@article{123558b33bd54704a81dbba6124d1f1f,

title = "Approximating probability distributions using small sample spaces",

abstract = "We formulate the notion of a {"}good approximation{"} to a probability distribution over a finite abelian group double-struck G sign. The quality of the approximating distribution is characterized by a parameter ε which is a bound on the difference between corresponding Fourier coefficients of the two distributions. It is also required that the sample space of the approximating distribution be of size polynomial in log |double-struck G sign| and 1/ε. Such approximations are useful in reducing or eliminating the use of randomness in certain randomized algorithms. We demonstrate the existence of such good approximations to arbitrary distributions. In the case of n random variables distributed uniformly and independently over the range {0, . . ., d - 1}, we provide an efficient construction of a good approximation. The approximation constructed has the property that any linear combination of the random variables (modulo d) has essentially the same behavior under the approximating distribution as it does under the uniform distribution over {0, . . ., d - 1}. Our analysis is based on Weil's character sum estimates. We apply this result to the construction of a non-binary linear code where the alphabet symbols appear almost uniformly in each non-zero code-word.",

author = "Yossi Azar and Rajeev Motwani and Joseph Naor",

note = "Funding Information: Mathematics Subject Classi cation (1991): 60C05, 60E15, 68Q22, 68Q25, 68R10, 94C12 * Part of this work was done while the author was at the Computer Science Department, Stanford University and supported by a Weizmann fellowship and Contract ONR N00014-88-K-0166. Funding Information: y Supported by an Alfred P. Sloan Research Fellowship, an IBM Faculty Development Award, grants from Mitsubishi Electric Laboratories and OTL, NSF Grant CCR-9010517, and NSF Young Investigator Award CCR-9357849, with matching funds from IBM, Schlumberger Foundation, Shell Foundation, and Xerox Corporation. Funding Information: z Part of this work was done while the author was visiting the Computer Science Department, Stanford University, and supported by Contract ONR N00014-88-K-0166, and by Grant No. 92-00225 from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel.",

year = "1998",

doi = "10.1007/PL00009813",

language = "אנגלית",

volume = "18",

pages = "151--171",

journal = "Combinatorica",

issn = "0209-9683",

publisher = "Janos Bolyai Mathematical Society",

number = "2",

}