Combinatorial Markov chains R. Gr ubel Leibniz Universit at - - PowerPoint PPT Presentation

Combinatorial Markov chains

• R. Gr¨

ubel Leibniz Universit¨ at Hannover

AofA, Menorca 2013

Combinatorial Markov chains

Combinatorial Markov chains

• A combinatorial family is a set F with a size function

φ : F → N such that Fn := {x ∈ F : φ(x) = n} is finite for all n ∈ N. We assume that F1 = {e}.

Combinatorial Markov chains

• A combinatorial family is a set F with a size function

φ : F → N such that Fn := {x ∈ F : φ(x) = n} is finite for all n ∈ N. We assume that F1 = {e}.

• A combinatorial Markov chain (CMC) is a Markov chain

X = (Xn)n∈N that is adapted to F: P(Xn ∈ Fn) = 1 for all n ∈ N.

Combinatorial Markov chains

• A combinatorial family is a set F with a size function

φ : F → N such that Fn := {x ∈ F : φ(x) = n} is finite for all n ∈ N. We assume that F1 = {e}.

• A combinatorial Markov chain (CMC) is a Markov chain

X = (Xn)n∈N that is adapted to F: P(Xn ∈ Fn) = 1 for all n ∈ N. AofA context: Consider an sequential algorithm that transforms a sequence (ηn)n∈N of input values into a sequence (xn)n∈N of combinatorial objects such that xn ∈ Fn, xn+1 depends on xn and ηn+1 only. For i.i.d. input this algorithm generates a CMC.

Growth / Transition structure

Growth / Transition structure

On F we have a relation ‘֒ →’ with x ֒ → y = ⇒ φ(x) + 1 = φ(y).

Growth / Transition structure

On F we have a relation ‘֒ →’ with x ֒ → y = ⇒ φ(x) + 1 = φ(y). This turns F into a directed graph with edges E(F) = {(x, y) : x, y ∈ F, x ֒ → y}. Such structures (F, ֒ →) appear elsewhere, e.g. as Bratelli diagrams.

Growth / Transition structure

On F we have a relation ‘֒ →’ with x ֒ → y = ⇒ φ(x) + 1 = φ(y). This turns F into a directed graph with edges E(F) = {(x, y) : x, y ∈ F, x ֒ → y}. Such structures (F, ֒ →) appear elsewhere, e.g. as Bratelli diagrams. More assumptions:

• weak connectedness – there is a path from the root to each

node, ∀ x ∈ F ∃ yi ∈ Fi, i = 1, . . . , n := φ(x) : y1 = e, yn = x, (yi−1, yi) ∈ E(F) for i = 2, . . . , n

• the transition probabilities p(x, y) = P(Xn+1 = y|Xn = x) are

adapted to this structure, p(x, y) > 0 ⇐ ⇒ x ֒ → y,

Example 1: Random walk, urns, and special numbers

Example 1: Random walk, urns, and special numbers

F = N0 × N0, φ(i, j) = i + j + 1, (i, j) ֒ → (i + 1, j), (i, j) ֒ → (i, j + 1)

Example 1: Random walk, urns, and special numbers

F = N0 × N0, φ(i, j) = i + j + 1, (i, j) ֒ → (i + 1, j), (i, j) ֒ → (i, j + 1)

Example 1: Random walk, urns, and special numbers

F = N0 × N0, φ(i, j) = i + j + 1, (i, j) ֒ → (i + 1, j), (i, j) ֒ → (i, j + 1) CMC p

• (i, j), (i + 1, j)
• L(Xn)

north-east random walk

• param. θ ∈ (0, 1)

i+j

i

• θi(1 − θ)j

" θ = 1/2 ∼ binomial coefficients record walk 1/(i + j + 2) ∼ Stirling(1) numbers Friedman urn (j + 1)/(i + j + 2) ∼ Euler numbers P´

• lya urn

(i + 1)/(i + j + 2) uniform distribution

Example 2: Partitions (of integers)

Example 2: Partitions (of integers)

Fn :=

• λ = (λ1, . . . , λk) ∈ N⋆ : k

i=1 λi = n, λ1 ≥ · · · ≥ λk

• ,

Fn ∋ λ = (λ1, . . . , λk) ֒ → µ = (µ1, . . . , µl) ∈ Fn+1 iff k ≤ l and λj ≤ µj.

Example 2: Partitions (of integers)

Fn :=

• λ = (λ1, . . . , λk) ∈ N⋆ : k

i=1 λi = n, λ1 ≥ · · · ≥ λk

• ,

Fn ∋ λ = (λ1, . . . , λk) ֒ → µ = (µ1, . . . , µl) ∈ Fn+1 iff k ≤ l and λj ≤ µj.

11 = 5 + 3 + 3 5/12 1/3 6/12 1/3 1/12 1/3

Example 2: Partitions (of integers)

Fn :=

• λ = (λ1, . . . , λk) ∈ N⋆ : k

i=1 λi = n, λ1 ≥ · · · ≥ λk

• ,

Fn ∋ λ = (λ1, . . . , λk) ֒ → µ = (µ1, . . . , µl) ∈ Fn+1 iff k ≤ l and λj ≤ µj.

11 = 5 + 3 + 3 5/12 1/3 6/12 1/3 1/12 1/3

• Thoma: p(λ, µ) = H(λ)/H(µ), hook length formula.

Example 2: Partitions (of integers)

Fn :=

• λ = (λ1, . . . , λk) ∈ N⋆ : k

i=1 λi = n, λ1 ≥ · · · ≥ λk

• ,

Fn ∋ λ = (λ1, . . . , λk) ֒ → µ = (µ1, . . . , µl) ∈ Fn+1 iff k ≤ l and λj ≤ µj.

11 = 5 + 3 + 3 5/12 1/3 6/12 1/3 1/12 1/3

• Thoma: p(λ, µ) = H(λ)/H(µ), hook length formula.
• The restaurant (or cycle) chain:

– If µj = λj + 1, p(λ, µ) = λj · #{i ≥ j : λi = λj}/(n + 1). – If µ = (λ1, . . . , λk, 1): p(λ, µ) = 1/(n + 1)

Example 3: Binary trees

Example 3: Binary trees

Let V = {0, 1}⋆ be the set of nodes, with prefix relation ’≤’. Fn :=

• x ⊂ V : #x = n, (u ∈ x, v ≤ u) ⇒ v ∈ x
• ,

x ֒ → y iff #y = #x + 1 and x ⊂ y.

Example 3: Binary trees

Let V = {0, 1}⋆ be the set of nodes, with prefix relation ’≤’. Fn :=

• x ⊂ V : #x = n, (u ∈ x, v ≤ u) ⇒ v ∈ x
• ,

x ֒ → y iff #y = #x + 1 and x ⊂ y. BST chain: L(Xn+1|Xn = x) = unif

• {y ∈ Fn+1 : x ֒

→ y}

• .

