Doob-Martin compactification: h -transforms Call h : F → R harmonic if it has the ‘mean value property’: � h ( x ) = p ( x , y ) h ( y ) for all x ∈ F . y ∈ F Let H + resp. H b be the set of non-negative resp. bounded harmonic functions. For h ∈ H + , h � = 0, 1 p h ( x , y ) := 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 ) = p ( x , y ) h ( y ) for all x ∈ F . y ∈ F Let H + resp. H b be the set of non-negative resp. bounded harmonic functions. For h ∈ H + , h � = 0, 1 p h ( x , y ) := h ( x ) p ( x , y ) h ( y ) , x , y ∈ F , defines another transition probability on F . 1 = e and transitions p h we obtain another CMC With X h 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 ) = K ( x , α ) ν h ( d α ) ∂ F
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 ) = K ( x , α ) ν h ( d α ) ∂ F (conv) As n → ∞ , X n → 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 ) = K ( x , α ) ν h ( d α ) ∂ F (conv) As n → ∞ , X n → 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 ( X n ) n ∈ N with Martin kernel K , � � � ≤ K ( x , X m ) . E K ( x , X m − 1 ) � X m , . . . , X n
Doob-Martin compactification: The ‘why’ In a CMC ( X n ) n ∈ N with Martin kernel K , � � � ≤ K ( x , X m ) . E K ( x , X m − 1 ) � X m , . . . , X n Hence, for each n ∈ N , ( Y n k , F n k ) k =1 ,..., n with Y n F n k := K ( x , X n +1 − k ) , k = σ ( X n +1 − k , . . . , X n ) is a supermartingale.
Doob-Martin compactification: The ‘why’ In a CMC ( X n ) n ∈ N with Martin kernel K , � � � ≤ K ( x , X m ) . E K ( x , X m − 1 ) � X m , . . . , X n Hence, for each n ∈ N , ( Y n k , F n k ) k =1 ,..., n with Y n F n k := K ( x , X n +1 − k ) , k = σ ( X n +1 − k , . . . , X n ) is a supermartingale. For the number U n ( a , b ) of upcrossings of some interval [ a , b ] this means 1 1 n − a ) − = � − . b − aE ( Y n � EU n [ a , b ] ≤ K ( x , e ) − a b − a It follows that ( K ( x , X n )) n ∈ N converges almost surely.
Poisson boundary
Poisson boundary A measurable space ( E , B ) together with a family ν x , x ∈ F , of probability measures on ( E , B ) such that ν x ≪ ν e for all x ∈ F ‘is’ the Poisson boundary of ( X , F ) if L ∞ ( E , B , ν e ) ∼ = H b as Banach spaces, with the supremum norm on H b ,
Poisson boundary A measurable space ( E , B ) together with a family ν x , x ∈ F , of probability measures on ( E , B ) such that ν x ≪ ν e for all x ∈ F ‘is’ the Poisson boundary of ( X , F ) if L ∞ ( E , B , ν e ) ∼ = H b as Banach spaces, with the supremum norm on H b , via: For each φ ∈ L ∞ � (reprP) h ( x ) = φ ( y ) ν x ( dy ) , x ∈ F , E is in H b , and for each h ∈ H b there is a unique φ ∈ L ∞ such that this representation holds.
Tail boundary
Tail boundary Let T := � ∞ � � n =1 σ { X m : m ≥ n } be the tail σ -field of ( X n ) n ∈ N .
Tail boundary Let T := � ∞ � � n =1 σ { X m : m ≥ n } be the tail σ -field of ( X n ) n ∈ N . Call h : N × F → R space-time harmonic if � h ( n , x ) = p ( x , y ) h ( n + 1 , y ) for all x ∈ F , n ∈ N . y ∈ F
Tail boundary Let T := � ∞ � � n =1 σ { X m : m ≥ n } be the tail σ -field of ( X n ) n ∈ N . Call h : N × F → R space-time harmonic if � h ( n , x ) = p ( x , y ) h ( n + 1 , y ) for all x ∈ F , n ∈ N . y ∈ F Let H st b be the set of all bounded space-time harmonic functions.
