On Pólya Urn Scheme with Infinitely Many Colors
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1 On Pólya Urn Scheme with Infinitely Many Colors DEBLEENA THACKER Indian Statistical Institute, New Delhi Joint work with: ANTAR BANDYOPADHYAY, Indian Statistical Institute, New Delhi.
2 Genaralization of the Polya Urn scheme to infinitely many colors We introduce an urn with infinite but countably many colors/types of balls indexed by Z. D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 2 / 17
3 Genaralization of the Polya Urn scheme to infinitely many colors We introduce an urn with infinite but countably many colors/types of balls indexed by Z. In this case, the so called uniform selection of balls does not make sense. D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 2 / 17
4 Genaralization of the Polya Urn scheme to infinitely many colors We introduce an urn with infinite but countably many colors/types of balls indexed by Z. In this case, the so called uniform selection of balls does not make sense. The intial configuration of the urn U 0 is taken to be a probability vector and can be thought to be the proportion of balls of each color/type we start with. Then P (A ball of color j is selected at the first trial U 0 ) = U 0,j. We consider the replacement matrix R to be an infinite dimensional stochastic matrix. D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 2 / 17
5 Genaralization of the Polya Urn scheme to infinitely many colors We introduce an urn with infinite but countably many colors/types of balls indexed by Z. In this case, the so called uniform selection of balls does not make sense. The intial configuration of the urn U 0 is taken to be a probability vector and can be thought to be the proportion of balls of each color/type we start with. Then P (A ball of color j is selected at the first trial U 0 ) = U 0,j. We consider the replacement matrix R to be an infinite dimensional stochastic matrix. At each step n 1, the same procedure as that of Polya Urn Scheme is repeated. D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 2 / 17
6 Genaralization of the Polya Urn scheme to infinitely many colors We introduce an urn with infinite but countably many colors/types of balls indexed by Z. In this case, the so called uniform selection of balls does not make sense. The intial configuration of the urn U 0 is taken to be a probability vector and can be thought to be the proportion of balls of each color/type we start with. Then P (A ball of color j is selected at the first trial U 0 ) = U 0,j. We consider the replacement matrix R to be an infinite dimensional stochastic matrix. At each step n 1, the same procedure as that of Polya Urn Scheme is repeated. Let U n be the row vector denoting the number of balls of different colors at time n. D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 2 / 17
7 Genaralization of the Polya Urn scheme to infinitely many colors We introduce an urn with infinite but countably many colors/types of balls indexed by Z. In this case, the so called uniform selection of balls does not make sense. The intial configuration of the urn U 0 is taken to be a probability vector and can be thought to be the proportion of balls of each color/type we start with. Then P (A ball of color j is selected at the first trial U 0 ) = U 0,j. We consider the replacement matrix R to be an infinite dimensional stochastic matrix. At each step n 1, the same procedure as that of Polya Urn Scheme is repeated. Let U n be the row vector denoting the number of balls of different colors at time n. D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 2 / 17
8 Fundamental Recursion If the chosen ball turns out to be of j th color, then U n+1 is given by the equation U n+1 = U n + R j where R j is the jth row of the matrix R. This can also be written in the matrix notation as D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 3 / 17
9 Fundamental Recursion If the chosen ball turns out to be of j th color, then U n+1 is given by the equation U n+1 = U n + R j where R j is the jth row of the matrix R. This can also be written in the matrix notation as U n+1 = U n + I n+1 R (1) where I n = (..., I n, 1, I n,0, I n,1...) where I n,i = 1 for i = j and 0 elsewhere. We study this process for the replacement matrices R which arise out of the Random Walks on Z. D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 3 / 17
