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/ 8th Intl Conf
Lattice Path Combinatorics and Applications, Cal Poly Pomona
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. , gm,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
Persistent Random Walk
Walk on Integers: Xj , j = 0, 1, 2, . . .
Increments: εj := Xj − Xj−1
Transitions: P(εj+1 = 1|εj = 1) = P(εj+1 = −1|εj = −1) = a
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
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, X0 ∈ (−N, N), T := inf{j : |Xj | = N}.
Picture: "Last Visit" L := inf{j : |Xj | = 0}, "Meander" L ≤ j ≤ T
8
7
6
FORTUNE
5
4
3
2
1
0
0
5
10
15
NUMBER of BETS
20
25
Uncorrelated Case: Runs and Steps, a =
1
2
[M, 2015]
2 Parameter Fibonacci Recurrence: (F) vn+1 = βvn − xvn−1 .
Define {qn }, {wn }Pby (F): q0 = 0, q1 = 1; w0 = 1, w1 = 1;
t
=⇒
qn t n = 1−βt+xt
2 , wn = qn − xqn−1
Theorem
KN+1 := E{r R z L |H ≤ N} = c · r 2 z 2 (qN /wN ); x := 14 z 2 , β := 1 − x(r 2 − 1)
Denote R := #Runs(|Xj |, 0 ≤ j ≤ L)
Corollary
Both 12 L/N 2 → f , R/N 2 → f ; f (x) :=
√1
πx
ν −ν 2 /x ,
ν=−∞ (−1) e
P∞
x > 0.
R0 := #Runs(Xj , L ≤ j ≤ T ); L0 := T − L; ∆0 := 2R0 − L0 ; (Meander)
∆ := 2R − L − M; M := #Excursions to L; (Last Visit)
Corollary
∆0 /N → π4 sech2 (πx/2); (∆ + ∆0 )/N → 12 sech(πx/2), −∞ < x < ∞.
Persistence in Strata
General Persistent Symmetric Gambler’s Ruin:
P(εj+1 = 1|εj = 1, |Xj | = k ) = P(εj+1 = −1|εj = −1, |Xj | = k ) = ak
Two Persistence Parameters:
ak = a, 0 ≤ k < f ; ak = 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 6= a
Key: Explicitly Calculate KN := E{r R y V z L |H < N}.
Outline: Formula for KN = KN (r , y , z) =⇒
Formula for Last Visit Statistics G. F. (†).
KN not needed for Meander.
M := # Excursions Until Last Visit L.
Note:
M is a Geometric random variable:
pm := 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
X
R(m) , V :=
m=0
M
X
V(m) , L :=
m=0
M
X
L(m)
m=0
Last Visit Statistics G. F.
(†)
E{r R y V z L u M } =
∞
X
m=0
pm [u · KN ]m =
P(H ≥ N)
1 − u · KN P(H < N)
Upward and Downward Conditional G.F. gm,n
Focus on Construction for Meander
Γ0m,n := Conditional First Passage Path to Level n,
given X0 = m, & Path "One-Sided" : Xj ∈ [m, n]
L0m,n := #(Steps along Γ0m,n ). Similarly: R0m,n , V0m,n (Runs, Short Runs)
Define:
0
0
0
gm,n (r , y , z) := E{r Rm,n y Vm,n z Lm,n |ε1 = ε2 }.
Picture:
A Path for g5,0
Recurrence Relation for gm,n
Example : Decompose Downward Transition for gn,0 by:
Return to Level 1 After Each Future Maximum.
ρm,n := Probability of One-sided Transition
+
km
:= G.F. over (UD)` of Continuation Seq. (UD)` UU at Level m
−
kn := Corresponding G.F. at Level n with Roles of U, D, Reversed
+
z · hm
:= G.F. over Termination Seq. (UD)` D, at Level m
z · hn− := Corresponding G.F. at Level n with Roles of U, D, Reversed
`
P
+
−
λm,n := ∞
`=0 cm,n · ρm,n km gm,n · ρn,m kn gn,m , m < n; λn,m = λm,n
Recurrence
(g)
gn,0 = c · gn,1 · λ1,n · λ1,n−1 · · · λ1,3 · z · h1+
Denominators wn∗ of g0,n : Homogeneous case b = a
Consider First : b = a. Define Denominator Polys wn∗ = wn∗ (a)
vn = wn∗ satisfy
such that :
(F)
vn+1 = βvn − xvn−1 ,
w1∗ = 1, & wn∗ Serves as Denominator of gn := g0,n .
Here, x = xa and β = βa Determined by: (g), (F), and
(Interlacing) vn2 −vn+1 vn−1 = β −1 x n−1 (v2 v1 −v3 v0 ). . . . conseq. of (F)
w0∗ by back iteration; =⇒ wn∗ = (1 − w0∗ ) · qn (xa , βa ) + w0∗ · wn (xa , βa )
(g) =⇒
gn+1 = c · λn · gn2 /gn−1 ;
λn := λ0,n
Lemma
If b = a, Then: λn =
(wn∗ )2
∗ w∗ ;
wn−1
n+1
gn = c · r · z n τan−2 /wn∗ ; n ≥ 2.
xa := a2 z 2 τa2 , τa := 1 + (1 − a)2 r 2 z 2 y (1 − y ), βa is explicit too.
Numerator Polys qn∗ Satisfy (F), 3: q1∗ = y 2 , and
∗
∗
(Commutation) [w ∗ (a), q ∗ ]n := wn∗ qn+1
− qn∗ wn+1
= a2 z 2 xan−1 .
Denominators w m,n of gm,n : Full model
Definition
w m,m+` := w`∗ (a), m + ` ≤ f ;
w f −`,f +1 :=
1−b ∗
1−a w`+1 (a)
+
w m,m+` := w`∗ (b), f ≤ m
b−a ∗
1−a w` (a),
1≤`≤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 := βb w m,f +j − xb 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 −`,f +j =
qj∗ (b) wj∗ (b)
M
w`∗ (a)
∗
w`+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 −`,f +j = a2 r 2 z 4 (1 − a)(1 − b)xa`−2 · x(a, b) · xbj−1 , ` ≥ 2, j ≥ 1.
(2) λm.n = w m,n w m+1,n+1 /{w m,n+1 w m+1,n } and
gm,n has Closed Formula .
Joint Distribution of Excursion Stats: Case b = a
Theorem
Let b = a. Then KN+1 = E{r R y V z L | H ≤ N} = c · r 2 z 2
qN∗ (a)
wN∗ (a)
Corollary
p
Let b = a. Define αa := βa2 − 4xa .
Then,
(1)
K (r , y , z; a) := E{r R y V z L } = (1 − 12 βa − 12 αa )/(1 − a)
Define the Excursion Statistic U := #Long Runs = R − V
Corollary
Let Pa denote the Probability for Homogeneous case. Then, for n ≥ 2,
1−a
a Pa (L = 2n, R = 2k , U = `) = P1−a (L = 2n, L − R = 2k , U = `)
Proof: (1) =⇒ K (ru, 1/u, z; 12 ) − K (u/r , 1/u, rz; 12 ) = 12 z 2 (r 2 − 1). Note Taylor Expansion: 21 K (ru, 1/u, 2z; 21 ) = · · · +

