Laws Relating Runs, Long Runs and Steps in Gambler s Ruin with Persisitence in Two Strata

Transcription

1 Laws Relating Runs, Long Runs and Steps in Gambler s Ruin with Persisitence in Two Strata Gregory J. Morrow Department of Mathematics University of Colorado Colorado Springs Aug 20, 2015/ 8 th Intl Conf Lattice Path Combinatorics and Applications, Cal Poly Pomona

2 Outline 1 Introduction Persistent Random Walk and Excursion Gambler s Ruin Persistence in Strata 2 Conditional Joint G. F. of Excursion Statistics given Height Future Maxima Decomposition Recurrence Relation for One-sided G.F., g m,n Extend 2-Parameter Fibonacci Polys over Stratum Boundary 3 Applications Joint Distribution of Excursion Statistics: Homogeneous Case Meander Limit, with Order N Scaling Last Visit Limit, with Order N scaling

3 Persistent Random Walk Walk on Integers: X j, j = 0, 1, 2,... Increments: ε j := X j X j 1 Transitions: P(ε j+1 = 1 ε j = 1) = P(ε j+1 = 1 ε j = 1) = a

4 Excursion: Steps, Height, Runs, Short Runs Excursion: A Path of First Return to Zero by Random Walk Sample : #Steps: L = 10, Height: H = 3 #Runs: R = 6, #Short Runs: V = 3

5 Gambler s Ruin Initial Fortune (0, 2N). Unit Bets. Correlated Run of Fortune. Terminate Game at Step T when Fortune First Equals 2N or 0. Equivalently, X 0 ( N, N), T := inf{j : X j = N}. Picture: "Last Visit" L := inf{j : X j = 0}, "Meander" L j T FORTUNE NUMBER of BETS

6 Uncorrelated Case: Runs and Steps, a = 1 2 [M, 2015] 2 Parameter Fibonacci Recurrence: (F) v n+1 = βv n xv n 1. Define {q n }, {w n } by (F): q 0 = 0, q 1 = 1; w 0 = 1, w 1 = 1; = q n t n = t 1 βt+xt 2, w n = q n xq n 1 Theorem K N+1 := E{r R z L H N} = c r 2 z 2 (q N /w N ); x := 1 4 z2, β := 1 x(r 2 1) Denote R := #Runs( X j, 0 j L) Corollary Both 1 2 L/N2 f, R/N 2 f ; f (x) := 1 πx ν= ( 1)ν e ν2 /x, x > 0. R := #Runs(X j, L j T ); L := T L; := 2R L ; (Meander) := 2R L M; M := #Excursions to L; (Last Visit) Corollary /N π 4 sech2 (πx/2); ( + )/N 1 2sech(πx/2), < x <.

7 Persistence in Strata General Persistent Symmetric Gambler s Ruin: P(ε j+1 = 1 ε j = 1, X j = k) = P(ε j+1 = 1 ε j = 1, X j = k) = a k Two Persistence Parameters: a k = a, 0 k < f ; a k = b, f k < N; some f (0, N) Velocity Model: Velocities ±1, Deterministic Change in Persistence Interpretation: Change in Stratum Change in Medium cf. [Szasz and Toth, 1984] 1-D Random Environment Motivation: 3 Counting Stats, Non-trivial Scaling Limits for b a

8 Key: Explicitly Calculate K N := E{r R y V z L H < N}. Outline: Formula for K N = K N (r, y, z) = Formula for Last Visit Statistics G. F. ( ). Note: K N not needed for Meander. M := # Excursions Until Last Visit L. M is a Geometric random variable: p m := P(M = m) = P(H < N) m P(H N), m = 0, 1, 2,.... Sum Independent Copies of Excursion Stats, Obtain Last Visit Stats: R := M R (m), V := m=0 Last Visit Statistics G. F. ( ) E{r R y V z L u M } = M V (m), L := m=0 p m [u K N ] m = m=0 M m=0 L (m) P(H N) 1 u K N P(H < N)

9 Upward and Downward Conditional G.F. g m,n Focus on Construction for Meander Γ m,n := Conditional First Passage Path to Level n, given X 0 = m, & Path "One-Sided" : X j [m, n] L m,n := #(Steps along Γ m,n). Similarly: R m,n, V m,n (Runs, Short Runs) Define: g m,n (r, y, z) := E{r R m,ny V m,nz L m,n ε 1 = ε 2 }. Picture: A Path for g 5,0