10 We can generalize this process to general graphs on R d, d 1. Let G = (V, E) be a connected graph on R d with vertex set V which is countably infinite. The edges are taken to be bi-directional and there exists m N such that d(v) = m for every v V. Let the distribution of X 1 be given by P (X 1 = v) = p(v) for v B where B <. (2) where v B p(v) = 1. Let S n = n X i. i=1 Let R be the matrix/operator corresponding to the random walk S n and the urn process evolve according to R. In this case, the configuration U n of the process is a row vector with co-ordinates indexed by V. The dynamics is similar to that in one-dimension, that is an element is drawn at random, its type noted and returned to the urn. If the v th type is selected at the n + 1 st trial, then U n+1 = U n + e v R (3) where e v is a row vector with 1 at the v th co-ordinate and zero elsewhere. D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 4 / 17
11 We note the following, for all d 1 U n,v = n + 1. v V D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 5 / 17
12 We note the following, for all d 1 U n,v = n + 1. v V Hence Un n+1 is a random probability vector. For every ω Ω, we can define a random d-dimensional vector T n (ω) with law Un(ω) n+1. D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 5 / 17
13 We note the following, for all d 1 U n,v = n + 1. v V Hence Un n+1 is a random probability vector. For every ω Ω, we can define a random d-dimensional vector T n (ω) with law Un(ω) n+1. Also (E[Un,v]) v V n+1 is a probability vector. Therefore we can define a random vector Z n with law (E[Un,v]) v V n+1. D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 5 / 17
14 Previous work Literature is available only for finitely many types/ colors. It is known that the asymptotic behavior of the urn model depends on the qualitative properties (transience or recurrence) of the underlying Markov Chain of the replacement matrix. Svante Janson, Stochastic Processes, D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 6 / 17
15 Previous work Literature is available only for finitely many types/ colors. It is known that the asymptotic behavior of the urn model depends on the qualitative properties (transience or recurrence) of the underlying Markov Chain of the replacement matrix. Svante Janson, Stochastic Processes, Svante Janson, Probab Theory and Related Fields, D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 6 / 17
16 Previous work Literature is available only for finitely many types/ colors. It is known that the asymptotic behavior of the urn model depends on the qualitative properties (transience or recurrence) of the underlying Markov Chain of the replacement matrix. Svante Janson, Stochastic Processes, Svante Janson, Probab Theory and Related Fields, Arup Bose, Amites Dasgupta, Krishanu Maulik, Bernoulli, D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 6 / 17
17 Previous work Literature is available only for finitely many types/ colors. It is known that the asymptotic behavior of the urn model depends on the qualitative properties (transience or recurrence) of the underlying Markov Chain of the replacement matrix. Svante Janson, Stochastic Processes, Svante Janson, Probab Theory and Related Fields, Arup Bose, Amites Dasgupta, Krishanu Maulik, Bernoulli, Arup Bose, Amites Dasgupta, Krishanu Maulik, Journal of Applied Probability, D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 6 / 17
18 Previous work Literature is available only for finitely many types/ colors. It is known that the asymptotic behavior of the urn model depends on the qualitative properties (transience or recurrence) of the underlying Markov Chain of the replacement matrix. Svante Janson, Stochastic Processes, Svante Janson, Probab Theory and Related Fields, Arup Bose, Amites Dasgupta, Krishanu Maulik, Bernoulli, Arup Bose, Amites Dasgupta, Krishanu Maulik, Journal of Applied Probability, Amites Dasgupta, Krishanu Maulik, preprint. D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 6 / 17
19 Previous work Literature is available only for finitely many types/ colors. It is known that the asymptotic behavior of the urn model depends on the qualitative properties (transience or recurrence) of the underlying Markov Chain of the replacement matrix. Svante Janson, Stochastic Processes, Svante Janson, Probab Theory and Related Fields, Arup Bose, Amites Dasgupta, Krishanu Maulik, Bernoulli, Arup Bose, Amites Dasgupta, Krishanu Maulik, Journal of Applied Probability, Amites Dasgupta, Krishanu Maulik, preprint. T. W. Mullikan, Transactions of American Mathematical Society, D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 6 / 17
20 Previous work Literature is available only for finitely many types/ colors. It is known that the asymptotic behavior of the urn model depends on the qualitative properties (transience or recurrence) of the underlying Markov Chain of the replacement matrix. Svante Janson, Stochastic Processes, Svante Janson, Probab Theory and Related Fields, Arup Bose, Amites Dasgupta, Krishanu Maulik, Bernoulli, Arup Bose, Amites Dasgupta, Krishanu Maulik, Journal of Applied Probability, Amites Dasgupta, Krishanu Maulik, preprint. T. W. Mullikan, Transactions of American Mathematical Society, Shu-Teh C. Moy, The Annals of Mathematical Statistics, D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 6 / 17
