Erik A. van Doorn Department of Applied Mathematics University of Twente Enschede, The Netherlands

Transcription

1 REPRESENTATIONS FOR THE DECAY PARAMETER OF A BIRTH-DEATH PROCESS Erik A. van Doorn Department of Applied Mathematics University of Twente Enschede, The Netherlands University of Leeds 30 October 2014

2 Acknowledgment: Thanks to the London Mathematical Society for financial support through a Scheme 2 grant

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4 outline 1. birth-death processes 2. Karlin-McGregor representation 3. intermezzo: orthogonal polynomials 4. decay parameter: representations 5. decay parameter: bounds 1

5 birth-death process definition: birth-death process is Markov process {X(t), t 0} on {( 1, )0, 1,...} with absorbing state -1, birth rates λ n > 0 (n 0) and death rates µ n > 0 (n > 0) and µ 0 0 assumption: rates determine process uniquely notation: p ij (t) := Pr{X(t) = j X(0) = i} result (j 0): p j := p j := lim t p ij (t) π 0 := 1, π j := λ 0... λ j 1 µ 1... µ j j > 0 π j n π n if µ 0 = 0 and n π n < 0 otherwise 2

6 birth-death process i, j 0: p ij (t) := Pr{X(t) = j X(0) = i}, p j := lim t p ij (t) notation: α ij := sup{a > 0 : p ij (t) p j = O(e at ) as t } results (Kingman (1963)): and 1 α ij = lim t t log p ij(t) p j α := α 00 α ij note: equality if µ 0 > 0 inequality for at most one value of i or j if µ 0 = 0 interest: decay parameter α 3

7 birth-death process: Karlin-McGregor representation representation (Karlin-McGregor (1957)): p ij (t) = π j 0 e xt Q i (x)q j (x)ψ(dx), i, j 0 with and π 0 = 1, π j = λ 0... λ j 1 µ 1... µ j j > 0 λ n Q n+1 (x) = (λ n + µ n x)q n (x) µ n Q n 1 (x) Q 0 (x) = 1, λ 0 Q 1 (x) = λ 0 + µ 0 x definition decay parameter α: α = sup{a > 0 : p 00 (t) p 0 = O(e at ) as t } 4

8 birth-death process: Karlin-McGregor representation hence so that p 00 (t) = π 0 0 e xt Q 0 (x)q 0 (x)ψ(dx) = p 0 = lim t p 00 (t) = ψ({0}) p 00 (t) p 0 = 0+ e xt ψ(dx) 0 e xt ψ(dx) hence representation decay parameter α = where ξ 1 if ξ 1 > 0 ξ 2 if ξ 2 > ξ 1 = 0 0 if ξ 1 = ξ 2 = 0 ξ 1 = inf supp(ψ), ξ 2 = inf{supp(ψ)\{ξ 1 }} 5

9 birth-death process: Karlin-McGregor representation representation: p ij (t) = π j 0 e xt Q i (x)q j (x)ψ(dx) t = 0: π j 0 Q i(x)q j (x)ψ(dx) = δ ij {Q n (x)} orthogonal polynomial sequence with respect to measure ψ on [0, )! defining we have P n (x) := ( 1) n λ 0 λ 1... λ n 1 Q n (x) P n (x) = (x λ n 1 µ n 1 )P n 1 (x) λ n 2 µ n 1 P n 2 (x) P 0 (x) = 1, P 1 (x) = x λ 0 µ 0 6

10 birth-death process: Karlin-McGregor representation Associate Editor Stochastic Processes and Their Applications:... Van Doorn s paper is difficult to read because hardly anybody knows orthogonal polynomials anymore. 7

11 intermezzo: orthogonal polynomials 8

12 intermezzo: orthogonal polynomials definition: {P n (x), n = 0, 1,...} (monic, deg(p n ) = n) is orthogonal polynomial sequence (OPS) if there exists (Borel) measure ψ (of total mass 1) such that P n(x)p m (x)ψ(dx) = k n δ nm with k n > 0 Favard s theorem: {P n (x), n = 0, 1,...} is OPS there exist c n R, d n > 0 such that P n (x) = (x c n )P n 1 (x) d n P n 2 (x) P 0 (x) = 1, P 1 (x) = x c 1 remark: ψ not necessarily unique (Hamburger moment problem) 9

