Abstract :
We explore the problem of transforming n independent and identically biased {-1,1}-valued random variables X1,...,X n into a single {-1,1} random variable f(X1,...,X n), so that this result is as unbiased as possible. In general, no function f produces a completely unbiased result. We perform the first study of the relationship between the bias b of these Xi and the rate at which f(X1,...,Xn) can converge to an unbiased {-1,1} random variable (as n→∞). A {-1,1} random variable has bias b if E(Xi)=b. Fixing a bias b, we explore the rate at which the output bias |E(f(X1,...,X n))| can tend to zero for a function f:{-1,1}*→{-1,1}. This is accomplished by classifying the behavior of the natural normalized quantity Ξ(b)Δinff[limn→∞ n√(|E(f(X1,...,Xn))|] this infimum taken over all such f. We show that for rational b, Ξ(b)=(1/s), where (1+b/2)=(r/s) (r and s relatively prime). Developing the theory of uniform distribution of sequences to suit our problem, we then explore the case where b is irrational. We prove a new metrical theorem concerning multidimensional Diophantine approximation type from which we show that for (Lebesgue) almost all biases b, Ξ(b)=0. Finally, we show that algebraic biases exhibit curious “boundary” behavior, falling into two classes. Class 1. Those algebraics b for which Ξ(b)>0 and, furthermore, c1⩽Ξ(b)⩽c2 where c1 and c 2 are positive constants depending only on b´s algebraic characteristics. Class 2. Those algebraics b for which there exist n>0 and f: {-1,1}n→{-1,1} so that E(f(X1,...,Xn))=0. Notice that this classification excludes the possibility that n√(|E(f(X1,...,Xn ))| limits to zero (for algebraics). For rational and algebraic biases, we also study the computational problem by restricting f to be a polynomial time computable function. Finally, we discuss natural extensions where output distributions other than the uniform distribution on {-1,1} are sought
Keywords :
algebra; approximation theory; convergence of numerical methods; random processes; algebraic bias; algebraic biases; algebraic characteristics; biased source; boundary behavior; computational problem; converge rate; independent identically biased-valued random variables; metrical theorem; multidimensional Diophantine approximation; natural normalized quantity; optimally unbiased bits extraction; output bias; output distributions; polynomial time computable function; rational bias; unbiased random variable; uniform distribution; Algorithm design and analysis; Multidimensional systems; Network address translation; Polynomials; Random variables;
