P olya urns via the contraction method Ralph Neininger Institute - PowerPoint PPT Presentation

P´ olya urns via the contraction method Ralph Neininger Institute for Mathematics J.W. Goethe-University Frankfurt a.M. AofA 2013 Menorca, Spain joint work with M. Knape arXiv:1301.3404

P´ olya urns Initial configuration: r 0 red balls, b 0 blue balls. Replacement matrix: red blue red a b blue c d a, d ∈ N 0 ∪ {− 1 } , b, c ∈ N 0

Methods Main focus: # balls of each color, n → ∞ Calculations with moment generating function Calculations with moments Martingale methods Embedding into continuous time branching processes Counting urn histories + analytic combinatorics Here: Recursive approach a + b = c + d : “balanced urn”

Methods Main focus: # balls of each color, n → ∞ Calculations with moment generating function Calculations with moments Martingale methods Embedding into continuous time branching processes Counting urn histories + analytic combinatorics Here: Recursive approach a + b = c + d : “balanced urn”

Methods Main focus: # balls of each color, n → ∞ Calculations with moment generating function Calculations with moments Martingale methods Embedding into continuous time branching processes Counting urn histories + analytic combinatorics Here: Recursive approach a + b = c + d : “balanced urn”

red blue A discrete-time embedding red 1 4 blue 3 2

red blue A discrete-time embedding red 1 4 blue 3 2

red blue A discrete-time embedding red 1 4 blue 3 2

red blue A discrete-time embedding red 1 4 blue 3 2

red blue A discrete-time embedding red 1 4 blue 3 2

red blue A discrete-time embedding red 1 4 blue 3 2

red blue A discrete-time embedding red 1 4 blue 3 2

red blue A discrete-time embedding red 1 4 blue 3 2

red blue A discrete-time embedding red 1 4 blue 3 2

red blue A discrete-time embedding red 1 4 blue 3 2

red blue A discrete-time embedding red 1 4 blue 3 2

Embedding into trees Balls are held in leaves of the tree. Balanced urn: a + b = c + d =: K − 1. Branch degree: K . Recurrence: K � # red balls = # red balls in j -th subtree. j =1 Subtrees rooted by red / blue (ghost) balls behave differently.

Embedding into trees Balls are held in leaves of the tree. Balanced urn: a + b = c + d =: K − 1. Branch degree: K . Recurrence: K � # red balls = # red balls in j -th subtree. j =1 Subtrees rooted by red / blue (ghost) balls behave differently.

Embedding into trees Balls are held in leaves of the tree. Balanced urn: a + b = c + d =: K − 1. Branch degree: K . Recurrence: K � # red balls = # red balls in j -th subtree. j =1 Subtrees rooted by red / blue (ghost) balls behave differently.

Embedding into trees Balls are held in leaves of the tree. Balanced urn: a + b = c + d =: K − 1. Branch degree: K . Recurrence: K � # red balls = # red balls in j -th subtree. j =1 Subtrees rooted by red / blue (ghost) balls behave differently.

General recurrence R ( r ) n : # red balls when starting with red after n steps. R ( b ) n : # red balls when starting with blue after n steps. I ( n ) = ( I 1 , . . . , I K ): vector of # draws in each subtree. a + b = c + d = K − 1 balanced urn. a +1 K � � d R ( r ) R ( r ) , j R ( b ) , j = + n I j I j j =1 j = a +2 c K � � d R ( b ) R ( r ) , j R ( b ) , j = + n I j I j j =1 j = c +1

General recurrence R ( r ) n : # red balls when starting with red after n steps. R ( b ) n : # red balls when starting with blue after n steps. I ( n ) = ( I 1 , . . . , I K ): vector of # draws in each subtree. a + b = c + d = K − 1 balanced urn. a +1 K � � d R ( r ) R ( r ) , j R ( b ) , j = + n I j I j j =1 j = a +2 c K � � d R ( b ) R ( r ) , j R ( b ) , j = + n I j I j j =1 j = c +1

General recurrence R ( r ) n : # red balls when starting with red after n steps. R ( b ) n : # red balls when starting with blue after n steps. I ( n ) = ( I 1 , . . . , I K ): vector of # draws in each subtree. a + b = c + d = K − 1 balanced urn. a +1 K � � d R ( r ) R ( r ) , j R ( b ) , j = + n I j I j j =1 j = a +2 c K � � d R ( b ) R ( r ) , j R ( b ) , j = + n I j I j j =1 j = c +1

General recurrence R ( r ) n : # red balls when starting with red after n steps. R ( b ) n : # red balls when starting with blue after n steps. I ( n ) = ( I 1 , . . . , I K ): vector of # draws in each subtree. a + b = c + d = K − 1 balanced urn. a +1 K � � d R ( r ) R ( r ) , j R ( b ) , j = + n I j I j j =1 j = a +2 c K � � d R ( b ) R ( r ) , j R ( b ) , j = + n I j I j j =1 j = c +1

General recurrence R ( r ) n : # red balls when starting with red after n steps. R ( b ) n : # red balls when starting with blue after n steps. I ( n ) = ( I 1 , . . . , I K ): vector of # draws in each subtree. a + b = c + d = K − 1 balanced urn. a +1 K � � d R ( r ) R ( r ) , j R ( b ) , j = + n I j I j j =1 j = a +2 c K � � d R ( b ) R ( r ) , j R ( b ) , j = + n I j I j j =1 j = c +1