Example 3: Binary trees

Let V = {0, 1}⋆ be the set of nodes, with prefix relation ’≤’. Fn :=

• x ⊂ V : #x = n, (u ∈ x, v ≤ u) ⇒ v ∈ x
• ,

x ֒ → y iff #y = #x + 1 and x ⊂ y. BST chain: L(Xn+1|Xn = x) = unif

• {y ∈ Fn+1 : x ֒

→ y}

• .

1 2 3 4 5 6 8 7 9 8 1 9 3 2 5 7 6 4

Boundaries: Background

Boundaries: Background

potential theory Markov processes Laplace operator transition probabilities harmonic functions martingales Dirichlet problem Brownian motion electrical networks random walks on graphs

Boundaries: Background

potential theory Markov processes Laplace operator transition probabilities harmonic functions martingales Dirichlet problem Brownian motion electrical networks random walks on graphs Doob, Hunt, Dynkin, Meyer . . .; an ‘icon formula’ is

h(x) = Exφ(Bτ)

Boundaries: Background

potential theory Markov processes Laplace operator transition probabilities harmonic functions martingales Dirichlet problem Brownian motion electrical networks random walks on graphs Doob, Hunt, Dynkin, Meyer . . .; an ‘icon formula’ is

h(x) = Exφ(Bτ)

Recent textbook:

• W. Woess, Denumerable Markov chains. EMS 2009.

Boundaries: Background

potential theory Markov processes Laplace operator transition probabilities harmonic functions martingales Dirichlet problem Brownian motion electrical networks random walks on graphs Doob, Hunt, Dynkin, Meyer . . .; an ‘icon formula’ is

h(x) = Exφ(Bτ)

Recent textbook:

• W. Woess, Denumerable Markov chains. EMS 2009.

Connection to representation theory, random walks on groups: Vershik, Kaimanovich, Kerov (book), Gnedin, Okounkov, Woess . . .

Doob-Martin compactification: Construction

Doob-Martin compactification: Construction

Let X = (Xn)n∈N be a CMC on the combinatorial family F.

Doob-Martin compactification: Construction

Let X = (Xn)n∈N be a CMC on the combinatorial family F.

• Regard F as a discrete topological space.

Doob-Martin compactification: Construction

Let X = (Xn)n∈N be a CMC on the combinatorial family F.

• Regard F as a discrete topological space.
• Define the Martin kernel by

K(x, y) := P(Xn = y|Xm = x) P(Xn = y) , x ∈ Fm, y ∈ Fn.

Doob-Martin compactification: Construction

Let X = (Xn)n∈N be a CMC on the combinatorial family F.

• Regard F as a discrete topological space.
• Define the Martin kernel by

K(x, y) := P(Xn = y|Xm = x) P(Xn = y) , x ∈ Fm, y ∈ Fn.

• The Doob-Martin compactification ¯

F is the minimal compact enlargement of F that allows for a continuous extension of the (bounded) functions y → K(x, y), x ∈ F.

Doob-Martin compactification: Construction

Let X = (Xn)n∈N be a CMC on the combinatorial family F.

• Regard F as a discrete topological space.
• Define the Martin kernel by

K(x, y) := P(Xn = y|Xm = x) P(Xn = y) , x ∈ Fm, y ∈ Fn.

• The Doob-Martin compactification ¯

F is the minimal compact enlargement of F that allows for a continuous extension of the (bounded) functions y → K(x, y), x ∈ F. Then ∂F := ¯ F \ F is the Martin boundary.

Doob-Martin compactification: h-transforms

Doob-Martin compactification: h-transforms

Call h : F → R harmonic if it has the ‘mean value property’: h(x) =

• y∈F

p(x, y)h(y) for all x ∈ F.

Doob-Martin compactification: h-transforms

Call h : F → R harmonic if it has the ‘mean value property’: h(x) =

• y∈F

p(x, y)h(y) for all x ∈ F. Let H+ resp. Hb be the set of non-negative resp. bounded harmonic functions.

Doob-Martin compactification: h-transforms

Call h : F → R harmonic if it has the ‘mean value property’: h(x) =

• y∈F

p(x, y)h(y) for all x ∈ F. Let H+ resp. Hb be the set of non-negative resp. bounded harmonic functions. For h ∈ H+, h = 0, ph(x, y) := 1 h(x) p(x, y) h(y), x, y ∈ F, defines another transition probability on F.

Doob-Martin compactification: h-transforms

Call h : F → R harmonic if it has the ‘mean value property’: h(x) =

• y∈F

p(x, y)h(y) for all x ∈ F. Let H+ resp. Hb be the set of non-negative resp. bounded harmonic functions. For h ∈ H+, h = 0, ph(x, y) := 1 h(x) p(x, y) h(y), x, y ∈ F, defines another transition probability on F. With X h

1 = e and transitions ph we obtain another CMC

X h = (X h

n )n∈N, the h-transform of X.

Doob-Martin compactification: Results

Doob-Martin compactification: Results

(reprDM) The extensions x → K(x, α), α ∈ ∂F, are minimal harmonic, and all h ∈ H+ with h(e) = 1 are mixtures of these: h(x) =

• ∂F

K(x, α) νh(dα)

Doob-Martin compactification: Results

(reprDM) The extensions x → K(x, α), α ∈ ∂F, are minimal harmonic, and all h ∈ H+ with h(e) = 1 are mixtures of these: h(x) =

• ∂F

K(x, α) νh(dα) (conv) As n → ∞, Xn → X∞ ∈ ∂F almost surely, with L(X∞) the measure ν representing h ≡ 1.

Doob-Martin compactification: Results

(reprDM) The extensions x → K(x, α), α ∈ ∂F, are minimal harmonic, and all h ∈ H+ with h(e) = 1 are mixtures of these: h(x) =

• ∂F

K(x, α) νh(dα) (conv) As n → ∞, Xn → X∞ ∈ ∂F almost surely, with L(X∞) the measure ν representing h ≡ 1. (struct) The conditional distribution L(X|X∞ = α) is equal to the distribution of the h-transform L(X h) with h = K(·, α) (in particular, this is the distribution of a Markov chain).

Doob-Martin compactification: The ‘why’

Doob-Martin compactification: The ‘why’

In a CMC (Xn)n∈N with Martin kernel K, E

• K(x, Xm−1)
• Xm, . . . , Xn
• ≤ K(x, Xm).

Doob-Martin compactification: The ‘why’

In a CMC (Xn)n∈N with Martin kernel K, E

• K(x, Xm−1)
• Xm, . . . , Xn
• ≤ K(x, Xm).