Tail boundary Let T := � ∞ � � n =1 σ { X m : m ≥ n } be the tail σ -field of ( X n ) n ∈ N . Call h : N × F → R space-time harmonic if � h ( n , x ) = p ( x , y ) h ( n + 1 , y ) for all x ∈ F , n ∈ N . y ∈ F Let H st b be the set of all bounded space-time harmonic functions. For each h ∈ H st b � �� � h ( n , X n ) , σ { X m : m ≤ n } n ∈ N is a bounded martingale, hence h ( n , X n ) → Y ( h ) a.s., with Y ( h ) T -mesurable.
Tail boundary Let T := � ∞ � � n =1 σ { X m : m ≥ n } be the tail σ -field of ( X n ) n ∈ N . Call h : N × F → R space-time harmonic if � h ( n , x ) = p ( x , y ) h ( n + 1 , y ) for all x ∈ F , n ∈ N . y ∈ F Let H st b be the set of all bounded space-time harmonic functions. For each h ∈ H st b � �� � h ( n , X n ) , σ { X m : m ≤ n } n ∈ N is a bounded martingale, hence h ( n , X n ) → Y ( h ) a.s., with Y ( h ) T -mesurable. This leads to L ∞ (Ω , T , P ↾ T ) ∼ = H st b via (reprT) � � � h ( n , x ) = E Y ( h ) � X n = 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 , d ν x ν e = L ( X ∞ ) , ( α ) = K ( x , α ). d ν e
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 , d ν x ν e = L ( X ∞ ) , ( α ) = K ( x , α ). d ν e • 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 , d ν x ν e = L ( X ∞ ) , ( α ) = K ( x , α ). d ν e • 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 , X n ) 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 , d ν x ν e = L ( X ∞ ) , ( α ) = K ( x , α ). d ν e • 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 , X n ) for all n ∈ N . • CMCs have the space-time property, i.e. n = φ ( X n ), hence tail boundary and Poisson boundary coincide.
Boundaries for the P´ olya urn (Blackwell and Kendall, 1964)
Boundaries for the P´ olya urn (Blackwell and Kendall, 1964) Path counting leads to � k + l − i − j � ( i + j + 1)! k − i � � ( i , j ) , ( k , l ) = , i ≤ k , j ≤ l . K � k + l i ! j ! � k
Boundaries for the P´ olya urn (Blackwell and Kendall, 1964) Path counting leads to � k + l − i − j � ( i + j + 1)! k − i � � ( i , j ) , ( k , l ) = , i ≤ k , j ≤ l . K � k + l i ! j ! � k � � Inspection shows that K (( i , j ) , ( k n , l n )) n ∈ N with k n + l n → ∞ k n converges for all ( i , j ) ∈ F iff k n + l n → α ∈ [0 , 1],
Boundaries for the P´ olya urn (Blackwell and Kendall, 1964) Path counting leads to � k + l − i − j � ( i + j + 1)! k − i � � ( i , j ) , ( k , l ) = , i ≤ k , j ≤ l . K � k + l i ! j ! � k � � Inspection shows that K (( i , j ) , ( k n , l n )) n ∈ N with k n + l n → ∞ k n converges for all ( i , j ) ∈ F iff k n + l n → α ∈ [0 , 1], and then = ( i + j + 1)! α i (1 − α ) j , � � • K ( i , j ) , α 0 ≤ α ≤ 1, i ! j !
