Date & Time: Friday, July 08, 2011 16:00-17:00.
Venue: Ramanujan Hall
Title: Optimal points for a probability distribution on a Cantor set
Speaker: Mrinal Kanti Roychowdhury, The University of Texas-Pan American, USA
Abstract: It is well-known that a general homogeneous dyadic Cantor set C is generated by a set of two mappings S1 , S2 : [0, 1] → [0, 1] given by S1 (x) = rx and S2 (x) = rx + 1 − r, 1 where 0 < r < 2 . Then for a given probability vector (p1 , p2 ) (0 < p1 , p2 < 1) there exists a unique Borel probability measure P such that
−1 −1 P = p1 ◦ S1 + p2 P ◦ S2 .
Let X be a random variable with distribution P . Suppose we would like to send the infor- mation about X to some other place by sending some discrete number of points say n points, then the nth quantization error is given by
Vn = inf α min x − a 2 dP (x), a∈α
where the infimum is taken over all subsets α of Rd with card α ≤ n, and · denotes the Euclidean norm on Rd . It is known that if x 2 dμ(x) < ∞, then there is some set α for which the infimum is achieved and the set α is called the optimal set of n-means or n-optimal set. It can be shown that for a continuous probability measure P , an optimal set of n-means always has exactly n elements. Quantization dimension for the probability measure P is given by
2 log n D(P ) = lim , n→∞ − log Vn and it gives the rate how fast Vn goes to zero as n tends to infinity. −1 −1 In this talk, I will discuss for the probability distribution P := 1 P ◦ S1 + 1 P ◦ S2 2 2 with the support the classical Cantor set C generated by S1 (x) = 1 x and S2 (x) = 1 x + 2 , 3 3 3 the n-optimal points and the nth quantization error for any n ≥ 1, and the quantization dimension. Some open problems in this regard will also be pointed out.