[PPT] - Scaling limits of non-increasing Markov chains and applications to PowerPoint Presentation

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Scaling limits of non-increasing Markov chains and applications to random trees and coalescents

Bénédicte HAAS

Université Paris-Dauphine

based on joint works with Grégory MIERMONT (Orsay)

Bénédicte Haas SSP - March 2012 1 / 26

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Outline

1

Introduction

2

Scaling limits of non-increasing Markov chains

3

Applications:

scaling limits of Markov branching trees
number of collisions in Λ-coalescents
random walks with barriers

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Scaling limits: a basic example

I.i.d sequence of centered random variables Xi ∈ {−1, 1} : −1, 1, 1, 1, 1, −1, −1, −1, 1, −1, −1, 1, 1, 1, 1, ... Centered random walk: Sn = X1 + ... + Xn How does Sn behave when n is large ? (1) what is the growth rate ? (2) what is the limit after rescaling ? Central limit theorem: Sn √n

law

→ N(0, 1) Functional version:

DONSKER’S theorem (51):

„S[nt] √n , t ∈ [0, 1] «

law

→ (B(t), t ∈ [0, 1]) where B is a standard Brownian motion

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Another example: Galton-Watson trees

Galton-Watson processes are introduced in 1873 to study the extinction of family names

ancester=root generation 3 generation 2 generation 1

η: offspring distribution (proba. on Z+ = {0, 1, 2, ..}), such that η(1) < 1, with mean m Extinction probability = 1 in subcritical (m < 1) and critical (m = 1) cases ∈ [0, 1) in supercritical cases (m > 1)

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Large Galton-Watson trees

T GW

n

: critical GW tree conditioned to have n nodes Offspring distribution η: finite variance σ2 < ∞ ◮ H(U)

n

: height (=generation) of a node chosen uniformly at random H(U)

n

√n

law

→ R σ , where R has a Rayleigh distribution: P(R > x) = exp(−x2/2) (MEIR & MOON 78) ◮ Hn: height of the tree Hn √n

law

→ 2W σ , where W =maximum of a Brownian excursion with length 1 (KOLCHIN 86) With length 1 on each edge, what does the tree look like when n → ∞ ?

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Universal limit: the Brownian tree TBr

(picture by G. Miermont)

ALDOUS 93: T GW

n

√n

law

→ 2 σ TBr Compact (random) real tree, i.e. compact metric space with the tree property: ∀x, y ∈ TBr ∃! path from x to y almost surely: binary tree, self-similar, with Hausdorff dimension = 2

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Topology on the set of compact rooted real trees

Gromov-Hausdorff distance: let (T , ρ), (T ′, ρ′) be two compact rooted real trees dGH ` (T , ρ), (T ′, ρ′) ´ := inf ` dH(ϕ1(T ), ϕ2(T ′)) ∨ dZ(ϕ1(ρ), (ϕ2(ρ′)) ´ the infimum being on all isometric embeddings ϕ1 : T ֒ → Z and ϕ2 : T ′ ֒ → Z into a same metric space (Z, dZ). (T , ρ) and (T ′, ρ′) are equivalent if ∃ ϕ isometry: T ′ = ϕ(T ), ρ′ = ϕ(ρ) dGH: distance on the set of equivalence classes

ALDOUS 93:

T GW

n

√n

law

→

GH

2 σ TBr, jointly with the convergence of the uniform probability on the nodes of T GW

n

towards a probability measure on the leaves of TBr

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Large Galton-Watson trees: when the variance is infinite

Assume: η(k) = P(to have k children) ∼

k→∞ Ck −β, 1 < β < 2

Then (DUQUESNE 03): T GW

n

n1−1/β

law

→

GH C−1/βTβ

◮ “smaller" trees ◮ the limiting tree Tβ belongs to the family of stable Lévy trees (introduced by Duquesne, Le Gall, Le Jan):

each branching vertex branches in an infinite, countable number of subtrees
it is a self-similar tree, with Hausdorff dimension = β/(β − 1)

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Non-increasing Markov chains

(X(k), k ≥ 0): Z+-valued Markov chain, non-increasing (Xn(k), k ≥ 0): chain starting from Xn(0) = n

An Xn n k

Absorption time: An = inf{i : Xn(i) = Xn(j), ∀j ≥ i}, finite How behave Xn(·) n and An when n → ∞ ?

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Scaling limit

Assumption: macroscopic jumps are rare starting from n, the probability that the first jump is larger than ≥ nε behaves like ∼

n→∞

cε nγ for some γ > 0 (cε ր when ε ց)

More precisely, ∃ finite measure µ on [0, 1] such that, for ε ∈ (0, 1] P (n − Xn(1) ≥ nε) ∼ 1 nγ Z

[0,1−ε]

µ(dx) 1 − x , and E »n − Xn(1) n – ∼ µ ([0, 1]) nγ

Theorem (H.-MIERMONT 11) Then, ∃ time-continuous Markov process X∞ such that „Xn ([nγt]) n , t ≥ 0 «

law

→ (X∞(t), t ≥ 0), for the Skorokhod topology on the set D([0, ∞), [0, ∞)).

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The limit process X∞ is:

◮ self-similar: starting from X∞(0) = x, the process ` cX∞(c−γt), t ≥ 0 ´ is distributed as X∞ starting from X∞(0) = cx, ∀c > 0 ◮ starting from X∞(0) = 1, X∞ writes X∞ = exp(−ξρ) where

ξ is a subordinator
E[exp(−λξt)] = exp(−tφ(λ)), with

φ(λ) = µ({1})λ + Z

(0,1)

(1 − xλ)µ(dx) 1 − x + µ({0}), λ ≥ 0,

ρ is an acceleration of time:

ρ(t) = inf  u ≥ 0 : Z u exp(−γξr)dr ≥ t ff

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Absorption time

Consequently, X∞ starting from X∞(0) = 1 is absorbed at 0 at time Z ∞ exp(−γξr)dr

(almost surely finite)

Jointly with the previous convergence, we have An nγ

law

→ Z ∞ exp(−γξr)dr This is not an immediate consequence of the convergence of the whole process! Remark: extension of these results to regular variation assumptions.

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Application to Markov branching trees

(Tn, n ≥ 1) : Tn random rooted tree with n nodes Markov branching property: Tn :

(n = 17)

9 nods R 4 nods 3 nods 4 nods R

law

∼ T4

R 3 nods law

∼ T3

9 nods R

law

∼ T9 Conditional on “the root of Tn branches in p sub-trees with n1 ≥ ... ≥ np nodes”, these sub-trees are independent, with respective distributions those of Tn1, ..., Tnp Similar definition for sequences of trees indexed by the number of leaves Ex.: Galton-Watson trees conditioned to have n nodes (respectively n leaves)

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