Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=bold-italic xi equals left-brace xi Subscript n Baseline right-brace Subscript n greater-than-or-equal-to 0> <mml:semantics> <mml:mrow> <mml:mi mathvariant=bold-italic>ξ<!-- ξ --></mml:mi> <mml:mo>=</mml:mo> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:msub> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:msub> <mml:mo fence=false stretchy=false>}</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>n</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>boldsymbol {xi }={xi _n}_{nge 0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a Markov chain defined on a probability space <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis normal upper Omega comma script upper F comma double-struck upper P right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> <mml:mo>,</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=script>F</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>P</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(Omega ,mathscr {F},mathbb {P})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> valued in a discrete topological space <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=bold-italic upper S> <mml:semantics> <mml:mi mathvariant=bold-italic>S</mml:mi> <mml:annotation encoding=application/x-tex>boldsymbol {S}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that consists of a finite number of real <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d times d> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>×<!-- × --></mml:mo> <mml:mi>d</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>dtimes d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> matrices. As usual, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=bold-italic xi> <mml:semantics> <mml:mi mathvariant=bold-italic>ξ<!-- ξ --></mml:mi> <mml:annotation encoding=application/x-tex>boldsymbol {xi }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is called <italic>uniformly exponentially stable</italic> if there exist two constants <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C greater-than 0> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>C>0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=0 greater-than lamda greater-than 1> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>></mml:mo> <mml:mi>λ<!-- λ --></mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>0>lambda >1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <disp-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=StartLayout 1st Row double-struck upper P left-parenthesis double-vertical-bar xi 0 left-parenthesis omega right-parenthesis midline-horizontal-ellipsis xi Subscript n minus 1 Baseline left-parenthesis omega right-parenthesis double-vertical-bar less-than-or-equal-to upper C lamda Superscript n Baseline for-all n greater-than-or-equal-to 1 right-parenthesis equals 1 semicolon EndLayout> <mml:semantics> <mml:mtable rowspacing=3pt columnspacing=1em side=left displaystyle=true> <mml:mtr> <mml:mtd> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>P</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo fence=false stretchy=false>‖<!-- ‖ --></mml:mo> <mml:msub> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>ω<!-- ω --></mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>⋯<!-- ⋯ --></mml:mo> <mml:msub> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>n</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>ω<!-- ω --></mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo fence=false stretchy=false>‖<!-- ‖ --></mml:mo> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>C</mml:mi> <mml:msup> <mml:mi>λ<!-- λ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> <mml:mtext> </mml:mtext> <mml:mi mathvariant=normal>∀<!-- ∀ --></mml:mi> <mml:mi>n</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>;</mml:mo> </mml:mtd> </mml:mtr> </mml:mtable> <mml:annotation encoding=application/x-tex>begin{gather*} mathbb {P}left (|xi _0(omega )dotsm xi _{n-1}(omega )|le Clambda ^{n} forall nge 1right )=1; end{gather*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=bold-italic xi> <mml:semantics> <mml:mi mathvariant=bold-italic>ξ<!-- ξ --></mml:mi> <mml:annotation encoding=application/x-tex>boldsymbol {xi }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is called <italic>nonuniformly exponentially stable</italic> if there exist two random variables <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C left-parenthesis omega right-parenthesis greater-than 0> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>ω<!-- ω --></mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>C(omega )>0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=0 greater-than lamda left-parenthesis omega right-parenthesis greater-than 1> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>></mml:mo> <mml:mi>λ<!-- λ --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>ω<!-- ω --></mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>0>lambda (omega )>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <disp-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=StartLayout 1st Row double-struck upper P left-parenthesis double-vertical-bar xi 0 left-parenthesis omega right-parenthesis midline-horizontal-ellipsis xi Subscript n minus 1 Baseline left-parenthesis omega right-parenthesis double-vertical-bar less-than-or-equal-to upper C left-parenthesis omega right-parenthesis lamda left-parenthesis omega right-parenthesis Superscript n Baseline for-all n greater-than-or-equal-to 1 right-parenthesis equals 1 period EndLayout> <mml:semantics> <mml:mtable rowspacing=3pt columnspacing=1em side=left displaystyle=true> <mml:mtr> <mml:mtd> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>P</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo fence=false stretchy=false>‖<!-- ‖ --></mml:mo> <mml:msub> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>ω<!-- ω --></mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>⋯<!-- ⋯ --></mml:mo> <mml:msub> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>n</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>ω<!-- ω --></mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo fence=false stretchy=false>‖<!-- ‖ --></mml:mo> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>C</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>ω<!-- ω --></mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mi>λ<!-- λ --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>ω<!-- ω --></mml:mi> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> <mml:mtext> </mml:mtext> <mml:mi mathvariant=normal>∀<!-- ∀ --></mml:mi> <mml:mi>n</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mn>1.</mml:mn> </mml:mtd> </mml:mtr> </mml:mtable> <mml:annotation encoding=application/x-tex>begin{gather*} mathbb {P}left (|xi _0(omega )dotsm xi _{n-1}(omega )|le C(omega )lambda (omega )^{n} forall nge 1right )=1. end{gather*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> In this paper, we characterize the exponential stabilities of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=bold-italic xi> <mml:semantics> <mml:mi mathvariant=bold-italic>ξ<!-- ξ --></mml:mi> <mml:annotation encoding=application/x-tex>boldsymbol {xi }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> via its <italic>nonignorable periodic data</italic> whenever <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=bold-italic xi> <mml:semantics> <mml:mi mathvariant=bold-italic>ξ<!-- ξ --></mml:mi> <mml:annotation encoding=application/x-tex>boldsymbol {xi }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has a constant transition binary matrix. As an application, we construct a Lipschitz continuous <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper S normal upper L left-parenthesis 2 comma double-struck upper R right-parenthesis> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>S</mml:mi> <mml:mi mathvariant=normal>L</mml:mi> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>mathrm {SL}(2,mathbb {R})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-cocycle driven by a Markov chain with <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=2> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding=application/x-tex>2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-points state space, which is nonuniformly but not uniformly hyperbolic and which has constant Oseledeč splitting with respect to a canonical Markov measure." @default.
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• W2570667799 date "2017-01-09" @default.
• W2570667799 modified "2023-09-23" @default.
• W2570667799 title "Exponential stability of matrix-valued Markov chains via nonignorable periodic data" @default.
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