Boundaries for the P´ olya urn (Blackwell and Kendall, 1964) Path counting leads to � k + l − i − j � ( i + j + 1)! k − i � � ( i , j ) , ( k , l ) = , i ≤ k , j ≤ l . K � k + l i ! j ! � k � � Inspection shows that K (( i , j ) , ( k n , l n )) n ∈ N with k n + l n → ∞ k n converges for all ( i , j ) ∈ F iff k n + l n → α ∈ [0 , 1], and then = ( i + j + 1)! α i (1 − α ) j , � � • K ( i , j ) , α 0 ≤ α ≤ 1, i ! j ! • ∂ F ∼ = [0 , 1],
Boundaries for the P´ olya urn (Blackwell and Kendall, 1964) Path counting leads to � k + l − i − j � ( i + j + 1)! k − i � � ( i , j ) , ( k , l ) = , i ≤ k , j ≤ l . K � k + l i ! j ! � k � � Inspection shows that K (( i , j ) , ( k n , l n )) n ∈ N with k n + l n → ∞ k n converges for all ( i , j ) ∈ F iff k n + l n → α ∈ [0 , 1], and then = ( i + j + 1)! α i (1 − α ) j , � � • K ( i , j ) , α 0 ≤ α ≤ 1, i ! j ! • ∂ F ∼ = [0 , 1], • L ( X ∞ ) = L ( U , 1 − U ) with L ( U ) = unif(0 , 1),
Boundaries for the P´ olya urn (Blackwell and Kendall, 1964) Path counting leads to � k + l − i − j � ( i + j + 1)! k − i � � ( i , j ) , ( k , l ) = , i ≤ k , j ≤ l . K � k + l i ! j ! � k � � Inspection shows that K (( i , j ) , ( k n , l n )) n ∈ N with k n + l n → ∞ k n converges for all ( i , j ) ∈ F iff k n + l n → α ∈ [0 , 1], and then = ( i + j + 1)! α i (1 − α ) j , � � • K ( i , j ) , α 0 ≤ α ≤ 1, i ! j ! • ∂ 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´ olya urn (Blackwell and Kendall, 1964) Path counting leads to � k + l − i − j � ( i + j + 1)! k − i � � ( i , j ) , ( k , l ) = , i ≤ k , j ≤ l . K � k + l i ! j ! � k � � Inspection shows that K (( i , j ) , ( k n , l n )) n ∈ N with k n + l n → ∞ k n converges for all ( i , j ) ∈ F iff k n + l n → α ∈ [0 , 1], and then = ( i + j + 1)! α i (1 − α ) j , � � • K ( i , j ) , α 0 ≤ α ≤ 1, i ! j ! • ∂ 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 } ∞ , • x n → µ ∈ ∂ F iff # x n → ∞ and, for all u ∈ V , # x ( u ) → µ ( A u ) , A u := { v ∈ { 0 , 1 } ∞ : u ≤ v } , # x
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 } ∞ , • x n → µ ∈ ∂ F iff # x n → ∞ and, for all u ∈ V , # x ( u ) → µ ( A u ) , A u := { v ∈ { 0 , 1 } ∞ : u ≤ v } , # x • 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 } ∞ , • x n → µ ∈ ∂ F iff # x n → ∞ and, for all u ∈ V , # x ( u ) → µ ( A u ) , A u := { v ∈ { 0 , 1 } ∞ : u ≤ v } , # x • 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 ( A u ) = 2 −| u | ).
From boundaries to AofA
From boundaries to AofA Rough idea: Instead of functionals Φ n ( X n ) consider X n directly.
From boundaries to AofA Rough idea: Instead of functionals Φ n ( X n ) consider X n directly. Boundary theory may give X n → X ∞ a.s., where X ∞ generates the tail σ -field T .
From boundaries to AofA Rough idea: Instead of functionals Φ n ( X n ) consider X n directly. Boundary theory may give X n → X ∞ a.s., where X ∞ generates the tail σ -field T . Suppose that Φ n ( X n ) → Z a.s., with E | Z | < ∞ . Then Z n := E [ Z |F n ] = Ψ n ( X n ) → Z a.s. by L´ evy’s martingale convergence theorem.
From boundaries to AofA Rough idea: Instead of functionals Φ n ( X n ) consider X n directly. Boundary theory may give X n → X ∞ a.s., where X ∞ generates the tail σ -field T . Suppose that Φ n ( X n ) → Z a.s., with E | Z | < ∞ . Then Z n := E [ Z |F n ] = Ψ n ( X n ) → 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 Φ n ( X n ) consider X n directly. Boundary theory may give X n → X ∞ a.s., where X ∞ generates the tail σ -field T . Suppose that Φ n ( X n ) → Z a.s., with E | Z | < ∞ . Then Z n := E [ Z |F n ] = Ψ n ( X n ) → 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 Ψ n , • prove Φ n − Ψ n → 0.
Boundary theory approach: Examples
Boundary theory approach: Examples Throughout, ( X n ) n ∈ N is the BST chain.
Boundary theory approach: Examples Throughout, ( X n ) 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, ( X n ) 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, ( X n ) 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, ( X n ) 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, ( X n ) 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( X n ) := � u ∈ X n | u | , a n := E IPL( X n ) = 2( n + 1) H n − 4 n .
