Scaling limits of non-increasing Markov chains
and applications
Bénédicte HAAS
Université Paris-Dauphine
Joint work with Grégory MIERMONT (Orsay)
Applications to random walks with a barrier, Markov branching trees, number of
collisions in Λ-coalescents
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Setting
Main object: non-increasing Markov chain with values in Z+ = {0, 1, 2, ..}
• pi,j , i, j ≥ 0 : probabilities transition:
pi,j ≥ 0,
Pi
j=0
pi,j = 1 and pi,j = 0 for j > i
• (Xn (k ), k ≥ 0) : Markov chain starting at Xn (0) = n
• absorption time An := inf{i : Xn (i) = Xn (j), ∀j ≥ i} < ∞ a.s.
Behavior of the process
B. Haas
Xn (·)
and An as n → ∞ ?
n
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Example 1: random walks with a barrier
(Sk , k ≥ 0) Z+ -valued random walk: for Yi , i ≥ 1 i.i.d Z+ -valued,
Sk = Y1 + ... + Yk
(S0 = 0)
• behavior as n → ∞ of the walk killed at n (Sk ∧ n, k ≥ 0) ?
• behavior of the walk ignoring jumps that would make it exceed n ?
(defined recursively by: S̃k +1 = S̃k + Yk +1 1{S̃k +Yk +1 ≤n} )
• behavior of the walk Ŝ with jumps conditioned not to make it exceed n ?
In all cases:
Xn (k ) = n − Sk ∧ n,
X̃n (k ) = n − S̃k
and X̂n (k ) = n − Ŝk ,
k ≥1
define non-increasing, Z+ -valued Markov chains
Remark: Xn is absorbed at 0 (but not necessarily X̃n and X̂n )
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Example 2 : Markov branching trees
(Tn , n ≥ 1) Markov branching sequence of trees: Tn rooted tree with n leaves
For each k ∈ N = {1, 2, ...}, Pk : proba. on the set of partitions of k =
Pp
↓ seq. of positive integers (k1 , .., kp ) : i=1 ki = k ,
Pk ((k )) < 1; for k = 1, cemetery point ∆: P1 (∆) := 1 − P1 ((1))
Rules:
• a urn with k balls splits in p urns with k1 , .., kp balls with proba. Pk ((k1 , .., kp ))
• 6= urns split independently
Starting from a urn with n balls → gives a rooted tree Tn with n leaves, ∀n
Question: how does Tn grow with n ?
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Example 2: Markov branching trees
Ex.: T GW : critical Galton-Watson tree
TnGW ∼law T GW |{T GW has n leaves}
when the reproduction law ξ has finite variance σ:
law
n−1/2 TnGW → 2(σξ(0))−1 TBr
(KORTCHEMSKI 11, R IZZOLO 11)
TBr : Aldous’ continuum Brownian tree
General Markov branching sequence: useful tool: marked ball
Xn (k ): size of the urn containing ◦ after k steps → ↓ Markov chain
An = inf{k : Xn (k ) = 0}=height of a typical leaf in Tn +1
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Main result: hypothesis
Non-increasing Z+ -valued Markov chain, Xn version starting at n
Hypothesis (H)
∃γ > 0 and ` : N → (0, ∞) a function slowly varying at ∞ and µ a finite
measure on [0, 1] (µ([0, 1]) > 0) such that
Z
Xn (1)
Xn (1)
1−
→
f (x)µ(dx)
nγ `(n)E f
n
n
[0,1]
for all continuous functions f : [0, 1] → R.
i.e., starting from n, “macroscopic" (with size proportional to n) jumps are rare
n − Xn (1)
µ([0, 1])
E
∼ γ
n
n `(n)
and
Z
1
µ(dx)
P (n − Xn (1) ≥ nε) ∼ γ
n `(n) [0,1−ε] 1 − x
for a.e. 0 < ε ≤ 1.