Hence, for each n ∈ N, (Y n

k , Fn k )k=1,...,n with

Y n

k := K(x, Xn+1−k),

Fn

k = σ(Xn+1−k, . . . , Xn)

is a supermartingale.

Doob-Martin compactification: The ‘why’

In a CMC (Xn)n∈N with Martin kernel K, E

• K(x, Xm−1)
• Xm, . . . , Xn
• ≤ K(x, Xm).

Hence, for each n ∈ N, (Y n

k , Fn k )k=1,...,n with

Y n

k := K(x, Xn+1−k),

Fn

k = σ(Xn+1−k, . . . , Xn)

is a supermartingale. For the number Un(a, b) of upcrossings of some interval [a, b] this means EUn[a, b] ≤ 1 b − aE(Y n

n − a)− =

1 b − a

• K(x, e) − a

−. It follows that (K(x, Xn))n∈N converges almost surely.

Poisson boundary

Poisson boundary

A measurable space (E, B) together with a family νx, x ∈ F, of probability measures on (E, B) such that νx ≪ νe for all x ∈ F ‘is’ the Poisson boundary of (X, F) if L∞(E, B, νe) ∼ = Hb as Banach spaces, with the supremum norm on Hb,

Poisson boundary

A measurable space (E, B) together with a family νx, x ∈ F, of probability measures on (E, B) such that νx ≪ νe for all x ∈ F ‘is’ the Poisson boundary of (X, F) if L∞(E, B, νe) ∼ = Hb as Banach spaces, with the supremum norm on Hb, via: For each φ ∈ L∞ (reprP) h(x) =

• E

φ(y) νx(dy), x ∈ F, is in Hb, and for each h ∈ Hb there is a unique φ ∈ L∞ such that this representation holds.

Tail boundary

Tail boundary

Let T := ∞

n=1 σ

• {Xm : m ≥ n}
• be the tail σ-field of (Xn)n∈N.

Tail boundary

Let T := ∞

n=1 σ

• {Xm : m ≥ n}
• be the tail σ-field of (Xn)n∈N.

Call h : N × F → R space-time harmonic if h(n, x) =

• y∈F

p(x, y) h(n + 1, y) for all x ∈ F, n ∈ N.

Tail boundary

Let T := ∞

n=1 σ

• {Xm : m ≥ n}
• be the tail σ-field of (Xn)n∈N.

Call h : N × F → R space-time harmonic if h(n, x) =

• y∈F

p(x, y) h(n + 1, y) for all x ∈ F, n ∈ N. Let Hst

b be the set of all bounded space-time harmonic functions.

Tail boundary

Let T := ∞

n=1 σ

• {Xm : m ≥ n}
• be the tail σ-field of (Xn)n∈N.

Call h : N × F → R space-time harmonic if h(n, x) =

• y∈F

p(x, y) h(n + 1, y) for all x ∈ F, n ∈ N. Let Hst

b be the set of all bounded space-time harmonic functions.

For each h ∈ Hst

b

h(n, Xn), σ

• {Xm : m ≤ n}
• n∈N

is a bounded martingale, hence h(n, Xn) → Y (h) a.s., with Y (h) T -mesurable.

Tail boundary

Let T := ∞

n=1 σ

• {Xm : m ≥ n}
• be the tail σ-field of (Xn)n∈N.

Call h : N × F → R space-time harmonic if h(n, x) =

• y∈F

p(x, y) h(n + 1, y) for all x ∈ F, n ∈ N. Let Hst

b be the set of all bounded space-time harmonic functions.

For each h ∈ Hst

b

h(n, Xn), σ

• {Xm : m ≤ n}
• n∈N

is a bounded martingale, hence h(n, Xn) → Y (h) a.s., with Y (h) T -mesurable. This leads to L∞(Ω, T , P ↾ T ) ∼ = Hst

b via

(reprT) h(n, x) = E

• Y (h)
• Xn = x
• .

Boundaries: General relations

Boundaries: General relations

• From the topological Martin boundary ∂F we obtain the

measure-theoretic Poisson boundary via E = ∂F, B the Borel subsets of E, νe = L(X∞), dνx dνe (α) = K(x, α).

Boundaries: General relations

• From the topological Martin boundary ∂F we obtain the

measure-theoretic Poisson boundary via E = ∂F, B the Borel subsets of E, νe = L(X∞), dνx dνe (α) = K(x, α).

• The Martin boundary does not change when passing to an

h-transform, the Poisson boundary may do so.

Boundaries: General relations

• From the topological Martin boundary ∂F we obtain the

measure-theoretic Poisson boundary via E = ∂F, B the Borel subsets of E, νe = L(X∞), dνx dνe (α) = K(x, α).

• The Martin boundary does not change when passing to an

h-transform, the Poisson boundary may do so.

• The tail boundary is the Poisson boundary of the space-time

chain (X st

n )n∈N, with X st n = (n, Xn) for all n ∈ N.

Boundaries: General relations

• From the topological Martin boundary ∂F we obtain the

measure-theoretic Poisson boundary via E = ∂F, B the Borel subsets of E, νe = L(X∞), dνx dνe (α) = K(x, α).

• The Martin boundary does not change when passing to an

h-transform, the Poisson boundary may do so.

• The tail boundary is the Poisson boundary of the space-time

chain (X st

n )n∈N, with X st n = (n, Xn) for all n ∈ N.

• CMCs have the space-time property, i.e. n = φ(Xn), hence

tail boundary and Poisson boundary coincide.

Boundaries for the P´

• lya urn (Blackwell and Kendall, 1964)

Boundaries for the P´

• lya urn (Blackwell and Kendall, 1964)

Path counting leads to

K

• (i, j), (k, l)
• =

k+l−i−j

k−i

• k+l

k

• (i + j + 1)!

i! j! , i ≤ k, j ≤ l.

Boundaries for the P´

• lya urn (Blackwell and Kendall, 1964)

Path counting leads to

K

• (i, j), (k, l)
• =

k+l−i−j

k−i

• k+l

k

• (i + j + 1)!

i! j! , i ≤ k, j ≤ l.

Inspection shows that

• K((i, j), (kn, ln))
• n∈N with kn + ln → ∞

converges for all (i, j) ∈ F iff

kn kn+ln → α ∈ [0, 1],

Boundaries for the P´

• lya urn (Blackwell and Kendall, 1964)

Path counting leads to

K

• (i, j), (k, l)
• =

k+l−i−j

k−i

• k+l

k

• (i + j + 1)!

i! j! , i ≤ k, j ≤ l.