10 Recurrence Relation for g m,n Example : Decompose Downward Transition for g n,0 by: Return to Level 1 After Each Future Maximum. ρ m,n := Probability of One-sided Transition k m + := G.F. over (UD) l of Continuation Seq. (UD) l UU at Level m kn := Corresponding G.F. at Level n with Roles of U, D, Reversed z h m + := G.F. over Termination Seq. (UD) l D, at Level m z hn := Corresponding G.F. at Level n with Roles of U, D, Reversed λ m,n := l=0 ( cm,n ρ m,n k + m g m,n ρ n,m k n g n,m ) l, m < n; λn,m = λ m,n Recurrence (g) g n,0 = c g n,1 λ 1,n λ 1,n 1 λ 1,3 z h + 1

11 Denominators w n of g 0,n : Homogeneous case b = a Consider First : b = a. Define Denominator Polys w n = w n (a) Lemma v n = w n satisfy (F) v n+1 = βv n xv n 1, such that : w 1 = 1, & w n Serves as Denominator of g n := g 0,n. Here, x = x a and β = β a Determined by: (g), (F), and (Interlacing) v 2 n v n+1 v n 1 = β 1 x n 1 (v 2 v 1 v 3 v 0 ).... conseq. of (F) w 0 by back iteration; = w n = (1 w 0 ) q n(x a, β a ) + w 0 w n(x a, β a ) (g) = g n+1 = c λ n g 2 n/g n 1 ; λ n := λ 0,n If b = a, Then: λ n = (w n )2 wn 1 w n+1 ; g n = c r z n τa n 2 /wn ; n 2. x a := a 2 z 2 τ 2 a, τ a := 1 + (1 a) 2 r 2 z 2 y(1 y), β a is explicit too. Numerator Polys qn Satisfy (F), : q1 = y 2, and (Commutation) [w (a), q ] n := wn qn+1 q nwn+1 = a2 z 2 xa n 1.

12 Denominators w m,n of g m,n : Full model Definition w m,m+l := wl (a), m + l f ; w m,m+l := wl (b), f m w f l,f +1 := 1 b 1 a w l+1 b a (a) + 1 a w l (a), 1 l f w m,f +2 := β(a, b)w m,f +1 x(a, b)w m,f, m f 1 w m,f +j+1 := β b w m,f +j x b w m,f +j 1, m f 1, j 2 Define Downward Denominator w n,m by switching the roles of a and b. Lemma w f l,f +j = [ q j (b) wj (b) ] [ ] w M l (a) wl+1 (a), M an explicit 2 2 matrix. Define Interlacing Bracket [ w ] m.n := w m,n w m+1,n+1 w m,n+1 w m+1,n, m n 2 Lemma (1) [ w ] f l,f +j = a 2 r 2 z 4 (1 a)(1 b)x l 2 a x(a, b) x j 1 b, l 2, j 1. (2) λ m.n = w m,n w m+1,n+1 /{w m,n+1 w m+1,n } and g m,n has Closed Formula.

13 Joint Distribution of Excursion Stats: Case b = a Theorem ( ) Let b = a. Then K N+1 = E{r R y V z L H N} = c r 2 z 2 q N (a) wn (a) Corollary Let b = a. Define α a := β 2 a 4x a. Then, (1) K (r, y, z; a) := E{r R y V z L } = (1 1 2 β a 1 2 α a)/(1 a) Define the Excursion Statistic U := #Long Runs = R V Corollary Let P a denote the Probability for Homogeneous case. Then, for n 2, 1 a a P a(l = 2n, R = 2k, U = l) = P 1 a (L = 2n, L R = 2k, U = l) Proof: (1) = Note Taylor Expansion: z 16 K (ru, 1/u, z; 1 2 ) K (u/r, 1/u, rz; 1 2 ) = 1 2 z2 (r 2 1). 1 2 K (ru, 1/u, 2z; 1 2 ) = + (r 2 + r 4 + r 6 + r 8 + r 10 + r 12 + r 14 )u 2 + (10r r r r r 12 )u 3 + (10r r r r r 12 )u 4 + (36r r r 10 )u 5 + (6r r 8 + 6r 10 )u 6 + 2r 8 u 7 ) +

14 Scaling Limit of Order N: Meander of Gambler s Ruin Denote V := #Short Runs over Meander; ) R, L : Runs, Steps; X N (L := 2 a b (1 a)(1 b) R 1 + (1 a)(1 b) V /N. Theorem Let f ηn for some fixed 0 < η < 1. Denote σ 1 := a + b 2 2ab, and σ 2 := b + a 2 2ab. Write κ 1 := ησ 1 1 b and κ 2 := (1 η)σ 2 1 a. Then lim N E{e itx N } = ˆϕ(t), ˆϕ(t) := (bκ 1 σ 2 +aκ 2 σ 1 )t aσ 1 cosh(κ 1 t) sinh(κ 2 t)+bσ 2 sinh(κ 1 t) cosh(κ 2 t)+i(b a) 2 sinh(κ 1 t) sinh(κ 2 t) Put Y 1,N := (R 1 (1 a) V ) /N; Y 2,N := (L 1 (1 a) R ) /N Y 1,N. As a consequence of the Proof of Theorem, Corollary Let b = a. Then lim N E{e isy 1,N +ity 2,N } = (1 a)s 2 + at 2 / sinh( (1 a)s 2 + at 2 )

