MS&E 321 Spring Stochastic Systems June 1, 2013 Prof. Peter W. Glynn Page 1 of 5

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1 M&E 321 prig tochastic ystems Jue 1, 2013 Prof. Peter W. Gly Page 1 of 5 ectio 6: Harris Recurrece Cotets 6.1 Harris Recurret Markov Chais tochastic Lyapuov Fuctios Harris Recurret Markov Chais We have previously developed a fairly complete steady-state theory for discrete state space Markov chais. I particular, we established that whe X = (X : 0) is a irreducible Markov chai o discrete state space, the X has a steady-state (i the sese of the law of large umbers) precisely whe X has a statioary distributio. Oe problem with this theory is that we caot establish stability without essetially solvig the liear system π = πp. We ow wish to develop recurrece theory that permits us to establish stability eve for models for which the statioary equatios caot be explicitly solved. Ideally, this theory will also permit us to verify stability for Markov chais evolvig i o-compact state spaces. Defiitio A Markov chai X = (X : 0) with state space is said to be recurret i the sese of Harris if there exists A, λ > 0, m 1, ad a distributio ϕ o such that i.) P x (T A < ) = 1, x, where T A = if{ 0 : X A}; ii.) P x (X m ) λ ϕ( ), x A. Remark If is discrete, the X is Harris recurret if ad oly if there exists z such that P x (τ(z) < ) = 1 for x, where τ(z) = if{ 1 : X = z}. Note that coditio i.) guaratees that X visits A ifiitely ofte a.s. Usig the same coi flip idea, coditio ii.) esures that every time X visits A (ot havig visited A i the previous m time steps), there is a probability of λ of a successful coi toss (i.e. a heads ). It follow that there will be ifiitely may times T 1, T 2, at which X has distributio ϕ. Each such radom time T i iitiates a cycle havig a idetical distributio. Furthermore, X Ti m is idepedet of X Ti. Hece, if m = 1, the resultig cycles are iid, where if m > 1, the cycles are correlated. However, the degree of correlatio is modest, ad the followig theorem is easily established. Theorem Let X = (X : 0) is Harris recurret. Put τ i = T i T i 1. If Eτ 1 <, the X possesses a uique statioary distributio π ad for each o-egative f, as. 1 f(x 1 ) 1 π(dx)f(x) a.s

2 Remark A sufficiet coditio guarateeig Eτ 1 < is to require that where τ A = if{ 1 : X A}. sup E x τ A <, (6.1.1) To illustrate what is ivolved i verifyig coditio ii.), suppose that P x (X m B) = p m (x, y)ξ(dy) (6.1.2) for some distributio z( ) ad trasitio desity p m. uppose that ϕ(b) = φ(y)ξ(dy) B for some desity φ( ). (The Rado-Nikodym theorem actually guaratees that ϕ must take this form i the presece of coditio ii.).) If p m (, y) is cotiuous ad positive, with A compact, the we ca take φ(y) = if p m(x, y), ad λ = B if p m(x, z)ξ(dz). Hece i some geerality, it follows that ay compact set A satisfies coditio ii.). (Cautio: The above aalysis assumes that the m-step trasitio probabilities P x (X m ) ca be represeted as i (6.1.2), with a trasitio desity that is positive ad cotiuous. This must be verified separately for each example. If this fails to be true, oe must verify coditio ii.) from first priciples.) 6.2 tochastic Lyapuov Fuctios It remais to provide a techique for verifyig coditio i.) ad coditio (6.1.1). Propositio Let w : [1, ) for which there exists r < 1 such that E x w(x 1 ) rw(x) for x A c. The, for x A c. E x τ A (1 r) 1 w(x) Proof: Via use of operator/fuctio orms with weight fuctio w( ), we coclude that T A 1 E x for x A c. But sice w(x) 1 for x, w(x j ) (1 r) 1 w(x) τ A τ A 1 2 w(x j ),