Inspection shows that

• K((i, j), (kn, ln))
• n∈N with kn + ln → ∞

converges for all (i, j) ∈ F iff

kn kn+ln → α ∈ [0, 1], and then

• K
• (i, j), α
• = (i + j + 1)!

i! j! αi (1 − α)j, 0 ≤ α ≤ 1,

Boundaries for the P´

• lya urn (Blackwell and Kendall, 1964)

Path counting leads to

K

• (i, j), (k, l)
• =

k+l−i−j

k−i

• k+l

k

• (i + j + 1)!

i! j! , i ≤ k, j ≤ l.

Inspection shows that

• K((i, j), (kn, ln))
• n∈N with kn + ln → ∞

converges for all (i, j) ∈ F iff

kn kn+ln → α ∈ [0, 1], and then

• K
• (i, j), α
• = (i + j + 1)!

i! j! αi (1 − α)j, 0 ≤ α ≤ 1,

• ∂F ∼

= [0, 1],

Boundaries for the P´

• lya urn (Blackwell and Kendall, 1964)

Path counting leads to

K

• (i, j), (k, l)
• =

k+l−i−j

k−i

• k+l

k

• (i + j + 1)!

i! j! , i ≤ k, j ≤ l.

Inspection shows that

• K((i, j), (kn, ln))
• n∈N with kn + ln → ∞

converges for all (i, j) ∈ F iff

kn kn+ln → α ∈ [0, 1], and then

• K
• (i, j), α
• = (i + j + 1)!

i! j! αi (1 − α)j, 0 ≤ α ≤ 1,

• ∂F ∼

= [0, 1],

• L(X∞) = L(U, 1 − U) with L(U) = unif(0, 1),

Boundaries for the P´

• lya urn (Blackwell and Kendall, 1964)

Path counting leads to

K

• (i, j), (k, l)
• =

k+l−i−j

k−i

• k+l

k

• (i + j + 1)!

i! j! , i ≤ k, j ≤ l.

Inspection shows that

• K((i, j), (kn, ln))
• n∈N with kn + ln → ∞

converges for all (i, j) ∈ F iff

kn kn+ln → α ∈ [0, 1], and then

• K
• (i, j), α
• = (i + j + 1)!

i! j! αi (1 − α)j, 0 ≤ α ≤ 1,

• ∂F ∼

= [0, 1],

• L(X∞) = L(U, 1 − U) with L(U) = unif(0, 1),
• the h-transform with h = K(·, α) is the north-east random

walk with parameter θ = α.

Boundaries for the P´

• lya urn (Blackwell and Kendall, 1964)

Path counting leads to

K

• (i, j), (k, l)
• =

k+l−i−j

k−i

• k+l

k

• (i + j + 1)!

i! j! , i ≤ k, j ≤ l.

Inspection shows that

• K((i, j), (kn, ln))
• n∈N with kn + ln → ∞

converges for all (i, j) ∈ F iff

kn kn+ln → α ∈ [0, 1], and then

• K
• (i, j), α
• = (i + j + 1)!

i! j! αi (1 − α)j, 0 ≤ α ≤ 1,

• ∂F ∼

= [0, 1],

• L(X∞) = L(U, 1 − U) with L(U) = unif(0, 1),
• the h-transform with h = K(·, α) is the north-east random

walk with parameter θ = α. Note: The Poisson boundary for the NE random walk is trivial.

Boundaries for the BST chain (Evans, Gr., Wakolbinger, 2012)

Boundaries for the BST chain (Evans, Gr., Wakolbinger, 2012)

For a tree x ⊂ V = {0, 1}⋆ let x(u) := {v ∈ V : u + v ∈ x} be the subtree rooted at u.

Boundaries for the BST chain (Evans, Gr., Wakolbinger, 2012)

For a tree x ⊂ V = {0, 1}⋆ let x(u) := {v ∈ V : u + v ∈ x} be the subtree rooted at u. Then

• ∂F ‘is’ the space of probability measures µ on {0, 1}∞,

Boundaries for the BST chain (Evans, Gr., Wakolbinger, 2012)

For a tree x ⊂ V = {0, 1}⋆ let x(u) := {v ∈ V : u + v ∈ x} be the subtree rooted at u. Then

• ∂F ‘is’ the space of probability measures µ on {0, 1}∞,
• xn → µ ∈ ∂F iff #xn → ∞ and, for all u ∈ V,

#x(u) #x → µ(Au), Au := {v ∈ {0, 1}∞ : u ≤ v},

Boundaries for the BST chain (Evans, Gr., Wakolbinger, 2012)

For a tree x ⊂ V = {0, 1}⋆ let x(u) := {v ∈ V : u + v ∈ x} be the subtree rooted at u. Then

• ∂F ‘is’ the space of probability measures µ on {0, 1}∞,
• xn → µ ∈ ∂F iff #xn → ∞ and, for all u ∈ V,

#x(u) #x → µ(Au), Au := {v ∈ {0, 1}∞ : u ≤ v},

• L(X∞) ‘can be given’,

Boundaries for the BST chain (Evans, Gr., Wakolbinger, 2012)

For a tree x ⊂ V = {0, 1}⋆ let x(u) := {v ∈ V : u + v ∈ x} be the subtree rooted at u. Then

• ∂F ‘is’ the space of probability measures µ on {0, 1}∞,
• xn → µ ∈ ∂F iff #xn → ∞ and, for all u ∈ V,

#x(u) #x → µ(Au), Au := {v ∈ {0, 1}∞ : u ≤ v},

• L(X∞) ‘can be given’,
• the h-transform with h = K(·, µ) is the DST chain with

parameter µ (generalizes the classical digital search tree, which has µ(Au) = 2−|u|).

From boundaries to AofA

From boundaries to AofA

Rough idea: Instead of functionals Φn(Xn) consider Xn directly.

From boundaries to AofA

Rough idea: Instead of functionals Φn(Xn) consider Xn directly. Boundary theory may give Xn → X∞ a.s., where X∞ generates the tail σ-field T .

From boundaries to AofA

Rough idea: Instead of functionals Φn(Xn) consider Xn directly. Boundary theory may give Xn → X∞ a.s., where X∞ generates the tail σ-field T . Suppose that Φn(Xn) → Z a.s., with E|Z| < ∞. Then Zn := E[Z|Fn] = Ψn(Xn) → Z a.s. by L´ evy’s martingale convergence theorem.

From boundaries to AofA

Rough idea: Instead of functionals Φn(Xn) consider Xn directly. Boundary theory may give Xn → X∞ a.s., where X∞ generates the tail σ-field T . Suppose that Φn(Xn) → Z a.s., with E|Z| < ∞. Then Zn := E[Z|Fn] = Ψn(Xn) → Z a.s. by L´ evy’s martingale convergence theorem. As Z is T -measurable, we have Z = Φ(X∞) for some Φ.