15 Extend Numerator Polynomials: q n Definition Define q n = q n (r, y, z; a, b) for all n 1 by: (1) q n := qn(a), 1 n < f ; (2) q f := 1 b 1 a q b a f (a) + 1 a q f 1 (a); (3) q f +1 := β(a, b)q f x(a, b)q f 1 ; (4) q f +j+1 := β b q f +j x b q f +j 1, j 1. Lemma Let M 2 2 be as before. If j 1, q f +j 1 = [ q j (b) w j (b) ] M [ q f 1 (a) q f (a) ] Lemma [ w, q ] n := w n,0 q n+1 q n w n+1,0 = M [w (a), q (a)] f 1 [w (b), q (b)] n f Theorem K N+1 (r, y, z; a, b) = E{r R y V z L H N} = c r 2 z 2 ( ) q N /w 1,N+1

16 Scaling Limit: Last Visit portion of Gambler s Ruin Denote( V := #Short Runs until L; M := #Excursions ) until L; X N := L 2 a b (1 a)(1 b) R + 1 (1 a)(1 b) V a(b a) (1 a)(1 b) M /N Theorem Let f ηn for some fixed 0 < η < 1. Let σ j, κ j, and ˆϕ(t) as before. Then lim N E{e itx N } = ˆψ(t)/ ˆϕ(t), ˆψ(t) := abσ 1 σ 2 abσ 1 σ 2 cosh(κ 1 t) cosh(κ 2 t)+a 2 σ 2 1 sinh(κ 1t) sinh(κ 2 t)+iaσ 1 (b a) 2 cosh(κ 1 t) sinh(κ 2 t) Proof of "Last Visit" Thm Involves Second Order Cancellation in Denominator of Characteristic Function Expansion. Mathematica applied throughout to make "Direct Calculations". Key Closed Formula for g m,n Established by Induction Combine Results for Last Visit and Meander Corollary lim N E{e it(x N+X N ) } = ˆψ(t)

17 Conclusion The Future Maxima Decomposition is amenable to the study of the conditional Joint Generating Function, given the Height of a Random Walk Excursion, even with Persistence in Strata Applications: Distributional Symmetry for Runs, Long Runs, and Steps in Homogeneous case. New Order N Scaling Limits for Both Last Visit and Meander, and thereby for entire Gambler s ruin process Outlook One may attempt analogous results for the "Non-sequential" setting, where the ArcSine Law is already known for a fixed interval [0, N] of Steps along the x-axis.

18 Bibliography N.G. de Bruijn, D.E. Knuth, and S.O. Rice, The average height of planted plane trees, Graph Theory and Computing, Ronald C. Read, ed., Academic Press, New York (1972), p E. Deutsch, Dyck path enumeration, Discrete Mathematics 204 (1999) W. Feller, An Introduction to Probability Theory and Its Applications, Vol. I, 3rd ed. Wiley, New York (1968). P. Flagolet and R. Sedgewick, Analystic Combinatorics. Cambridge University Press (2009). C. Mohan, The gambler s ruin problem with correlation, Biometrika 42 (1955) G.J. Morrow, Laws relating runs and steps in gambler s ruin, Stoch. Proc. Appl. 125 (2015) D. Szasz, B. Toth, Persistent random walks in a one-dimensional random environment, J. Stat. Phys 37 (1984)

19 Addendum: Formula for K (r, u, z) := K (ru, 1/u, z; 1 2 ) Recall the Joint G.F. of Excursions Stats R, V, L in Homogeneous Case Corollary Let b = a. Define α a := β 2 a 4x a. Then (1) K (r, y, z; a) := E{r R y V z L } = (1 1 2 β a 1 2 α a)/(1 a) Assume now in Addition that a = 1 2. Then by Direct Calculation the Joint G.F. of R, U, L is given by K (r, u, z) := 1 ( z 2 + 4r 2 z 2 + r 2 z 4 2r 2 uz 4 + r 2 u 2 z 4 S ), With Main Term S given by: S := (4 + 2z + 2rz + rz 2 ruz 2 )(4 + 2z 2rz rz 2 + ruz 2 ) (4 2z + 2rz rz 2 + ruz 2 )(4 2z 2rz + rz 2 ruz 2 ). It Holds that : S(r, u, z) = S(1/r, u, rz).