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An optical central limit theorem

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Abstract

The central limit theorem of statistics states that the arithmetic average of N (finite) random variables obeys a normal probability law. A requirement of the theorem is that the probability density common to the random variables have finite variance. This is not the case in optics, where for example the probability density on a photon position xi in an image can obey a sinc2 x law. The variance of this law is infinite. Then what probability density is obeyed by the arithmetic average x¯ of the xi in the optical case? It is shown that the law is Cauchy, i.e., of the form px¯(x)=(a/π)(x2+a2)1, a = const. Surprisingly, both the Cauchy form of the law and its free parameter a are entirely independent of the aberrations of the imaging system.

© 2003 Optical Society of America

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