From boundaries to AofA

Rough idea: Instead of functionals Φn(Xn) consider Xn directly. Boundary theory may give Xn → X∞ a.s., where X∞ generates the tail σ-field T . Suppose that Φn(Xn) → Z a.s., with E|Z| < ∞. Then Zn := E[Z|Fn] = Ψn(Xn) → Z a.s. by L´ evy’s martingale convergence theorem. As Z is T -measurable, we have Z = Φ(X∞) for some Φ. Boundary theory approach:

• Try to guess Φ,
• work out Ψn,
• prove Φn − Ψn → 0.

Boundary theory approach: Examples

Boundary theory approach: Examples

Throughout, (Xn)n∈N is the BST chain.

Boundary theory approach: Examples

Throughout, (Xn)n∈N is the BST chain. (1) Internal path length: BTA recovers R´ egnier’s result. Essentially, it ‘replaces inspiration by perspiration’. Provides representation of the limit.

Boundary theory approach: Examples

Throughout, (Xn)n∈N is the BST chain. (1) Internal path length: BTA recovers R´ egnier’s result. Essentially, it ‘replaces inspiration by perspiration’. Provides representation of the limit. (2) Wiener index: BTA works well.

Boundary theory approach: Examples

Throughout, (Xn)n∈N is the BST chain. (1) Internal path length: BTA recovers R´ egnier’s result. Essentially, it ‘replaces inspiration by perspiration’. Provides representation of the limit. (2) Wiener index: BTA works well. (3) Height and fill level: As Z is a constant, BTA is useless.

Boundary theory approach: Examples

Throughout, (Xn)n∈N is the BST chain. (1) Internal path length: BTA recovers R´ egnier’s result. Essentially, it ‘replaces inspiration by perspiration’. Provides representation of the limit. (2) Wiener index: BTA works well. (3) Height and fill level: As Z is a constant, BTA is useless. (4) Profiles: BTA ‘provides insight’.

Boundary theory approach: Examples

Throughout, (Xn)n∈N is the BST chain. (1) Internal path length: BTA recovers R´ egnier’s result. Essentially, it ‘replaces inspiration by perspiration’. Provides representation of the limit. (2) Wiener index: BTA works well. (3) Height and fill level: As Z is a constant, BTA is useless. (4) Profiles: BTA ‘provides insight’. (5) BTA may lead to interesting functionals (silhouette → metric silhouette).

Internal path length (re. . .revisited)

Internal path length (re. . .revisited)

IPL(Xn) :=

u∈Xn |u|,

an := E IPL(Xn) = 2(n + 1)Hn − 4n.

Internal path length (re. . .revisited)

IPL(Xn) :=

u∈Xn |u|,

an := E IPL(Xn) = 2(n + 1)Hn − 4n.

• Rewrite IPL in terms of subtree sizes,

IPL(Xn) =

• u∈Xn

|Xn(u)| − n.

Internal path length (re. . .revisited)

IPL(Xn) :=

u∈Xn |u|,

an := E IPL(Xn) = 2(n + 1)Hn − 4n.

• Rewrite IPL in terms of subtree sizes,

IPL(Xn) =

• u∈Xn

|Xn(u)| − n.

• Guess the limit functional,

ΦP(X∞) =

• u∈V

X∞(Au)C(ξu), with ξu = X∞(Au0)

X∞(Au) , C(s) = 1 + 2s log(s) + 2(1 − s) log(1 − s).

Internal path length (re. . .revisited)

IPL(Xn) :=

u∈Xn |u|,

an := E IPL(Xn) = 2(n + 1)Hn − 4n.

• Rewrite IPL in terms of subtree sizes,

IPL(Xn) =

• u∈Xn

|Xn(u)| − n.

• Guess the limit functional,

ΦP(X∞) =

• u∈V

X∞(Au)C(ξu), with ξu = X∞(Au0)

X∞(Au) , C(s) = 1 + 2s log(s) + 2(1 − s) log(1 − s).

• Calculate E[X∞(Au)|Fn] and E[ξu|Fn] to obtain the martingale

projection of ΦP(X∞).

Internal path length (re. . .revisited)

IPL(Xn) :=

u∈Xn |u|,

an := E IPL(Xn) = 2(n + 1)Hn − 4n.

• Rewrite IPL in terms of subtree sizes,

IPL(Xn) =

• u∈Xn

|Xn(u)| − n.

• Guess the limit functional,

ΦP(X∞) =

• u∈V

X∞(Au)C(ξu), with ξu = X∞(Au0)

X∞(Au) , C(s) = 1 + 2s log(s) + 2(1 − s) log(1 − s).

• Calculate E[X∞(Au)|Fn] and E[ξu|Fn] to obtain the martingale

projection of ΦP(X∞).

Theorem

As n → ∞, IPL(Xn) − an n + 1 → ΦP(X∞) a.s. and in L2.

Internal path length (re. . .revisited)

IPL(Xn) :=

u∈Xn |u|,

an := E IPL(Xn) = 2(n + 1)Hn − 4n.

• Rewrite IPL in terms of subtree sizes,

IPL(Xn) =

• u∈Xn

|Xn(u)| − n.

• Guess the limit functional,

ΦP(X∞) =

• u∈V

X∞(Au)C(ξu), with ξu = X∞(Au0)

X∞(Au) , C(s) = 1 + 2s log(s) + 2(1 − s) log(1 − s).

• Calculate E[X∞(Au)|Fn] and E[ξu|Fn] to obtain the martingale

projection of ΦP(X∞).

Theorem

As n → ∞, IPL(Xn) − an n + 1 → ΦP(X∞) a.s. and in L2.

R´ egnier 1989, R¨

• sler 1991,· · · , Bindjeme and Fill 2012, Gr. 2012

The Wiener index: Convergence in distribution

The Wiener index: Convergence in distribution

The Wiener index of a graph G = (V , E) is the sum of all node distances, WI(G) := 1 2

• (u,v)∈V ×V

d(u, v).

The Wiener index: Convergence in distribution

The Wiener index of a graph G = (V , E) is the sum of all node distances, WI(G) := 1 2

• (u,v)∈V ×V

d(u, v). Neininger (2002): For the BST sequence (Xn)n∈N, Wn := 1 n2 WI(Xn) − 2 log n converges in distribution as n → ∞.

The Wiener index: Convergence in distribution

The Wiener index of a graph G = (V , E) is the sum of all node distances, WI(G) := 1 2

• (u,v)∈V ×V

d(u, v). Neininger (2002): For the BST sequence (Xn)n∈N, Wn := 1 n2 WI(Xn) − 2 log n converges in distribution as n → ∞. Proof uses the contraction method (R¨

• sler, R¨

uschendorf, Neininger).