Sizes of subtrees I ( n ) = ( I 1 , . . . , I K ): vector of # draws in each subtree. Label balls by their subtree. P´ olya urn for the labels. Initial condition: (1 , 1 , . . . , 1) Replacement matrix: diag( K − 1 , K − 1 , . . . , K − 1) 1 a . s . → ( D 1 , . . . , D K ) nI n − With � � 1 1 ( D 1 , . . . , D K ) ∼ Dirichlet K − 1 , . . . , . K − 1

Sizes of subtrees I ( n ) = ( I 1 , . . . , I K ): vector of # draws in each subtree. Label balls by their subtree. P´ olya urn for the labels. 2 4 5 6 Initial condition: (1 , 1 , . . . , 1) 3 3 3 3 3 1 1 1 1 1 1 Replacement matrix: 3 3 3 3 3 3 diag( K − 1 , K − 1 , . . . , K − 1) 1 nI ( n ) a . s . → ( D 1 , . . . , D K ) − With � � 1 1 ( D 1 , . . . , D K ) ∼ Dirichlet K − 1 , . . . , . K − 1

Sizes of subtrees I ( n ) = ( I 1 , . . . , I K ): vector of # draws in each subtree. Label balls by their subtree. P´ olya urn for the labels. 2 4 5 6 Initial condition: (1 , 1 , . . . , 1) 3 3 3 3 3 1 1 1 1 1 1 Replacement matrix: 3 3 3 3 3 3 diag( K − 1 , K − 1 , . . . , K − 1) 1 nI ( n ) a . s . → ( D 1 , . . . , D K ) − With � � 1 1 ( D 1 , . . . , D K ) ∼ Dirichlet K − 1 , . . . , . K − 1

Sizes of subtrees I ( n ) = ( I 1 , . . . , I K ): vector of # draws in each subtree. Label balls by their subtree. P´ olya urn for the labels. 2 4 5 6 Initial condition: (1 , 1 , . . . , 1) 3 3 3 3 3 1 1 1 1 1 1 Replacement matrix: 3 3 3 3 3 3 diag( K − 1 , K − 1 , . . . , K − 1) 1 nI ( n ) a . s . → ( D 1 , . . . , D K ) − With � � 1 1 ( D 1 , . . . , D K ) ∼ Dirichlet K − 1 , . . . , . K − 1

Sizes of subtrees I ( n ) = ( I 1 , . . . , I K ): vector of # draws in each subtree. Label balls by their subtree. P´ olya urn for the labels. 2 4 5 6 Initial condition: (1 , 1 , . . . , 1) 3 3 3 3 3 1 1 1 1 1 1 Replacement matrix: 3 3 3 3 3 3 diag( K − 1 , K − 1 , . . . , K − 1) 1 nI ( n ) a . s . → ( D 1 , . . . , D K ) − With � � 1 1 ( D 1 , . . . , D K ) ∼ Dirichlet K − 1 , . . . , . K − 1

Normalization a +1 K � � d R ( r ) R ( r ) , j R ( b ) , j = + n I j I j j =1 j = a +2 c K � � d R ( b ) R ( r ) , j R ( b ) , j = + n I j I j j =1 j = c +1 Normalization := R ( r ) − E R ( r ) := R ( b ) − E R ( b ) X ( r ) n n X ( b ) n n , n n n γ L ( n ) n γ L ( n ) Modified equations � I j � γ L ( I j ) � I j � γ L ( I j ) a +1 K � � d X ( r ) L ( n ) X ( r ) , j L ( n ) X ( b ) , j = + n I j I j n n j =1 j = a +2 d X ( b ) = similarly n

Normalization a +1 K � � d R ( r ) R ( r ) , j R ( b ) , j = + n I j I j j =1 j = a +2 c K � � d R ( b ) R ( r ) , j R ( b ) , j = + n I j I j j =1 j = c +1 Normalization := R ( r ) − E R ( r ) := R ( b ) − E R ( b ) X ( r ) n n X ( b ) n n , n n n γ L ( n ) n γ L ( n ) Modified equations � I j � γ L ( I j ) � I j � γ L ( I j ) a +1 K � � d X ( r ) L ( n ) X ( r ) , j L ( n ) X ( b ) , j = + n I j I j n n j =1 j = a +2 d X ( b ) = similarly n

Normalization a +1 K � � d R ( r ) R ( r ) , j R ( b ) , j = + n I j I j j =1 j = a +2 c K � � d R ( b ) R ( r ) , j R ( b ) , j = + n I j I j j =1 j = c +1 Normalization := R ( r ) − E R ( r ) := R ( b ) − E R ( b ) X ( r ) n n X ( b ) n n , n n n γ L ( n ) n γ L ( n ) Modified equations � I j � γ L ( I j ) � I j � γ L ( I j ) a +1 K � � d X ( r ) L ( n ) X ( r ) , j L ( n ) X ( b ) , j + T ( n ) = + n r I j I j n n j =1 j = a +2 d X ( b ) = similarly n

Fixed-point equations � I j � γ L ( I j ) � I j � γ L ( I j ) a +1 K � � d X ( r ) L ( n ) X ( r ) , j L ( n ) X ( b ) , j + T ( n ) = + n r I j I j n n j =1 j = a +2 d X ( b ) = similarly n Limit a +1 K � � d j X ( r ) , j + D γ D γ X ( r ) j X ( b ) , j = j =1 j = a +2 c K � � d j X ( r ) , j + D γ D γ X ( b ) j X ( b ) , j = j =1 j = c +1 Calls for recursive methods: Contraction method