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Published in:
2014 | OriginalPaper | Chapter
20. Ergodic Theory
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Abstract
Laws of large numbers, e.g., for independent and identically distributed random variables X
1,X
2,… , state that \(\lim_{n\rightarrow\infty} \frac{1}{n} \sum_{i=1}^{n} X_{i} =E [ X_{1} ]\) converges almost surely. Hence averaging over one realization of many random variables is equivalent to averaging over all possible realizations of one random variable. In the terminology of statistical physics this means that the time average, or path (Greek: odos) average, equals the space average. The space in space average is the probability space in mathematical terminology, and in physics it is considered the space of admissible states with a certain energy (Greek: ergon). Combining the Greek words gives rise to the name ergodic theory, which studies laws of large numbers for possibly dependent, but stationary, random variables.