13 intermezzo: orthogonal polynomials {P n (x), n = 0, 1,...} is OPS with zeros x ni zeros of P n (x) real and distinct: x n1 < x n2 <... < x nn interlacing property: x n+1,i < x ni < x n+1,i+1 hence exist, and ξ i := lim n x ni ξ i ξ i+1 < 10

14 intermezzo: orthogonal polynomials {P n (x), n = 0, 1,...} is OPS satisfying P n (x) = (x c n )P n 1 (x) d n P n 2 (x) P 0 (x) = 1, P 1 (x) = x c 1 theorem: if ξ 1 > there exists a (unique) orthogonalizing measure ψ for {P n (x)}, that is, P m(x)p n (x)ψ(dx) = k n δ mn such that ξ 1 = lim n x n1 = inf supp(ψ) 11

15 intermezzo: orthogonal polynomials on [0, ) {P n (x), n = 0, 1,...} is OPS satisfying P n (x) = (x c n )P n 1 (x) d n P n 2 (x) P 0 (x) = 1, P 1 (x) = x c 1 theorem: (i) ξ 1 0 {P n } OPS w.r.t measure ψ on [0, ) there exist numbers λ n > 0, µ n+1 > 0 (n 0) and µ 0 0 such that c 1 = λ 0 + µ 0, and, for n > 1, c n+1 = λ n + µ n d n+1 = λ n 1 µ n (ii) if there exist... and µ 0 > 0 then ψ({0}) = 0 12

16 intermezzo: orthogonal polynomials on [0, ) summary: {P n (x), n = 0, 1,...} satisfies P n (x) = (x λ n µ n )P n 1 (x) λ n 1 µ n P n 2 (x) P 0 (x) = 1, P 1 (x) = x λ 0 µ 0 with λ n > 0, µ n+1 > 0 and µ 0 0 {P n (x), n = 0, 1,...} is OPS with respect to measure ψ with support in [0, ) (with ψ({0}) = 0 if µ 0 > 0), x n1 decreasing in n, and ξ 1 = lim n x n1 = inf supp(ψ) 13

17 birth-death processes: decay parameter representation: p ij (t) = π j 0 e xt Q i (x)q j (x)ψ(dx) p 00 (t) p 0 = 0+ e xt ψ(dx) interest: decay parameter where α = inf{supp(ψ)\{0}} = ξ 1 if ξ 1 > 0 ξ 2 if ξ 2 > ξ 1 = 0 0 if ξ 1 = ξ 2 = 0 ξ 1 = lim n x n1 = inf supp(ψ) ξ 2 = lim n x n2 = inf{supp(ψ)\{ξ 1 }} 14

18 birth-death processes: decay parameter p 00 (t) p 0 = decay parameter α = inf{supp(ψ)\{0}} = 0+ e xt ψ(dx) ξ 1 if ξ 1 > 0 ξ 2 if ξ 2 > ξ 1 = 0 0 if ξ 1 = ξ 2 = 0 recall: µ 0 > 0 p 0 = 0 ψ({0}) = 0 hence α = ξ 2 if µ 0 = 0 and ψ({0}) > 0 ξ 1 otherwise 15

19 birth-death processes: decay parameter problem: determine decay parameter α = ξ 2 if µ 0 = 0 and ψ({0}) > 0 ξ 1 otherwise approach if µ 0 = 0: dual process definition: given λ n, µ n+1 (n 0) and µ 0 = 0 the dual process has rates λ n := µ n+1, µ n+1 := λ n+1, µ 0 := λ 0 (> 0) then {P n (x)} OPS w.r.t ψ and so that supp(ψ ) = supp(ψ)\{0} α = inf{supp(ψ)\{0}} = ξ 1 16

20 decay parameter: representations assumption: µ 0 > 0 hence with α = ξ 1 = inf supp(ψ) = lim n x n1 x n1 < x n2 <... < x nn zeros of P n (x): P n (x) = (x λ n 1 µ n 1 )P n 1 (x) λ n 2 µ n 1 P n 2 (x) P 0 (x) = 1, P 1 (x) = x λ 0 µ 0 1st approach: orthogonal polynomial systems 17

21 decay parameter: representations ξ 1 = inf supp(ψ) = lim n x n1 1st approach: orthogonal polynomial systems example 1: λ n = λ, µ n = µ then ξ 1 = ( λ µ) 2 example 2: λ n = λ, µ n = µ n + 1 then ξ 1 = λ 1 2 ( ) µ 2 + 4λµ µ 18

