Urn models with multiple drawings Markus Kuba joint work with May-Ru Chen; Hosam Mahmoud and Alois Panholzer AofA 2013 Cala Galdana, Menorca, Spain 30.05.2013
Content 1 Urn models - Introduction Urn models with multiple drawings 2 3 Analysis using Analytic Combinatorics AofA 2013, Menorca 2/26
Urn models - Introduction
Urn models - Introduction
P´ olya-Eggenberger urns Urn contains n white and m black balls. Every discrete time steps a ball is drawn at random: n m p white = n + m , p black = n + m . Color inspection: White - a white and b black balls are added/removed; Black - c white and d black balls are added/removed; a , b , c , d ∈ Z . 2 × 2 ball replacement matrix � a � b M = c d AofA 2013, Menorca 4/26
P´ olya-Eggenberger urns Urn contains n white and m black balls. Every discrete time steps a ball is drawn at random: n m p white = n + m , p black = n + m . Color inspection: White - a white and b black balls are added/removed; Black - c white and d black balls are added/removed; a , b , c , d ∈ Z . 2 × 2 ball replacement matrix � a � b M = c d AofA 2013, Menorca 4/26
P´ olya-Eggenberger urns Urn contains n white and m black balls. Every discrete time steps a ball is drawn at random: n m p white = n + m , p black = n + m . Color inspection: White - a white and b black balls are added/removed; Black - c white and d black balls are added/removed; a , b , c , d ∈ Z . 2 × 2 ball replacement matrix � a � b M = c d AofA 2013, Menorca 4/26
P´ olya-Eggenberger urns Urn contains n white and m black balls. Every discrete time steps a ball is drawn at random: n m p white = n + m , p black = n + m . Color inspection: White - a white and b black balls are added/removed; Black - c white and d black balls are added/removed; a , b , c , d ∈ Z . 2 × 2 ball replacement matrix � a � b M = c d AofA 2013, Menorca 4/26
P´ olya-Eggenberger urns Tenable urns 1 The process of drawing and adding/removing balls can be continued ad infinitum . We start with W 0 ∈ N 0 white and B 0 ∈ N 0 black balls: configuration W n , B n after n draws? Diminishing urns 2 The process of drawing and adding/removing balls stops after a finite number of steps. We start with n ∈ N 0 white and m ∈ N 0 black balls, define so-called absorbing states A : what is the probability of reaching a state a ∈ A ? Generalizations 3 More than two colors. AofA 2013, Menorca 5/26
P´ olya-Eggenberger urns P´ olya urn (tenable urns) � 1 0 � M = 0 1 Urn contains n white and m black balls: (m,n+1) n m+n m (m,n) (m+1,n) m+n AofA 2013, Menorca 6/26
P´ olya-Eggenberger urns P´ olya urn (tenable urns) � 1 0 � M = 0 1 Urn contains n white and m black balls: (m,n +1 ) n m+n m (m,n) (m+1,n) m+n AofA 2013, Menorca 6/26
P´ olya-Eggenberger urns P´ olya urn (tenable urns) � 1 0 � M = 0 1 Urn contains n white and m black balls: (m,n+1) n m+n m (m,n) (m +1 ,n) m+n AofA 2013, Menorca 6/26
P´ olya-Eggenberger urns 2000-today: several approaches - very interesting developments Analytic Combinatorics (Symbolic methods, generating functions, etc.) F LAJOLET , G ABARR ´ O AND P EKARI ; B RENNAN AND P RODINGER ; S TADJE ; D UMAS , F LAJOLET AND P UYHAUBERT ; H WANG , K. AND P ANHOLZER ; F LAJOLET AND M ORCRETTE ; M AHMOUD AND M ORCRETTE ; M ORCRETTE ; . . . Probabilistic methods (stochastic processes, martingales) K INGMAN 2 ; K INGMAN AND V OLKOV ; M AHMOUD x ; P ITTEL ; J ANSON 2 ; C HAUVIN , P OUYANNE ET AL . 3 ; C HEN AND W EI ; R ENLUND ; . . . Contraction method (for balanced urns) N EININGER AND K NAPE ; C HAUVIN , P OUYANNE AND M AILLER ; . . . . . . � a b � An urn model M = is called balanced : a + b = c + d = σ . c d Consequently: T n = W n + B n = T 0 + n · σ . AofA 2013, Menorca 7/26