21 Previous work Literature is available only for finitely many types/ colors. It is known that the asymptotic behavior of the urn model depends on the qualitative properties (transience or recurrence) of the underlying Markov Chain of the replacement matrix. Svante Janson, Stochastic Processes, Svante Janson, Probab Theory and Related Fields, Arup Bose, Amites Dasgupta, Krishanu Maulik, Bernoulli, Arup Bose, Amites Dasgupta, Krishanu Maulik, Journal of Applied Probability, Amites Dasgupta, Krishanu Maulik, preprint. T. W. Mullikan, Transactions of American Mathematical Society, Shu-Teh C. Moy, The Annals of Mathematical Statistics, Shu-Teh C. Moy, Journal of Mathematics and Mechanics, D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 6 / 17
22 Main Result Theorem Let the process evolve according to a random walk on R d with bounded increments. ( Let the process ) begin with ( a single ball of type 0. For ) X 1 = X (1) 1, X(2) 1... X (d) 1, let µ = E[X (1) 1 ], E[X(2) 1 ],... E[X(d) 1 ] and Σ = [σ ij ] d d where σ i,j = E[X (i) 1 X(j) 1 ]. Let B be such that Σ is positive definite. Then Z n µ log n log n d N(0, Σ) as n (4) where N(0, Σ) denotes the d-dimensional Gaussian with mean vector 0 and variance-covariance matrix Σ. Furthermore there exists a subsequence {n k } such that as k almost surely T nk µ log n log n d N(0, Σ) (5) D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 7 / 17
23 Interesting Examples Corollary Let d 1 and we consider the SSRW. Let the process begin with a single ball of type 0. If Z n be the random d-dimensional vector corresponding to the probability distribution (E[Un,v]) v Z d n+1, then Z n log n d N ( 0, d 1 I d ) as n (6) where I d is the d-dimensional identity matrix. Furthemore, there exists a subsequence {n k } such that almost surely as k T nk nk d N ( 0, d 1 I d ). (7) D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 8 / 17
24 Corollary Let d = 1 and P (X 1 = 1) = 1. Let U 0 = 1 {0}. If Z n be the random variable corresponding to the probability mass function (E[U n,k]) k Z n+1, then Z n log n log n d N(0, 1) as n. (8) Also there exists a subsequence n k such that almost surely as k T nk log n k nk d N(0, 1). (9) D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 9 / 17
25 Figure: Triangular Lattice Corollary Let the urn model evolve according to the random walk on triangular lattice on R 2 and the process begin with a single particle of type 0, then as n ( Z n d N 0, 1 ) log n 2 I 2. (10) D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 10 / 17
26 Corollary (continued) Furthermore, there exists a subsequence {n k } such that as k, ( T log nk d N 0, 1 ) nk 2 I 2 (11) D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 11 / 17
27 Conclusion The SSRW is recurrent for d 2 and transient for d 3. D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 12 / 17
28 Conclusion The SSRW is recurrent for d 2 and transient for d 3. In both cases, with a scaling of log n the asymptotic behavior of the models are similar. D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 12 / 17
29 Conclusion The SSRW is recurrent for d 2 and transient for d 3. In both cases, with a scaling of log n the asymptotic behavior of the models are similar. On Z, the random walks are recurrent or transient depending on E[X 1 ] = 0 or not. Asypmtotically both behave similarly upto centering and scaling. D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 12 / 17
30 Conclusion The SSRW is recurrent for d 2 and transient for d 3. In both cases, with a scaling of log n the asymptotic behavior of the models are similar. On Z, the random walks are recurrent or transient depending on E[X 1 ] = 0 or not. Asypmtotically both behave similarly upto centering and scaling. We conjecture that in the infinite type/ color case, the asymptotic behavior of the process is not determined completely by the underlying Markov Chain of the operator, but by the qualitative properties of the underlying graph. D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 12 / 17
31 Proof of the Main Theorem We present the proof for SSRW on d = 2 for notational simplicity. We use the martingale methods for the proof. D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 13 / 17
32 Proof of the Main Theorem We present the proof for SSRW on d = 2 for notational simplicity. We use the martingale methods for the proof. For every t = (t 1, t 2 ) R 2, e(t) = 1 4 e u,t is an eigen value for u N(0) the operator R where 0 stands for the origin in Z 2 and.,. stands for the inner product. D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 13 / 17