22 decay parameter: representations let R n+1 = ξ 1 = inf supp(ψ) = lim n x n1 λ 0 µ 0 λ 0 0 µ 1 λ 1 µ 1 λ λ n 1 µ n 1 λ n 1 0 µ n λ n µ n truncated q-matrix of birth-death process, then P n (x) = det(xi n + R n ), n > 0 so zeros of P n (x) are eigenvalues of R n 19

23 decay parameter: representations more generally, let a j > 0 and ξ 1 = inf supp(ψ) = lim n x n1 T n+1 := λ 0 + µ 0 λ 0 µ 1 /a 1 0 a 1 λ 1 + µ 1 λ 1 µ 2 /a λ n 1 + µ n 1 λ n 1 µ n /a n 0 a n λ n + µ n then P n (x) = det(xi n T n ) 2nd approach: eigenvalue techniques 20

24 decay parameter: representations theorem so that x n1 = max a>0 min 0 i<n x n1 = min a>0 max 0 i<n { { ξ 1 = max a>0 inf i 0 ξ 1 = lim n min a>0 max 0 i n λ i + µ i (1 δ i,n 1 )a i+1 λ i 1µ i a i λ i + µ i (1 δ i,n 1 )a i+1 λ i 1µ i a i { λ i + µ i a i+1 λ i 1µ i a i where λ 1 = 0, a 0 = 1, a = (a 1, a 2,...) { λ i + µ i (1 δ in )a i+1 λ i 1µ i a i } } } } proof: Geršgorin discs, Collatz-Wielandt formula, Perron-Frobenius theory 21

25 decay parameter: representations theorem x n1 = max h min 1 i<n 1 2 c i + c i+1 (c i+1 c i ) 2 + 4d i+1 (1 h i )h i+1 where c i+1 = λ i + µ i, d i+1 = λ i 1 µ i and h = (h 1,..., h n ), h 1 = 0, h n = 1, 0 < h i < 1 (1 < i < n) so that ξ 1 = max b inf i { c i + c i+1 (c i+1 c i ) 2 + 4d i+1 /b i } where b = (b 1, b 2,...) is a chain sequence proof: ovals of Cassini 22

26 decay parameter: representations theorem x n1 = min h 0 n i=0 where h = (h 0,..., h n ), h 0 = 0, ξ 1 = inf h 0 lim n inf ( ) h 2 i (λ i + µ i ) 2h i 1 h i λ i µ i+1 n i=1 where h = (h 0, h 1,...), h 0 = 0, ni=1 h 2 i = 1, whence ( h 2 i (λ i + µ i ) 2h i 1 h i λ i µ i+1 ) i=1 h 2 i = 1 proof: symmetrize T n by suitable choice of a i, then Courant-Fischer theorem x n1 = min y 0 yt n y T yy T 23

27 decay parameter: representations setting: µ 0 0 p 00 (t) p 0 = α = inf{supp(ψ)\{0}} = 0+ e xt ψ(dx) ξ 2 if µ 0 = 0 and ψ({0}) > 0 ξ 1 otherwise notation: and π 0 = 1, K n = π n = λ 0... λ n 1 µ 1... µ n, n > 0 n i=0 π i K = π i i=0 L n = n i=1 1 µ i π i L = i=1 1 µ i π i 24

28 intermezzo: birth-death processes setting: µ 0 0 p 00 (t) p 0 = α = inf{supp(ψ)\{0}} = 0+ e xt ψ(dx) ξ 2 if µ 0 = 0 and ψ({0}) > 0 ξ 1 otherwise results (Karlin and McGregor (1957)): µ 0 = 0 ψ({0}) = K 1 0 µ 0 = 0 and K = (0, ) ψ(dx) x = L µ 0 > 0 ψ({0}) = 0 and (0, ) ψ(dx) x 1 µ 0 25

29 intermezzo: birth-death processes recall: K = i=0 π i and L = i=1 1 µ i π i assumption ψ unique K + L = results for µ 0 = 0: process is for µ 0 > 0: absorption is positive recurrent if K < and L = null recurrent if K = and L = transient if L < certain and E(time) < if K < and L = certain and E(time) = if K = and L = not certain if L < 26