The Wiener index: BTA

The Wiener index: BTA

• Rewrite Wiener index in terms of subtree sizes,

WI(Xn) = n

• u∈Xn

|Xn(u)| −

• u∈Xn

|Xn(u)|2.

The Wiener index: BTA

• Rewrite Wiener index in terms of subtree sizes,

WI(Xn) = n

• u∈Xn

|Xn(u)| −

• u∈Xn

|Xn(u)|2.

• Guess the limit functional ΦW ,

ΦW (X∞) = 2γ − 3 + ΦP(X∞) − Z∞, Z∞ :=

• u∈V

X∞(Au)2.

The Wiener index: BTA

• Rewrite Wiener index in terms of subtree sizes,

WI(Xn) = n

• u∈Xn

|Xn(u)| −

• u∈Xn

|Xn(u)|2.

• Guess the limit functional ΦW ,

ΦW (X∞) = 2γ − 3 + ΦP(X∞) − Z∞, Z∞ :=

• u∈V

X∞(Au)2.

• Show that EZ∞ < ∞ and calculate E[X∞(Au)2|Fn] to obtain

the martingale projection of ΦW (X∞).

The Wiener index: BTA

• Rewrite Wiener index in terms of subtree sizes,

WI(Xn) = n

• u∈Xn

|Xn(u)| −

• u∈Xn

|Xn(u)|2.

• Guess the limit functional ΦW ,

ΦW (X∞) = 2γ − 3 + ΦP(X∞) − Z∞, Z∞ :=

• u∈V

X∞(Au)2.

• Show that EZ∞ < ∞ and calculate E[X∞(Au)2|Fn] to obtain

the martingale projection of ΦW (X∞).

Theorem

As n → ∞, Wn → ΦW (X∞) a.s. and in L2.

Profiles (from R´

egnier to Jabbour-Hattab)

Profiles (from R´

egnier to Jabbour-Hattab) Count the external nodes of x at height k: Ψk(x) := #

• u ∈ V : |u| = k, u /

∈ x, ¯ u ∈ x

• .

Profiles (from R´

egnier to Jabbour-Hattab) Count the external nodes of x at height k: Ψk(x) := #

• u ∈ V : |u| = k, u /

∈ x, ¯ u ∈ x

• .

Chauvin, Drmota, Jabbour-Hattab (2001) and Chauvin, Klein, Marckert, Roualt (2005) used the martingales Mn(z) :=

n−1

• j=1

j + 1 j + 2z

∞

• k=1

Ψk(Xn) zk, z ∈ C, to obtain a.s. asymptotics for k → Ψk(Xn) as n → ∞.

Profiles (from R´

egnier to Jabbour-Hattab) Count the external nodes of x at height k: Ψk(x) := #

• u ∈ V : |u| = k, u /

∈ x, ¯ u ∈ x

• .

Chauvin, Drmota, Jabbour-Hattab (2001) and Chauvin, Klein, Marckert, Roualt (2005) used the martingales Mn(z) :=

n−1

• j=1

j + 1 j + 2z

∞

• k=1

Ψk(Xn) zk, z ∈ C, to obtain a.s. asymptotics for k → Ψk(Xn) as n → ∞.

• Mn(z) > 0 if z ∈ [0, ∞),

Profiles (from R´

egnier to Jabbour-Hattab) Count the external nodes of x at height k: Ψk(x) := #

• u ∈ V : |u| = k, u /

∈ x, ¯ u ∈ x

• .

Chauvin, Drmota, Jabbour-Hattab (2001) and Chauvin, Klein, Marckert, Roualt (2005) used the martingales Mn(z) :=

n−1

• j=1

j + 1 j + 2z

∞

• k=1

Ψk(Xn) zk, z ∈ C, to obtain a.s. asymptotics for k → Ψk(Xn) as n → ∞.

• Mn(z) > 0 if z ∈ [0, ∞),
• Mn(z) = h(Xn) for some harmonic function h : F → (0, ∞),

Profiles (from R´

egnier to Jabbour-Hattab) Count the external nodes of x at height k: Ψk(x) := #

• u ∈ V : |u| = k, u /

∈ x, ¯ u ∈ x

• .

Chauvin, Drmota, Jabbour-Hattab (2001) and Chauvin, Klein, Marckert, Roualt (2005) used the martingales Mn(z) :=

n−1

• j=1

j + 1 j + 2z

∞

• k=1

Ψk(Xn) zk, z ∈ C, to obtain a.s. asymptotics for k → Ψk(Xn) as n → ∞.

• Mn(z) > 0 if z ∈ [0, ∞),
• Mn(z) = h(Xn) for some harmonic function h : F → (0, ∞),
• the corresponding h-transform has a simple interpretation via a

marked spine construction used by Lyons, Pemantle and Peres and others in the Bienaym´ e-Galton-Watson context,

Profiles (from R´

egnier to Jabbour-Hattab) Count the external nodes of x at height k: Ψk(x) := #

• u ∈ V : |u| = k, u /

∈ x, ¯ u ∈ x

• .

Chauvin, Drmota, Jabbour-Hattab (2001) and Chauvin, Klein, Marckert, Roualt (2005) used the martingales Mn(z) :=

n−1

• j=1

j + 1 j + 2z

∞

• k=1

Ψk(Xn) zk, z ∈ C, to obtain a.s. asymptotics for k → Ψk(Xn) as n → ∞.

• Mn(z) > 0 if z ∈ [0, ∞),
• Mn(z) = h(Xn) for some harmonic function h : F → (0, ∞),
• the corresponding h-transform has a simple interpretation via a

marked spine construction used by Lyons, Pemantle and Peres and others in the Bienaym´ e-Galton-Watson context,

• (2z)−1M∞(z) = dPh/dP for z ∈ (c−, c+).

Profiles (from R´

egnier to Jabbour-Hattab) Count the external nodes of x at height k: Ψk(x) := #

• u ∈ V : |u| = k, u /

∈ x, ¯ u ∈ x

• .

Chauvin, Drmota, Jabbour-Hattab (2001) and Chauvin, Klein, Marckert, Roualt (2005) used the martingales Mn(z) :=

n−1

• j=1

j + 1 j + 2z

∞

• k=1

Ψk(Xn) zk, z ∈ C, to obtain a.s. asymptotics for k → Ψk(Xn) as n → ∞.

• Mn(z) > 0 if z ∈ [0, ∞),
• Mn(z) = h(Xn) for some harmonic function h : F → (0, ∞),
• the corresponding h-transform has a simple interpretation via a

marked spine construction used by Lyons, Pemantle and Peres and others in the Bienaym´ e-Galton-Watson context,

• (2z)−1M∞(z) = dPh/dP for z ∈ (c−, c+).