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Main result: chain’s behavior
Theorem
Assume (H). Then
Xn (bnγ `(n)tc)
law
,t ≥ 0
→ X∞ = exp(−ξρ )
n→∞
n
where
• ξ is a subordinator with Laplace exponant
Z
µ(dx)
φ(λ) = µ({0}) + µ({1})λ +
(1 − x λ )
, λ ≥ 0,
1−x
(0,1)
• ρ is the time-change ρ(t) = inf {u ≥ 0 :
Ru
0
exp(−γξr )dr ≥ t}
Rks.: • E[exp(−λξt )] = exp(−tφ(λ)), λ, t ≥ 0
• convergence holds for the Skorokhod topology on the set of càdlàg
functions from [0, ∞) to [0, ∞).
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The limiting process X∞
X∞ is a non-increasing self-similar Markov process with index 1/γ, that is
the law of (cXc −γ t , t ≥ 0) under Px is Pcx , ∀c, x > 0,
where Px is the distribution of X∞ conditional on X∞ (0) = x.
Reciprocally,
• each non-increasing self-similar Markov process is, under P1 , of the form
exp(−ξρ ) for some subordinator ξ and γ > 0 (L AMPERTI 72)
• each can obtained as the scaling limit of some non-increasing Z+ -valued
Markov chain
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Main result: absorption time’s behavior
Fact: inf{t ≥ 0 : X∞ (t) = 0} =
R∞
0
exp(−γξr )dr < ∞ a.s.
Theorem
Assume (H). Then, jointly with the previous convergence,
Z ∞
An
law
→
exp(−γξr )dr .
nγ `(n) n→∞ 0
this is not a direct corollary of the previous theorem, since the application:
non-increasing càdlàg function → its absorption time
is not continuous.
Also, under (H):
»„
E
An
nγ `(n)
«p –
n→∞
«p –
∞
»„Z
→ E
exp(−γξr )dr
,
∀p ≥ 0
0
and when p ∈ Z+ ,
»„Z ∞
«p –
p!
E
exp(−γξr )dr
= Qp
(by C ARMONA -P ETIT-YOR 97)
0
i=1 φ(γi)
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Outline of proof
Assume (H)
• let Yn (t) := n−1 Xn (bnγ `(n)tc), then (Yn , n ≥ 1) is tight
law
• let Y 0 be a possible limit: ∃ a subsequence (nk , k ≥ 1) s.t. Ynk → Y 0
Ru
Ru
let τYn (t) := inf{u : 0 Yn −γ (r )dr > t}, τY 0 (t) := inf{u : 0 Y 0−γ (r )dr > t}
and Z 0 (t) = Y 0 (τY 0 (t))
R u 0γ
0
(r )dr > t})
Fact: Y 0 (t) = Z 0 (τY−1
0 (t))= Z (inf{u : 0 Z
• for all λ ≥ 0, and n ≥ 1, let Gn (λ) := E (Xn (1)/n)λ . Then,
 γ
−1
bn `(n)τYn (t)c−1
Y
(λ)
Mn (t) := Znλ (t) 
GXn (i) (λ) , t ≥ 0
Zn (t) := Yn (τYn (t))
i=0
is a martingale (consequence of the Markov property of Xn )
(λ) law
• Mnk → (Z 0 )λ exp(φ(λ)·), which is also a martingale
• ⇒ ln Z 0 is a Lévy process
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Application: random walks with a barrier
(Yi , i ≥ 1) i.i.d., qn := P(Y1 = n), n ∈ Z+ ,
• Sk =
Pk
i=1
P
n
qn = 1
Yi , S0 = 0
• S̃k +1 = S̃k + Yk +1 1{S̃k +Yk +1 ≤n} , S̃0 = 0
• Ŝ walk with jumps conditioned not to make it exceed n
Then let
Xn (k ) = n − Sk ∧ n,
X̃n (k ) = n − S̃k
and X̂n (k ) = n − Ŝk , k ≥ 1
P
Respective transition probabilities, letting q i := n>i qn , and 0 ≤ j ≤ i
• pi,j = qi−j + 1{j=0} q i
• p̃i,j = qi−j + 1{j=i} q i
• p̂i,j = qi−j /(1 − q i )
Remark: if E[Y1 ] < ∞, An , Ãn , Ân grow at speed n (law of large numbers)
for CLT, see I KSANOV, M ÖHLE 08, VAN C UTSEM , Y CART 94
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Paris-Bath Meeting
11 / 20