P´ olya-Eggenberger urns 2000-today: several approaches - very interesting developments Analytic Combinatorics (Symbolic methods, generating functions, etc.) F LAJOLET , G ABARR ´ O AND P EKARI ; B RENNAN AND P RODINGER ; S TADJE ; D UMAS , F LAJOLET AND P UYHAUBERT ; H WANG , K. AND P ANHOLZER ; F LAJOLET AND M ORCRETTE ; M AHMOUD AND M ORCRETTE ; MORCRETTE ; . . . Probabilistic methods (stochastic processes, martingales) K INGMAN 2 ; K INGMAN AND V OLKOV ; M AHMOUD x ; P ITTEL ; J ANSON 2 ; C HAUVIN , P OUYANNE ET AL . 3 ; C HEN AND W EI ; R ENLUND ; . . . Contraction method (for balanced urns) NEININGER AND K NAPE ; C HAUVIN , P OUYANNE AND MAILLER ; . . . . . . � a b � An urn model M = is called balanced : a + b = c + d = σ . c d Consequently: T n = W n + B n = T 0 + n · σ . AofA 2013, Menorca 7/26
P´ olya-Eggenberger urns 2000-today: several approaches - very interesting developments Analytic Combinatorics (Symbolic methods, generating functions, etc.) ; ; ; ; ; ; ; . . . Probabilistic methods (stochastic processes, martingales) ; ; ; ; ; ; ; ; . . . Contraction method (for balanced urns) ; ;. . . . . . � a b � An urn model M = is called balanced : a + b = c + d = σ . c d Consequently: T n = W n + B n = T 0 + n · σ . AofA 2013, Menorca 7/26
Urn models - Multiple drawings
Urn models - Multiple drawings
Urn models with multiple drawings Previously : Urn contains w white and b black balls. w b Draw at random a single ball: p white = w + b , p black = w + b . New model: We draw m � 1 balls without replacement , 1 � m � w k b m − k , p { k times white, ( m − k ) times black } = ( b + w ) m k with x s = x ( x − 1 ) . . . ( x − s + 1 ) . Depending on the drawn multiset of white/black balls we add/remove balls. C HEN AND W EI 2005: Generalized P´ olya urn M AHMOUD 2008: Tenable balanced linear urns ( m = 2 ). R ENLUND 2010: Stochastic approximation for tenable urns. AofA 2013, Menorca 9/26
Urn models with multiple drawings � c � 0 P´ olya urn: M = , c ∈ N . 0 c Generalization: If we draw { W k S m − k } we add k · c white and ( m − k ) · c black balls, with c ∈ N . ( m + 1 ) × 2 -matrix mc 0 ( m − 1 ) c c M = . . . . . . c ( m − 1 ) c 0 mc Urn is balanced T n = W n + B n = nmc + T 0 . AofA 2013, Menorca 10/26
Urn models with multiple drawings � c � 0 P´ olya urn: M = , c ∈ N . 0 c Generalization: If we draw { W k S m − k } we add k · c white and ( m − k ) · c black balls, with c ∈ N . ( m + 1 ) × 2 -matrix mc 0 ( m − 1 ) c c M = . . . . . . c ( m − 1 ) c 0 mc Urn is balanced T n = W n + B n = nmc + T 0 . AofA 2013, Menorca 10/26
Urn models with multiple drawings � 0 � c Friedman urn: M = , c ∈ N . c 0 Generalization: If we draw { W k S m − k } we add ( m − k ) · c white and k · c black balls, with c ∈ N . ( m + 1 ) × 2 -matrix 0 mc c ( m − 1 ) c . . M = . . . . ( m − 1 ) c c mc 0 AofA 2013, Menorca 11/26
Urn models with multiple drawings � 0 � c Friedman urn: M = , c ∈ N . c 0 Generalization: If we draw { W k S m − k } we add ( m − k ) · c white and k · c black balls, with c ∈ N . ( m + 1 ) × 2 -matrix 0 mc c ( m − 1 ) c . . M = . . . . ( m − 1 ) c c mc 0 AofA 2013, Menorca 11/26
Urn models - Results
Urn models - Results
Urn models with multiple drawings Theorem ( C HEN AND W EI 2005) For the generalized P´ olya urn the number of white balls W n after n draws satisfies W n = W n a. s. → W ∞ ; − − T n W ∞ is absolutely continuous. Questions ( C HEN AND W EI ): (1) Is W ∞ beta-distributed ? (2) Explicit results concerning W n and W ∞ ; AofA 2013, Menorca 13/26
Urn models with multiple drawings Theorem ( C HEN AND W EI 2005) For the generalized P´ olya urn the number of white balls W n after n draws satisfies W n = W n a. s. → W ∞ ; − − T n W ∞ is absolutely continuous. Questions ( C HEN AND W EI ): (1) Is W ∞ beta-distributed ? (2) Explicit results concerning W n and W ∞ ; AofA 2013, Menorca 13/26
Urn models with multiple drawings Theorem ( C HEN AND K.) The expectation and the variance of W n are given by n ) − E ( W n ) 2 via E ( W n ) = W 0 T 0 ( nmc + T 0 ) , and V ( W n ) = E ( W 2 � n − 1 + λ 1 �� n − 1 + λ 2 � � E ( W 2 W 2 n n n ) = 0 � n − 1 + T 0 �� n − 1 + T 0 − 1 � mc mc n n � ℓ + T 0 �� ℓ + T 0 − 1 n − 1 ℓ + T 0 − m � + W 0 c 2 m mc mc � � ℓ + 1 ℓ + 1 mc , ℓ + T 0 − 1 � ℓ + λ 1 �� ℓ + λ 2 T 0 � ℓ = 0 mc ℓ + 1 ℓ + 1 with λ 1 , λ 2 given by λ 1,2 = − 1 2 + mc + T 0 ± 1 � 1 + 4 mc ( 1 + c ) 2 . mc AofA 2013, Menorca 14/26
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