BTA ‘explains’ the martingale and gives an interpretation of the limit, but does not (easily) lead to a new proof of the known results.

From AofA to boundaries

From AofA to boundaries

Folklore: In specific cases the state space compactification can generally be obtained directly by using the algebraic or geometric structure.

From AofA to boundaries

Folklore: In specific cases the state space compactification can generally be obtained directly by using the algebraic or geometric structure. Here: The BST chain X is generated by an algorithm that gives X as a deterministic function of the input sequence η = (ηi)i∈N.

From AofA to boundaries

Folklore: In specific cases the state space compactification can generally be obtained directly by using the algebraic or geometric structure. Here: The BST chain X is generated by an algorithm that gives X as a deterministic function of the input sequence η = (ηi)i∈N. In particular, X∞ must be a function of η: For each u ∈ V, let τu := inf{n ∈ N : Xn ∋ u}, be the entrance time of the node and let η(i:0) := 0 < η(i:1) < · · · < η(i:i) < η(i:i+1) := 1 be the order statistics associated with η1, . . . , ηi, i = τu − 1.

From AofA to boundaries

Folklore: In specific cases the state space compactification can generally be obtained directly by using the algebraic or geometric structure. Here: The BST chain X is generated by an algorithm that gives X as a deterministic function of the input sequence η = (ηi)i∈N. In particular, X∞ must be a function of η: For each u ∈ V, let τu := inf{n ∈ N : Xn ∋ u}, be the entrance time of the node and let η(i:0) := 0 < η(i:1) < · · · < η(i:i) < η(i:i+1) := 1 be the order statistics associated with η1, . . . , ηi, i = τu − 1. Then X∞(Au) = η(i:j+1) − η(i:j) with η(i:j) < ηi+1 < η(i:j+1).

From AofA to boundaries

Folklore: In specific cases the state space compactification can generally be obtained directly by using the algebraic or geometric structure. Here: The BST chain X is generated by an algorithm that gives X as a deterministic function of the input sequence η = (ηi)i∈N. In particular, X∞ must be a function of η: For each u ∈ V, let τu := inf{n ∈ N : Xn ∋ u}, be the entrance time of the node and let η(i:0) := 0 < η(i:1) < · · · < η(i:i) < η(i:i+1) := 1 be the order statistics associated with η1, . . . , ηi, i = τu − 1. Then X∞(Au) = η(i:j+1) − η(i:j) with η(i:j) < ηi+1 < η(i:j+1). Conclusion: Add ‘algorithmic’ to the above structural elements.

From AofA to boundaries

Folklore: In specific cases the state space compactification can generally be obtained directly by using the algebraic or geometric structure. Here: The BST chain X is generated by an algorithm that gives X as a deterministic function of the input sequence η = (ηi)i∈N. In particular, X∞ must be a function of η: For each u ∈ V, let τu := inf{n ∈ N : Xn ∋ u}, be the entrance time of the node and let η(i:0) := 0 < η(i:1) < · · · < η(i:i) < η(i:i+1) := 1 be the order statistics associated with η1, . . . , ηi, i = τu − 1. Then X∞(Au) = η(i:j+1) − η(i:j) with η(i:j) < ηi+1 < η(i:j+1). Conclusion: Add ‘algorithmic’ to the above structural elements. Also: A representation may lead to convergence in stronger topologies.

Metric binary trees: Generalities

Metric binary trees: Generalities

Turn x ⊂ V into a metric space (x, d) :

• assign a positive value w(u) to each non-root node u of x,

Metric binary trees: Generalities

Turn x ⊂ V into a metric space (x, d) :

• assign a positive value w(u) to each non-root node u of x,
• for u = (u1, . . . , uk) ∈ x, u = ∅, regard w(u) as the distance

d(u, ¯ u) of u and its direct ancestor ¯ u := (u1, . . . , uk−1),

Metric binary trees: Generalities

Turn x ⊂ V into a metric space (x, d) :

• assign a positive value w(u) to each non-root node u of x,
• for u = (u1, . . . , uk) ∈ x, u = ∅, regard w(u) as the distance

d(u, ¯ u) of u and its direct ancestor ¯ u := (u1, . . . , uk−1),

• for u, v ∈ x let d(u, v) be the sum of the distances along the

unique path from u to v.

Metric binary trees: Generalities

Turn x ⊂ V into a metric space (x, d) :

• assign a positive value w(u) to each non-root node u of x,
• for u = (u1, . . . , uk) ∈ x, u = ∅, regard w(u) as the distance

d(u, ¯ u) of u and its direct ancestor ¯ u := (u1, . . . , uk−1),

• for u, v ∈ x let d(u, v) be the sum of the distances along the

unique path from u to v. Special cases:

• w ≡ 1 leads to the canonical graph distance,

Metric binary trees: Generalities

Turn x ⊂ V into a metric space (x, d) :

• assign a positive value w(u) to each non-root node u of x,
• for u = (u1, . . . , uk) ∈ x, u = ∅, regard w(u) as the distance

d(u, ¯ u) of u and its direct ancestor ¯ u := (u1, . . . , uk−1),

• for u, v ∈ x let d(u, v) be the sum of the distances along the

unique path from u to v. Special cases:

• w ≡ 1 leads to the canonical graph distance,
• with w(u) = wx(u) := #x(u)/#x we obtain the

(normalized) subtree size metric.

Metric binary trees: Generalities

Turn x ⊂ V into a metric space (x, d) :

• assign a positive value w(u) to each non-root node u of x,
• for u = (u1, . . . , uk) ∈ x, u = ∅, regard w(u) as the distance

d(u, ¯ u) of u and its direct ancestor ¯ u := (u1, . . . , uk−1),

• for u, v ∈ x let d(u, v) be the sum of the distances along the

unique path from u to v. Special cases:

• w ≡ 1 leads to the canonical graph distance,
• with w(u) = wx(u) := #x(u)/#x we obtain the

(normalized) subtree size metric.

• For ρ > 0, define the weighted subtree size metric via

wρ(u) = wρ,x(u) := ρ|u| wx(u).

Metric binary trees: Generalities

Turn x ⊂ V into a metric space (x, d) :

• assign a positive value w(u) to each non-root node u of x,
• for u = (u1, . . . , uk) ∈ x, u = ∅, regard w(u) as the distance

d(u, ¯ u) of u and its direct ancestor ¯ u := (u1, . . . , uk−1),

• for u, v ∈ x let d(u, v) be the sum of the distances along the

unique path from u to v. Special cases:

• w ≡ 1 leads to the canonical graph distance,
• with w(u) = wx(u) := #x(u)/#x we obtain the

(normalized) subtree size metric.

