Asymptotic properties of the occupation measure in a multidimensional skip-free Markov-modulated random walk

Toshihisa Ozawa ()
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Toshihisa Ozawa: Komazawa University

Queueing Systems: Theory and Applications, 2021, vol. 97, issue 1, No 6, 125-161

Abstract: Abstract We consider a discrete-time d-dimensional process $$\{{\varvec{X}}_n\}=\{(X_{1,n},X_{2,n},\ldots ,X_{d,n})\}$$ { X n } = { ( X 1 , n , X 2 , n , … , X d , n ) } on $${\mathbb {Z}}^d$$ Z d with a background process $$\{J_n\}$$ { J n } on a countable set $$S_0$$ S 0 , where individual processes $$\{X_{i,n}\},i\in \{1,2,\ldots ,d\},$$ { X i , n } , i ∈ { 1 , 2 , … , d } , are skip free. We assume that the joint process $$\{{\varvec{Y}}_n\}=\{({\varvec{X}}_n,J_n)\}$$ { Y n } = { ( X n , J n ) } is Markovian and that the transition probabilities of the d-dimensional process $$\{{\varvec{X}}_n\}$$ { X n } vary according to the state of the background process $$\{J_n\}$$ { J n } . This modulation is assumed to be space homogeneous. We refer to this process as a d-dimensional skip-free Markov-modulated random walk. For $${\varvec{y}}, {\varvec{y}}'\in {\mathbb {Z}}_+^d\times S_0$$ y , y ′ ∈ Z + d × S 0 , consider the process $$\{{\varvec{Y}}_n\}_{n\ge 0}$$ { Y n } n ≥ 0 starting from the state $${\varvec{y}}$$ y and let $${\tilde{q}}_{{\varvec{y}},{\varvec{y}}'}$$ q ~ y , y ′ be the expected number of visits to the state $${\varvec{y}}'$$ y ′ before the process leaves the nonnegative area $${\mathbb {Z}}_+^d\times S_0$$ Z + d × S 0 for the first time. For $${\varvec{y}}=({\varvec{x}},j)\in {\mathbb {Z}}_+^d\times S_0$$ y = ( x , j ) ∈ Z + d × S 0 , the measure $$({\tilde{q}}_{{\varvec{y}},{\varvec{y}}'}; {\varvec{y}}'=({\varvec{x}}',j')\in {\mathbb {Z}}_+^d\times S_0)$$ ( q ~ y , y ′ ; y ′ = ( x ′ , j ′ ) ∈ Z + d × S 0 ) is called an occupation measure. Our primary aim is to obtain the asymptotic decay rate of the occupation measure as $${\varvec{x}}'$$ x ′ goes to infinity in a given direction. We also obtain the convergence domain of the matrix moment generating function of the occupation measure.

Keywords: Markov-modulated random walk; Markov additive process; Occupation measure; Asymptotic decay rate; Moment generating function; Convergence domain; 60J10; 60K25 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s11134-020-09673-9

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