Multi-drawing, multi-colour Pólya urns – Cécile Mailler – ArXiV:1611.09090 joint work with Nabil Lassmar and Olfa Selmi (Monastir, Tunisia) October 11th, 2017 Cécile Mailler (Prob-L@B) Multi-drawing, multi-colour Pólya urns October 11th, 2017 1 / 22
Happy birthday, Henning!! Cécile Mailler (Prob-L@B) Multi-drawing, multi-colour Pólya urns October 11th, 2017 2 / 22
Professorship @ Bath!! Deadline for applications: 01/01/2018. Cécile Mailler (Prob-L@B) Multi-drawing, multi-colour Pólya urns October 11th, 2017 3 / 22
Multi-drawing, multi-colour Pólya urns – Cécile Mailler – ArXiV:1611.09090 joint work with Nabil Lassmar and Olfa Selmi (Monastir, Tunisia) October 11th, 2017 Cécile Mailler (Prob-L@B) Multi-drawing, multi-colour Pólya urns October 11th, 2017 4 / 22
Introduction Classical Pólya’s urns The “classical” Pólya urn model Two parameters: the replacement matrix R = ( a d ) b c and the initial composition U 0 = ( U 0 , 1 U 0 , 2 ) Cécile Mailler (Prob-L@B) Multi-drawing, multi-colour Pólya urns October 11th, 2017 5 / 22
Introduction Classical Pólya’s urns The “classical” Pólya urn model Two parameters: uniformly at random the replacement matrix R = ( a d ) b c and the initial composition U 0 = ( U 0 , 1 U 0 , 2 ) Cécile Mailler (Prob-L@B) Multi-drawing, multi-colour Pólya urns October 11th, 2017 5 / 22
Introduction Classical Pólya’s urns The “classical” Pólya urn model Two parameters: the replacement matrix R = ( a d ) b c and the initial composition U 0 = ( U 0 , 1 U 0 , 2 ) Cécile Mailler (Prob-L@B) Multi-drawing, multi-colour Pólya urns October 11th, 2017 5 / 22
Introduction Classical Pólya’s urns The “classical” Pólya urn model Two parameters: the replacement matrix R = ( a d ) b c and the initial composition U 0 = ( U 0 , 1 U 0 , 2 ) Same for d -colours! Cécile Mailler (Prob-L@B) Multi-drawing, multi-colour Pólya urns October 11th, 2017 5 / 22
Introduction Classical Pólya’s urns The “classical” Pólya urn model Two parameters: the replacement matrix R = ( a d ) b c and the initial composition U 0 = ( U 0 , 1 U 0 , 2 ) Same for d -colours! Questions: How does U n behave when n is large? How does this asymptotic behaviour depend on R and U 0 ? Cécile Mailler (Prob-L@B) Multi-drawing, multi-colour Pólya urns October 11th, 2017 5 / 22
Introduction Classical Pólya’s urns Asymptotic theorems Perron-Frobenius: If R is irreducible, then its spectral radius λ 1 is positive, and a simple eigenvalue of R . And there exists an eigenvector u 1 with positive coordinates such that t Ru 1 = λ 1 u 1 . λ 2 is the eigenvalue of R with the second largest real part, and σ = Re λ 2 / λ 1 . Theorem (see, e.g. [Athreya & Karlin ’68] [Janson ’04]): Assume that R is irreducible and ∑ d i = 1 U 0 , i > 0, then, U n / n → u 1 ( n → ∞ ) almost surely; furthermore, when n → ∞ , ▸ if σ < 1 / 2 , then n − 1 / 2 ( U n − nu 1 ) → N ( 0 , Σ 2 ) in distribution; ▸ if σ = 1 / 2 , then ( n log n ) − 1 / 2 ( U n − nu 1 ) → N ( 0 , Θ 2 ) in distribution; ▸ if σ > 1 / 2 , then n − σ ( U n − nu 1 ) cv. a.s. to a finite random variable. Cécile Mailler (Prob-L@B) Multi-drawing, multi-colour Pólya urns October 11th, 2017 6 / 22