Application: random walks with a barrier
• assume q n = n−γ `(n), for some γ ∈ (0, 1) and ` regularly varying at ∞
• let ξ be a subordinator with Laplace exponent
R∞
γe−y dy
φ(λ) = 0 (1 − e−λy ) (1−e
−y )γ+1 ,
λ ≥ 0,
• let e(1) be an exponential distribution with parameter 1, independent of ξ
(k)
• let ξt
= ξt + ∞1{e(1)≤t} , t ≥ 0
(k)
• let X∞ and X∞ be the 1/γ-self-similar Markov processes associated to
ξ and ξ (k) respectively
Theorem
Then, we have the joint convergence
law
(k)
, X∞ , X∞
n−1 Xn (b·/q n c), n−1 X̃n (b·/q n c), n−1 X̂n (b·/q n c) → X∞
R∞
R∞
law R e(1)
q n An , Ãn , Ân → 0 exp(−γξr )dr , 0 exp(−γξr )dr , 0 exp(−γξr )dr
Remark: I KSANOV, M ÖHLE 08 show this asymptotic behavior for Ân
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Application: Markov branching trees
• (Tn , n ≥ 1): Markov branching sequence of trees, Tn : n leaves
• Pn : proba. on the partitions of n, describing how a urn with n balls splits
• Xn (k ): size of the marked ball ◦ urn after k splitting events
Xn non-increasing Markov chain with transition probabilities
X
i
pk ,i =
Pk ((k1 , .., kp )) #{r : kr = i}, i ≤ k
k
Pp
(k1 ,..,kp ):
i=1
ki =k
Let S ↓ := {s = (s1 , s2 , ...) : s1 ≥ s2 ... ≥ 0,
P
si = 1}.
Hypothesis (H’)
∀ continuous f : S ↓ → R, ∃γ > 0, ` slowly varying, η finite measure on S ↓ s.t.
Z
X
kp
k1
k1
γ
n `(n)
Pn ((k1 , ..., kp )) f
, ..., , 0, ..
1−
→
f (s)η(ds)
n
n
n n→∞ S ↓
Pp
i=1
ki =n
Fact: (H’) ⇒ (H) for the related transition probabilities ⇒ a typical leaf grows
at speed nγ `(n)
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Application: Markov branching trees
Theorem
Assume that (Pn , n ≥ 1) satisfies (H’). Then, ∃ a continuum random tree Tγ,η :
Tn
law
→ Tγ,η
nγ `(n) n→∞
(for the Gromov-Hausdorff topology).
A similar result holds for sequences of trees indexed by vertices
Corollary 1
Let Tn be uniformly distributed among the set of rooted trees, unordered and
unlabeled, with n vertices (Pólya trees). Then,
Tn law
→ cTBr
n1/2 n→∞
for some deterministic c > 0.
TBr : Aldous’ continuum Brownian tree
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Paris-Bath Meeting
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Application to Markov Branching Trees
Let T GW be a critical Galton-Watson tree, with a reproduction law, ξ, with finite
variance σ
Corollary 2 (Aldous 93)
Let TnGW,v be distributed as T GW conditioned to have n vertices. Then,
TnGW,v law 2
→
TBr
n1/2 n→∞ σ
Corollary 3 (Kortchemski 11, Rizzolo 11)
Let TnGW,l be distributed as T GW conditioned to have n leaves. Then,
TnGW,l
n1/2
B. Haas
law
→
n→∞
2
TBr
σξ(0)
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Application: number of collisions in Λ-coalescents
Introduced and studied first by P ITMAN 99 and S AGITOV 99
Λ: finite measure on [0, 1]
Λn -coalescent:
continuous-time Markov chain on the set of partitions of {1, ..., n}:
when there is i present blocks, j of them coalesce in a unique block (to give a
total number of blocks equal to i − j + 1) at rate
Z
i
gi,j =
x j−2 (1 − x)i−j Λ(dx), 2 ≤ j ≤ i.