Internal path length (re . . . revisited) IPL( X n ) := � u ∈ X n | u | , a n := E IPL( X n ) = 2( n + 1) H n − 4 n . • Rewrite IPL in terms of subtree sizes, � | X n ( u ) | − n . IPL( X n ) = u ∈ X n
Internal path length (re . . . revisited) IPL( X n ) := � u ∈ X n | u | , a n := E IPL( X n ) = 2( n + 1) H n − 4 n . • Rewrite IPL in terms of subtree sizes, � | X n ( u ) | − n . IPL( X n ) = u ∈ X n • Guess the limit functional, � Φ P ( X ∞ ) = X ∞ ( A u ) C ( ξ u ) , with u ∈ V ξ u = X ∞ ( A u 0 ) X ∞ ( A u ) , C ( s ) = 1 + 2 s log( s ) + 2(1 − s ) log(1 − s ) .
Internal path length (re . . . revisited) IPL( X n ) := � u ∈ X n | u | , a n := E IPL( X n ) = 2( n + 1) H n − 4 n . • Rewrite IPL in terms of subtree sizes, � | X n ( u ) | − n . IPL( X n ) = u ∈ X n • Guess the limit functional, � Φ P ( X ∞ ) = X ∞ ( A u ) C ( ξ u ) , with u ∈ V ξ u = X ∞ ( A u 0 ) X ∞ ( A u ) , C ( s ) = 1 + 2 s log( s ) + 2(1 − s ) log(1 − s ) . • Calculate E [ X ∞ ( A u ) |F n ] and E [ ξ u |F n ] to obtain the martingale projection of Φ P ( X ∞ ).
Internal path length (re . . . revisited) IPL( X n ) := � u ∈ X n | u | , a n := E IPL( X n ) = 2( n + 1) H n − 4 n . • Rewrite IPL in terms of subtree sizes, � | X n ( u ) | − n . IPL( X n ) = u ∈ X n • Guess the limit functional, � Φ P ( X ∞ ) = X ∞ ( A u ) C ( ξ u ) , with u ∈ V ξ u = X ∞ ( A u 0 ) X ∞ ( A u ) , C ( s ) = 1 + 2 s log( s ) + 2(1 − s ) log(1 − s ) . • Calculate E [ X ∞ ( A u ) |F n ] and E [ ξ u |F n ] to obtain the martingale projection of Φ P ( X ∞ ). As n → ∞ , IPL ( X n ) − a n → Φ P ( X ∞ ) a.s. and in L 2 . Theorem n + 1
Internal path length (re . . . revisited) IPL( X n ) := � u ∈ X n | u | , a n := E IPL( X n ) = 2( n + 1) H n − 4 n . • Rewrite IPL in terms of subtree sizes, � | X n ( u ) | − n . IPL( X n ) = u ∈ X n • Guess the limit functional, � Φ P ( X ∞ ) = X ∞ ( A u ) C ( ξ u ) , with u ∈ V ξ u = X ∞ ( A u 0 ) X ∞ ( A u ) , C ( s ) = 1 + 2 s log( s ) + 2(1 − s ) log(1 − s ) . • Calculate E [ X ∞ ( A u ) |F n ] and E [ ξ u |F n ] to obtain the martingale projection of Φ P ( X ∞ ). As n → ∞ , IPL ( X n ) − a n → Φ P ( X ∞ ) a.s. and in L 2 . Theorem n + 1 R´ egnier 1989, R¨ osler 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 � d ( u , v ) . 2 ( u , v ) ∈ V × 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 � d ( u , v ) . 2 ( u , v ) ∈ V × V Neininger (2002): For the BST sequence ( X n ) n ∈ N , W n := 1 n 2 WI( X n ) − 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 � d ( u , v ) . 2 ( u , v ) ∈ V × V Neininger (2002): For the BST sequence ( X n ) n ∈ N , W n := 1 n 2 WI( X n ) − 2 log n converges in distribution as n → ∞ . Proof uses the contraction method (R¨ osler, R¨ uschendorf, Neininger).
The Wiener index: BTA
The Wiener index: BTA • Rewrite Wiener index in terms of subtree sizes, � � | X n ( u ) | 2 . WI( X n ) = n | X n ( u ) | − u ∈ X n u ∈ X n
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