Introduction Classical Pólya’s urns Asymptotic theorems Theorem (see, e.g. [Athreya & Karlin ’68] [Janson ’04]): Assume that R is irreducible and ∑ d i = 1 U 0 , i > 0, then, U n / n → u 1 ( n → ∞ ) almost surely; furthermore, when n → ∞ , ▸ if σ < 1 / 2 , then n − 1 / 2 ( U n − nu 1 ) ⇒ N( 0 , Σ 2 ) in distribution; ▸ if σ = 1 / 2 , then ( n log n ) − 1 / 2 ( U n − nu 1 ) ⇒ N( 0 , Θ 2 ) in distribution; ▸ if σ > 1 / 2 , then n − σ ( U n − nu 1 ) cv. a.s. to a finite random variable. A few remarks: Both Σ and Θ don’t depend on the initial composition. Cécile Mailler (Prob-L@B) Multi-drawing, multi-colour Pólya urns October 11th, 2017 7 / 22
Introduction Classical Pólya’s urns Asymptotic theorems Theorem (see, e.g. [Athreya & Karlin ’68] [Janson ’04]): Assume that R is irreducible and ∑ d i = 1 U 0 , i > 0, then, U n / n → u 1 ( n → ∞ ) almost surely; furthermore, when n → ∞ , ▸ if σ < 1 / 2 , then n − 1 / 2 ( U n − nu 1 ) ⇒ N( 0 , Σ 2 ) in distribution; ▸ if σ = 1 / 2 , then ( n log n ) − 1 / 2 ( U n − nu 1 ) ⇒ N( 0 , Θ 2 ) in distribution; ▸ if σ > 1 / 2 , then n − σ ( U n − nu 1 ) cv. a.s. to a finite random variable. A few remarks: Both Σ and Θ don’t depend on the initial composition. It actually applies to a largest class of urns: R can be reducible as long as there is a Perron-Frobenius-like eigenvalue. Cécile Mailler (Prob-L@B) Multi-drawing, multi-colour Pólya urns October 11th, 2017 7 / 22
Introduction Classical Pólya’s urns Asymptotic theorems Theorem (see, e.g. [Athreya & Karlin ’68] [Janson ’04]): Assume that R is irreducible and ∑ d i = 1 U 0 , i > 0, then, U n / n → u 1 ( n → ∞ ) almost surely; furthermore, when n → ∞ , ▸ if σ < 1 / 2 , then n − 1 / 2 ( U n − nu 1 ) ⇒ N( 0 , Σ 2 ) in distribution; ▸ if σ = 1 / 2 , then ( n log n ) − 1 / 2 ( U n − nu 1 ) ⇒ N( 0 , Θ 2 ) in distribution; ▸ if σ > 1 / 2 , then n − σ ( U n − nu 1 ) cv. a.s. to a finite random variable. A few remarks: Both Σ and Θ don’t depend on the initial composition. It actually applies to a largest class of urns: R can be reducible as long as there is a Perron-Frobenius-like eigenvalue. The non-Perron-Frobenius-like cases are much less understood (see, e.g. [Janson ’05]). Cécile Mailler (Prob-L@B) Multi-drawing, multi-colour Pólya urns October 11th, 2017 7 / 22
Introduction Multi-drawing Pólya urns Multi-drawing d -colour Pólya urns Three parameters: an integer m ≥ 1, the initial composition U 0 , and the replacement rule R ∶ Σ m → N d , where ( d ) m = { v ∈ N d ∶ v 1 + ... + v d = m } . ( d ) Σ Start with U 0 , i balls of colour i in the urn ( ∀ 1 ≤ i ≤ d ). At step n , pick m balls in the urn (with or without replacement), denote by ξ n + 1 ∈ Σ ( d ) m the composition of the set drawn; then set U n + 1 = U n + R ( ξ n + 1 ) . Cécile Mailler (Prob-L@B) Multi-drawing, multi-colour Pólya urns October 11th, 2017 8 / 22