j
[0,1]
Ex.: • Λ = δ0 : Kingman’s coalescent (blocks coalesce 2 by 2 at rate 1)
• Λ = Leb[0,1] : Bolthausen-Snitzman coalescent
• Λ(dx) = x a−1 (1 − x)b−1 dx, 0 < x < 1: Beta(a, b)-coalescent, a, b > 0
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Application: number of collisions in Λ-coalescents
Starting from the trivial partition {{1}, {2}, ..., {n}}:
• Xn (k ) :=number of blocks in the coalescent after k collisions, k ≥ 0
• Xn : non-increasing Z+ -Markov chain with transition probabilities
Z
gi,j
1
i
pi,j =
=
x j−2 (1 − x)i−j Λ(dx)
j
gi
gi
[0,1]
where gi =
Pi
j=2
gi,j is the total rate of coalescence of i blocks.
• An = inf{k : Xn (k ) = 1}=total number of collisions of the process
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Application: number of collisions in Λ-coalescents
• assume that Λ({0}) = 0 and
Z 1
h(u) :=
x −2 Λ(dx) = u −γ `(1/u),
u ∈ (0, 1]
u
for γ ∈ (0, 1) and ` slowly varying at ∞
• let ξ be the subordinator with Laplace exponent
Z 1
1
φ(λ) =
(1 − (1 − x)λ )x −2 Λ(dx) ,
Γ(2 − γ) 0
and let X∞ be the 1/γ-self-similar Markov process associated with ξ.
Theorem
Under these assumptions, we have the joint convergence
Xn (bh(1/n) ·c)
n
An
h(1/n)
B. Haas
law
→
law
X∞
Z ∞
n→∞
0
n→∞
→
exp(−γξr )dr .
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Application: number of collisions in Beta-coalescents
Λ(dx) = x a−1 (1 − x)b−1 dx, 0 < x < 1, a, b > 0
For 1 < a < 2 and b > 0, let ξ be a subordinator with Laplace exponent
Z
2−a ∞
e−by
φ(λ) =
(1 − e−λy )
dy ,
Γ(a) 0
(1 − e−y )3−a
and X∞ be the associated 1/(2 − a)-self-similar Markov process.
Corollary
For the beta-coalescent β(a, b) with parameters 1 < a < 2 and b > 0 we
have the joint convergence
Xn (bn2−a ·c)
n
An
n2−a
law
→
law
X∞
Z ∞
n→∞
0
n→∞
→
exp(−(2 − a)ξr )dr .
Result previously proved by I KSANOV, M ÖHLE 08 when b = 1
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Application: number of collisions in Beta-coalescents
To compare, a list of previously known results:
• if 0 < a < 1, b > 0, the limit law of
An − (1 − a)n
n1/(2−a)
is a (2 − a)-stable distribution (G NEDIN , YAKUBOVICH 07, I KSANOV, M ÖHLE 08 when
b = 1)
• if a = b = 1 (Bolthausen-Sznitman coalescent), the limit law of
log2 (n)
An − log(n log(n))
n
is a 1-stable distribution (I KSANOV, M ÖHLE 07)
• if a = 2, b > 0, there exists c > 0 such that the limit law of
An − c log2 (n)
log3/2 (n)
is a centered normal distribution (I KSANOV, M ARYNYCH , M ÖHLE 09)
• if a > 2, b > 0, there exists c > 0 such that the limit law of
An − c log(n)
p
log(n)
is a centered normal distribution (G NEDIN , I KSANOV, M ÖHLE 08)
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