• For ρ > 0, define the weighted subtree size metric via

wρ(u) = wρ,x(u) := ρ|u| wx(u). Note: The first one is local, the others are global.

Metric binary trees: Pictures

Metric binary trees: Pictures

Map u = (u1, . . . , un) ∈ V to t = 2−1 + n

i=1(2ui − 1)2−i−1 ∈ (0, 1),

vertical distance is d(u, ∅).

Metric binary trees: Pictures

Map u = (u1, . . . , un) ∈ V to t = 2−1 + n

i=1(2ui − 1)2−i−1 ∈ (0, 1),

vertical distance is d(u, ∅).

canonical distance

Metric binary trees: Pictures

Map u = (u1, . . . , un) ∈ V to t = 2−1 + n

i=1(2ui − 1)2−i−1 ∈ (0, 1),

vertical distance is d(u, ∅).

canonical distance subtree size metric

Metric binary trees: Convergence

Metric binary trees: Convergence

Any measure µ on {0, 1}∞ defines a metric on V via d∞(¯ u, u) = µ(Au) for all u ∈ V, u = ∅, (recall that Au = {v ∈ {0, 1}∞ : u ≤ v}).

Metric binary trees: Convergence

Any measure µ on {0, 1}∞ defines a metric on V via d∞(¯ u, u) = µ(Au) for all u ∈ V, u = ∅, (recall that Au = {v ∈ {0, 1}∞ : u ≤ v}). Convergence of trees ↔ convergence of metric spaces

Metric binary trees: Convergence

Any measure µ on {0, 1}∞ defines a metric on V via d∞(¯ u, u) = µ(Au) for all u ∈ V, u = ∅, (recall that Au = {v ∈ {0, 1}∞ : u ≤ v}). Convergence of trees ↔ convergence of metric spaces Rˆ

• le model:

Continuum random tree, Brownian excursion, Gromov convergence.

Metric binary trees: Convergence

Any measure µ on {0, 1}∞ defines a metric on V via d∞(¯ u, u) = µ(Au) for all u ∈ V, u = ∅, (recall that Au = {v ∈ {0, 1}∞ : u ≤ v}). Convergence of trees ↔ convergence of metric spaces Rˆ

• le model:

Continuum random tree, Brownian excursion, Gromov convergence. In view of Xn ↑ V we can work with (xn, dn) → (V, d∞) meaning that lim

n→∞ dn(u, v) = d∞(u, v) for all u, v ∈ V.

Metric binary trees: A phase transition

Metric binary trees: A phase transition Theorem

Let ρ0 = 1.26107 · · · be the smaller one of the two roots of the equation 2e log(ρ) = ρ, ρ > 0.

Metric binary trees: A phase transition Theorem

Let ρ0 = 1.26107 · · · be the smaller one of the two roots of the equation 2e log(ρ) = ρ, ρ > 0. (a) For ρ < ρ0, the metric space (V, dX∞,ρ) is compact w.p. 1.

Metric binary trees: A phase transition Theorem

Let ρ0 = 1.26107 · · · be the smaller one of the two roots of the equation 2e log(ρ) = ρ, ρ > 0. (a) For ρ < ρ0, the metric space (V, dX∞,ρ) is compact w.p. 1. (b) For ρ > ρ0, (V, dX∞,ρ) has infinite diameter w.p. 1.

Metric binary trees: A phase transition Theorem

Let ρ0 = 1.26107 · · · be the smaller one of the two roots of the equation 2e log(ρ) = ρ, ρ > 0. (a) For ρ < ρ0, the metric space (V, dX∞,ρ) is compact w.p. 1. (b) For ρ > ρ0, (V, dX∞,ρ) has infinite diameter w.p. 1. (c) For ρ < ρ0, the metric spaces (Xn, dXn,ρ) converge uniformly to (V, dX∞,ρ) as n → ∞ in the sense of sup

u,v∈Xn

• dXn,ρ(u, v) − dX∞,ρ(u, v)
• → 0 a. s. and in mean.

Metric binary trees: A phase transition Theorem

Let ρ0 = 1.26107 · · · be the smaller one of the two roots of the equation 2e log(ρ) = ρ, ρ > 0. (a) For ρ < ρ0, the metric space (V, dX∞,ρ) is compact w.p. 1. (b) For ρ > ρ0, (V, dX∞,ρ) has infinite diameter w.p. 1. (c) For ρ < ρ0, the metric spaces (Xn, dXn,ρ) converge uniformly to (V, dX∞,ρ) as n → ∞ in the sense of sup

u,v∈Xn

• dXn,ρ(u, v) − dX∞,ρ(u, v)
• → 0 a. s. and in mean.

(d) For ρ > ρ0, and with dXn,ρ(¯ u, u) := 0 for u / ∈ Xn, sup

u,v∈V

• dXn,ρ(u, v) − dX∞,ρ(u, v)
• = ∞ w.p. 1.

Outlook

Outlook

• F: Ulam-Harris trees, CMC: random recursive trees:

Roughly: Nesting leads from the P´

• lya urn to the BST chain,

and from the Chinese restaurant to RRT, see EGW 2012.

Outlook

• F: Ulam-Harris trees, CMC: random recursive trees:

Roughly: Nesting leads from the P´

• lya urn to the BST chain,

and from the Chinese restaurant to RRT, see EGW 2012.

• Graph asymptotics?

CMC style growth, see e.g. Janson and Severini 2012

Outlook

• F: Ulam-Harris trees, CMC: random recursive trees:

Roughly: Nesting leads from the P´

• lya urn to the BST chain,

and from the Chinese restaurant to RRT, see EGW 2012.

• Graph asymptotics?

CMC style growth, see e.g. Janson and Severini 2012

• Second order results via (struct)?

IPL: Convergence in distribution in Neininger 2013+.

Outlook

• F: Ulam-Harris trees, CMC: random recursive trees:

Roughly: Nesting leads from the P´

• lya urn to the BST chain,

and from the Chinese restaurant to RRT, see EGW 2012.

• Graph asymptotics?

CMC style growth, see e.g. Janson and Severini 2012

• Second order results via (struct)?

IPL: Convergence in distribution in Neininger 2013+.

• Specific questions.

Is the continuum random tree the boundary of the R´ emy chain?

Outlook

• F: Ulam-Harris trees, CMC: random recursive trees:

Roughly: Nesting leads from the P´

• lya urn to the BST chain,

and from the Chinese restaurant to RRT, see EGW 2012.

• Graph asymptotics?

CMC style growth, see e.g. Janson and Severini 2012

• Second order results via (struct)?

IPL: Convergence in distribution in Neininger 2013+.

• Specific questions.

Is the continuum random tree the boundary of the R´ emy chain?

• Functorial properties of the boundary constructions.