The class of tenable zerobalanced Pólya urn schemes: characterization and Gaussian phases Hosam M. Mahmoud The George Washington University Joint work with Sanaa kholfi Talk at 22 Workshop on Analysis of Algorithms June 13, 2011 Plan Pólya urn schemes Explain the title: Irreducible nondegenerate zero-balanced tenable schemes (the class Ck) Characterization The Markov chain Gaussian phases Examples Pólya Urns We have a starting urn of white and blue balls We have rules of evolution (ball addition matrix) add W B pick A = W a b B c d Intricate probabilities (depending on paths) P(White) = 3/8 W B P(White) = 4/7 1 2 A = 2 0 General setup Can be positive or negative (application in trees) The entries are generally random Who said analysis of algorithms cannot be colorful? References Norman Johnson and Samuel Kotz (1977). Urn Models and Their Applications. Wily, New York, USA. Hosam Mahmoud (2008). Pólya Urn Models. Chapman-Hall, Florida, USA. Tenable urns As an example of what we have in mind Where X = Bernoulli(p) X = 1 persists W W W B W ? Zero-Balance a b c d When a+b = c+d =K we say the urn is balanced, and the balance factor is K we look at urns with balance 0 Ehrenfest A model for the mixing of gases Reducible and Irreducible schemes The urn associated with coupon collection is reducible. If we start monochromatically with blue balls, white balls never appear. Communication problems between colors This urn is like is two noncommunicating Ehrenfest schemes Degeneracy Suppose we have a k-color scheme in which a color (say pink) is always an exact multiple of another (say crimson), say twice as many (initially, and the rules keep this proportion). We can combine the two colors into one (say the simply très chic red), study a scheme of dimensionality k-1, then restore the structure of each color. Characterization Can the matrix 3 -6 3 0 0 0 -1 0 1 0 0 0 0 0 0 be in our class? 0 0 0 0 0 2 0 0 -1 1 -2 0 0 1 -1 Characterization Examples Ehrenfest 3x3 deterministic 3x3 random B and B’ are independent Bernoulli (½) random variables Sufficient conditions Proof. Necessity These conditions are also necessary. If we assume the conditions, the replacement matrix must be in the form given. Markov chain v = (v1, …, vk) is a left eigenvector of E[A]. Phases Sublinear: log log n, log n, n¼, n½, n¾, n/log n Linear: 5n, (7 + 2 (-1)n) n Superlinear: n log n, n2, en Theorem 3 (main result) In the absence of a dominant color Dominant color three colors: initially, n – 2└log n┘, └ log n┘, └log n┘ X0(n) = α n + o (n) = 1 0 n + + o (n) 0 Remark: In the presence of dominant colors in critical cases different scale factors are needed for different colors and there may or may not be a single multivariate central limit theorem for all the colors . Stochastic recurrence Let 1j,r (n) be the indicator of picking color r at the jth step The mean where Martingales E[Xj | Fj-1] = Xj-1 Fair gambling: E[Xj | Fj-1] = Xj-1 + (-1) x ½ + (+1) x ½ The underlying matingale Recall that Centered martingale Martingale conditions Uniformly bounded differences Matrix norm calculation Conditional Lindeberg’s condition Conditional variance Deterministic approximations in each phase: Sublinear: Linear and superlinear: mean Examples Ehrenfest Sublinear: Ehrenfest in its linear and superlinear phases linear: Superlinear: A 3-color scheme with random replacements Sublinear superlinear An example with a degenerate scheme Reduced nondegenerate scheme An example with an initially dominant color When the row corresponding to the dominant color is deterministic The sublinear phase is delayed and the scales are different When the row has random entries Conclusion Analysis in phases reveals subtle dynamics and rates of convergence The class is a good candidate for “analytic urn methodology” The methodology may carry over to other reducible urns The methodology may carry over to other balanced urns.