3 yieldig the result. Usig first trasitio aalysis, we see that for x A, E x τ A = 1 + P (x, dy)e y τ A. A c I view of Propositio 6.2.1, we coclude that sup E x τ A 1 + (1 r) 1 sup E x w(x 1 ). Propositio uppose that there exists A, X > 0, m 1, λ > 0, a distributio ϕ, r < 1, ad w : [1, ) such that: i.) E x w(x 1 ) rw(x), x A c ; ii.) sup E x w(x 1 ) < ; iii.) P x (X m ) λϕ( ), x A. The, X possesses a uique statioary distributio π ad for each o-egative f, as. 1 f(x i ) π(dx)f(x) Remark The fuctio w( ) is called a stochastic Lyapuov fuctio. Example uppose that ( : 0) is a sequece of rv s describig reservoir storage i the presece of a liear release rule, so that +1 = + Z +1 a +1 for a > 0. Assume that (Z : 1) is a sequece of iid o-egative rv s for which EZ 1 <. Put w(x) = 1 + x for x 0. The, E x w( 1 ) = 1 + E x 1 = 1 + E(x + Z 1) 1 + a ( 1 + a 2 a.s ) 1 w(x) EZ 1 (1 + a) ax(1 + a) 1 (2 + a) 1 ( 1 + a 2) 1 w(x) for x (1 + a)(2 + a)/a + EZ 1 (2 + a)/a. Put A = [0, (1 + a)(2 + a)/a + EZ 1 (2 + a)/a]. Coditio ii.) of Propositio is easily verified here. Hece, it remais oly to show that coditio iii.) is satisfied. But this is straightforward to carry out if we assume that Z 1 has a cotiuous positive desity o [0, ). A weaker Lyapuov coditio is offered by our ext result. 3

4 Theorem uppose that there exists A, λ > 0, m 1, a distributio ϕ, ɛ > 0, ad g : [0, ) such that: I.) E x g(x 1 ) g(x) ɛ, x A c ; II.) sup E x g(x 1 ) < ; III.) P x (X m ) λϕ( ), x A. The, X possesses a uique statioary distributio π ad for each o-egative f, 1 f(x i ) f(x)π(dx) a.s. as. Proof: Let B = (B(x, dy) : x, y A c ) be the restrictio of P = (P (x, dy) : x, y ) to A c, so that B(x, dy) = P (x, dy) for x, y A c. Put e(x) = 1 for x A c. Note that coditio i.) implies that B(x, dy)g(y) = P (x, dy)g(y) P (x, dy)g(y) = E x s(x 1 ) g(x) ɛe(x), A c A c so that Hece, Bg g ɛe. ɛe g Bg. ice B is a o-egative operator, it follows that ɛb j e B j g B j+1 g. ummig over j {0, 1,, }, we get ɛ B j e g B +1 g g. edig, we coclude that But so yieldig the iequality ɛ B j e g. (B j e)(x) = P x (τ A > j), (B j e)(x) = P x (τ A > j) = E x τ A, E x τ A g(x)/ɛ for x A c. Cosequetly, P x (τ A < ) = 1 for x A c (ad hece for all x ). Furthermore, sup E x τ A 1 + sup E x g(x 1 ). The coclusio of the theorem therefore follows as for Propositio

5 Remark A ice physical way to thik about g is to view g(x) as represetig the potetial eergy associated with x. Coditio i.) asserts that, i expectatio, the potetial eergy has a tedecy to decrease by ɛ o A c. Hece, sice the system wishes to move to poits of lower potetial eergy, it follows that the system should evetually eter A. Remark A high level of igeuity may be required to fid a suitable Lyapuov fuctio. Oe approach to fidig a suitable g is to try some cadidate fuctios. Whe R d, typical cadidates to try are: a.) x p ; b.) exp(a x p ); c.) (log(1 + x )) p. 5