33 Proof of the Main Theorem We present the proof for SSRW on d = 2 for notational simplicity. We use the martingale methods for the proof. For every t = (t 1, t 2 ) R 2, e(t) = 1 4 e u,t is an eigen value for u N(0) the operator R where 0 stands for the origin in Z 2 and.,. stands for the inner product. The corresponding right eigen vectors are x (t) = (x t (v)) v Z 2 where x t (v) = e t,v. D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 13 / 17
34 Proof of the Main Theorem We present the proof for SSRW on d = 2 for notational simplicity. We use the martingale methods for the proof. For every t = (t 1, t 2 ) R 2, e(t) = 1 4 e u,t is an eigen value for u N(0) the operator R where 0 stands for the origin in Z 2 and.,. stands for the inner product. The corresponding right eigen vectors are x (t) = (x t (v)) v Z 2 where x t (v) = e t,v. U We have noted earlier that n n+1 is a random probability vector. The moment generating function for this vector is given by Un.x(t) n+1 for every t R 2. D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 13 / 17
35 Proof of the Main Theorem We present the proof for SSRW on d = 2 for notational simplicity. We use the martingale methods for the proof. For every t = (t 1, t 2 ) R 2, e(t) = 1 4 e u,t is an eigen value for u N(0) the operator R where 0 stands for the origin in Z 2 and.,. stands for the inner product. The corresponding right eigen vectors are x (t) = (x t (v)) v Z 2 where x t (v) = e t,v. U We have noted earlier that n n+1 is a random probability vector. The moment generating function for this vector is given by Un.x(t) n+1 for every t R 2. Using (1), it can be shown that M n (t) = Un.x(t) Π n(e(t)) is a non-negative n martingale, where Π n (β) = (1 + β j ). j=1 D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 13 / 17
36 Since we begin with one element of type 0, E [ M n (t) ] = Π n (e(t)). D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 14 / 17
37 Since we begin with one element of type 0, E [ M n (t) ] = Π n (e(t)). (12) Let us denote by E n the expectation vector (E[U n,v ]) v Z 2. The moment generating function for this vector is En.x(t) n+1 D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 14 / 17
38 Since we begin with one element of type 0, E [ M n (t) ] = Π n (e(t)). (12) Let us denote by E n the expectation vector (E[U n,v ]) v Z 2. The moment generating function for this vector is En.x(t) n+1 We will show that for a suitable δ > 0 and for all t [ δ, δ] 2 E n. x( t log n ) n + 1 e t (13) where for all x R 2, x 2 denontes the l 2 norm. D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 14 / 17
39 Since we begin with one element of type 0, E [ M n (t) ] = Π n (e(t)). (12) Let us denote by E n the expectation vector (E[U n,v ]) v Z 2. The moment generating function for this vector is En.x(t) n+1 We will show that for a suitable δ > 0 and for all t [ δ, δ] 2 E n. x( t log n ) n + 1 e t (13) where for all x R 2, x 2 denontes the l 2 norm. We know that where t n = t log n. E n. x (t n ) = Π n (e (t n )) (14) D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 14 / 17
40 We use the following fact due to Euler, except for β non-negative integer. 1 Γ(β + 1) = lim Π n (β) n n β D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 15 / 17
41 We use the following fact due to Euler, except for β non-negative integer. 1 Γ(β + 1) = lim Π n (β) n n β It is easy known that this convergence is uniform for all β [1 η, 1 + η] for a suitable choice of η. D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 15 / 17
42 We use the following fact due to Euler, except for β non-negative integer. 1 Γ(β + 1) = lim Π n (β) n n β It is easy known that this convergence is uniform for all β [1 η, 1 + η] for a suitable choice of η. Due to the uniform convergence, it follows immediately that t [ δ, δ] 2 lim n Π n (e (t n )) n e(tn) = 1. (15) /Γ(e (t n ) + 1) D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 15 / 17
43 Simplifying the left hand side of [13] we get Π n (e (t n )) n + 1 (16) D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 16 / 17
44 Simplifying the left hand side of [13] we get Π n (e (t n )) n + 1 (16) It is enough to show that lim log(n + 1) + e (t n) log n log(γ(e (t n ) + 1)) n = t 2 2. (17) 4 D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 16 / 17
45 Simplifying the left hand side of [13] we get Π n (e (t n )) n + 1 (16) It is enough to show that lim log(n + 1) + e (t n) log n log(γ(e (t n ) + 1)) n = t 2 2. (17) 4 Expanding e (t n ) into power series and noting that Γ(x) is continuous as a function of x we can prove (17). D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 16 / 17
46 Thank You! D. Thacker (Indian Statistical Institute) On Pólya Urn Schemes with Infinitelay Many Colors 17 / 17
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