 2

 (r + r 4 + r 6 + r 8 + r 10 + r 12 + r 14 )u 2 +
+·
z 16
(10r 4 + 16r 6 + 18r 8 + 16r 10 + 10r 12 )u 3 + (10r 4 + 46r 6 + 63r 8 + 46r 10 + 10r 12 )u 4


+ (36r 6 + 68r 8 + 36r 10 )u 5 + (6r 6 + 23r 8 + 6r 10 )u 6 + 2r 8 u 7 )
Scaling Limit of Order N: Meander of Gambler’s Ruin
Denote V 0:= #Short Runs over Meander; R0 , L0 : Runs, Steps;
2−a−b
1
XN0 := L0 − (1−a)(1−b)
R0 + (1−a)(1−b)
V 0 /N.
Theorem
√
Let f ∼
Denote σ1 := a + b2 − 2ab, and
√ηN for some fixed 0 < η < 1.
ησ1
2
σ2 := b + a2 − 2ab. Write κ1 := 1−b
and κ2 := (1−η)σ
1−a .
0
Then limN→∞ E{eitXN } = ϕ̂(t),
(bκ1 σ2 +aκ2 σ1 )t
ϕ̂(t) := aσ cosh(κ t) sinh(κ t)+bσ sinh(κ
t) cosh(κ t)+i(b−a)2 sinh(κ t) sinh(κ t)
1
0
Put Y1,N
:= R0 −
1
2
1
0
(1−a) V
2
1
2
0
:= L0 −
/N; Y2,N
1
1
0
(1−a) R
2
0 .
/N − Y1,N
As a consequence of the Proof of Theorem,
Corollary
Let b = a. Then
p
p
0
0
limN→∞ E{eisY1,N +itY2,N } = (1 − a)s2 + at 2 / sinh( (1 − a)s2 + at 2 )
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 qf (a)
+
b−a ∗
1−a qf −1 (a);
(3) q f +1 := β(a, b)q f − x(a, b)q f −1 ; (4) q f +j+1 := βb q f +j − xb q f +j−1 , j ≥ 1.
Lemma
Let M 2×2 be as before. If j ≥ 1,
q f +j−1 =
qj∗ (b) wj∗ (b)
M
qf∗−1 (a)
qf∗ (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
KN+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
Scaling Limit: Last Visit portion of Gambler’s Ruin
Denote V := #Short Runs until L; M := #Excursions
until L;
a(b−a)
2−a−b
1
XN := L − (1−a)(1−b) R + (1−a)(1−b) V − (1−a)(1−b) M /N
Theorem
Let f ∼ ηN for some fixed 0 < η < 1. Let σj , κj , and ϕ̂(t) as before.
Then limN→∞ E{eitXN } = ψ̂(t)/ϕ̂(t),
ψ̂(t) :=
abσ1 σ2
abσ1 σ2 cosh(κ1 t) cosh(κ2 t)+a2 σ12 sinh(κ1 t) sinh(κ2 t)+iaσ1 (b−a)2 cosh(κ1 t) sinh(κ2 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 gm,n Established by Induction
Combine Results for Last Visit and Meander
Corollary
0
limN→∞ E{eit(XN +XN ) } = ψ̂(t)
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.
Bibliography
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(1999) 167-202.
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University Press (2009).
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Addendum:
Formula for K (r , u, z) := K (ru, 1/u, z; 12 )
Recall the Joint G.F. of Excursions Stats R, V, L in Homogeneous Case
Corollary
p
Let b = a. Define αa := βa2 − 4xa .
Then
(1)
K (r , y , z; a) := E{r R y V z L } = (1 − 21 βa − 12 αa )/(1 − a)
Assume now in Addition that a = 12 .
Then by Direct Calculation the Joint G.F. of R, U, L is given by
1
16 − 4z 2 + 4r 2 z 2 + r 2 z 4 − 2r 2 uz 4 + r 2 u 2 z 4 − S ,
K (r , u, z) := 16
With Main
p Term S given by:
S := (4 + 2z + 2rz + rz 2 − ruz 2 )(4 + 2z − 2rz − rz 2 + ruz 2 )
p
× (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).