Introduction Multi-drawing Pólya urns Multi-drawing d -colour Pólya urns Three parameters: an integer m ≥ 1, the initial composition U 0 , and the replacement rule R ∶ Σ m → N d , where ( d ) m = { v ∈ N d ∶ v 1 + ... + v d = m } . ( d ) Σ Start with U 0 , i balls of colour i in the urn ( ∀ 1 ≤ i ≤ d ). At step n , pick m balls in the urn (with or without replacement), denote by ξ n + 1 ∈ Σ ( d ) m the composition of the set drawn; then set U n + 1 = U n + R ( ξ n + 1 ) . Z n , i = proportion of balls of colour i in the urn at time n ; T n = total number of balls in the urn at time n . Without replacement: With replacement: For all v ∈ Σ For all v ∈ Σ ( d ) ( d ) m , − 1 ∏ d m , P n ( ξ n + 1 = v ) = ( v 1 ... v d )∏ d P n ( ξ n + 1 = v ) = ( T n m ) i = 1 ( U n , i v i ) . m i = 1 Z v i n , i . Cécile Mailler (Prob-L@B) Multi-drawing, multi-colour Pólya urns October 11th, 2017 8 / 22
Introduction Multi-drawing Pólya urns Multi-drawing d -colour Pólya urns Three parameters: an integer m ≥ 1, the initial composition U 0 , and the replacement rule R ∶ Σ m → N d , where ( d ) m = { v ∈ N d ∶ v 1 + ... + v d = m } . ( d ) Σ Start with U 0 , i balls of colour i in the urn ( ∀ 1 ≤ i ≤ d ). At step n , pick m balls in the urn (with or without replacement), denote by ξ n + 1 ∈ Σ ( d ) m the composition of the set drawn; then set U n + 1 = U n + R ( ξ n + 1 ) . Z n , i = proportion of balls of colour i in the urn at time n ; T n = total number of balls in the urn at time n . Without replacement: With replacement: For all v ∈ Σ For all v ∈ Σ ( d ) ( d ) m , − 1 ∏ d m , P n ( ξ n + 1 = v ) = ( v 1 ... v d )∏ d P n ( ξ n + 1 = v ) = ( T n m ) i = 1 ( U n , i v i ) . m i = 1 Z v i n , i . Cécile Mailler (Prob-L@B) Multi-drawing, multi-colour Pólya urns October 11th, 2017 8 / 22
Introduction Stochastic approximation The method Embed the urn into continuous-time onto a multi-type branching processes. [Athreya & Karlin ’68, Janson ’04] Restrict to the “affine” case and use martingales. [Kuba & Mahmoud ’17, Kuba & Sulzbach ’16] Cécile Mailler (Prob-L@B) Multi-drawing, multi-colour Pólya urns October 11th, 2017 9 / 22
Introduction Stochastic approximation The method Embed the urn into continuous-time onto a multi-type branching processes. [Athreya & Karlin ’68, Janson ’04] Restrict ourselves to the “affine” case and use martingales. [Kuba & Mahmoud ’17, Kuba & Sulzbach ’16] Cécile Mailler (Prob-L@B) Multi-drawing, multi-colour Pólya urns October 11th, 2017 9 / 22
Introduction Stochastic approximation The method Embed the urn into continuous-time onto a multi-type branching processes. [Athreya & Karlin ’68, Janson ’04] Restrict ourselves to the “affine” case and use martingales. [Kuba & Mahmoud ’17, Kuba & Sulzbach ’16] Use stochastic approximation! Cécile Mailler (Prob-L@B) Multi-drawing, multi-colour Pólya urns October 11th, 2017 9 / 22
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