Self-averaging is a concept often discussed in statistical mechanics, probability theory, and various fields of physics and mathematics. It refers to a phenomenon in which the macroscopic properties of a system become independent of the microscopic details as the size of the system increases. In other words, the fluctuations in the microscopic behavior of individual components average out, leading to stable and predictable macroscopic behavior.
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A sequence of observables is self-averaging when its fluctuations relative to its typical macroscopic size vanish as the system grows. One common criterion for nonzero means is , which implies in probability by Chebyshev's inequality. For independent and identically distributed variables with mean and finite variance , the sample mean has relative variance . Strong dependence or broad tails can prevent this reduction; increasing size alone is not a proof.