30 decay parameter: bounds K = i=0 π i, L = i=1 1 µ i π i assumption: K + L = fact: K = L = ξ 1 = ξ 2 = 0 4 scenarios: 1. K < and µ 0 = 0 : ξ 2 > 0? 2. K < and µ 0 > 0 : ξ 1 > 0? 3. L < and µ 0 = 0 : ξ 1 > 0? 4. L < and µ 0 > 0 : ξ 1 > 0? facts: scenario 1: ξ 1 = 0 scenarios 2-4: ξ 1 = 0 ξ 2 = 0 27

31 decay parameter: bounds results: 1: if K < and µ 0 = 0, then ξ 1 = 0 and Chen (2000) ξ 2 > 0 sup n L n (K K n 1 ) < 2: if K < and µ 0 > 0, then Sirl et al. (2007) ξ 1 > 0 sup n L n (K K n 1 ) < 3 and 4: if L <, then Chen (2010) ξ 1 > 0 sup n K n (L L n ) < 28

32 decay parameter: bounds more detailed results: 1: if K < and µ 0 = 0, then ξ 1 = 0 and Chen (2000) 1 4R ξ 2 K R, R := sup n L n(k K n 1 ) 2: if K < and µ 0 > 0, then 1 4S ξ 1 1 S, Sirl et al. (2007) S := sup n (L n + µ 1 0 )(K K n 1 ) similar results in scenarios 3 and 4: Chen (2010) 29

33 decay parameter: bounds setting: µ 0 = 0 recall dual process: λ n = µ n+1 and µ n = λ n (µ 0 > 0) supp(ψ ) = supp(ψ)\{0} so that α = inf{supp(ψ)\{0}} = ξ1 moreover L < K < and L < K < hence solution scenario 3 solution scenario 2 solution scenario 4 solution scenario 1 30

34 decay parameter: bounds recall: x n1 < x n2 < < x nn zeros of P n (x), ξ i = lim n x ni let T n+1 := λ 0 + µ 0 λ0 µ λ0 µ 1 λ 1 + µ λ1 µ λ n 1 + µ n 1 λn 1 µ n 0 0 λn 1 µ n λ n + µ n then det(xi n T n ) = P n (x), n > 0 hence zeros of P n (x) are eigenvalues of symmetric matrix T n 31

35 decay parameter: bounds zeros x ni of P n (x) are eigenvalues of symmetric matrix T n ξ i = lim n x ni recall Courant-Fischer Theorem: x n1 = min y 0 yt n y T yy, x T n2 = max min dim V=n y V y 0 yt n y T yy T,... trick: or ξ i = ξ i = lim n x ni ξ i = ξ i+1 = lim n x n,i+1 x ni zeros of suitable sequence of birth-death polynomials { P n } 32

36 decay parameter: bounds scenario 2: K < and µ 0 > 0 recall (Sirl et al. (2007)): 1 4R ξ 1 1 R, R = sup n 0 n i=0 1 µ i π i π i i=n alternative proof: ξ 1 = ξ 1 = lim n x n1 x n1 smallest zero of P n (x) = P n(x) + λ n 1 P n 1 (x) where P n are polynomials corrsponding to dual process then { P n } is sequence of birth-death polynomials whence x n1 is smallest eigenvalue of symmetric matrix T n 33

37 decay parameter: bounds scenario 2: K < and µ 0 > 0 x n1 is smallest eigenvalue of symmetric matrix T n Courant-Fischer Theorem: x n1 = min u 0 n 1 u = (u 0, u 1,..., u n 1 ), u i R, so that ξ 1 = ξ 1 = lim n x n1 = inf u U µ i π i u 2 i i=0 n 1 2 i π i u j i=0 j=0 µ i π i u 2 i i=0 2 i π i u j i=0 j=0 34

38 decay parameter: bounds scenario 2: K < and µ 0 > 0 hence 1 ξ 1 = inf ξ 1 = inf u U A : µ i π i u 2 i i=0 2 i π i u j i=0 j=0 2 i π i u j i=0 j=0 A i=0 µ i π i u 2 i for all u weighted discrete Hardy s inequalities (Miclo (1999)): 1 4R ξ 1 1 R, R = sup n 0 n i=0 1 µ i π i π i i=n 35

39 summary interest: decay parameter of birth-death process results: 1. representations in terms of support of orthogonalizing measure 2. representations as limits of eigenvalues of n n sign-symmetric tridiagonal matrices 3. alternative proofs for findings of Chen and Sirl et al involving orthogonal-polynomial and eigenvalue techniques reference: EAvD, Representations for the decay parameter of a birth-death process based on the Courant-Fischer theorem. J. Appl. Probab. (2015), to appear 36