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		1.5.2— Self-Similarity Implies That the Moments Do Not Exist

		Self-similarity means that there are an ever larger number of smaller pieces that resemble the larger pieces. Thus there are a few big pieces, many medium sized pieces, and a huge number of small pieces.

		When we determine the average of the values of the sizes of the pieces, the most significant contribution to the average will come from either: (1) the frequent number of small values, in which case the average decreases as more data are included, or (2) the infrequent large values, in which case the average increases as more data are included.

		A moment is the average of a property raised to a power.

		The mean is the first moment. It is the average of the values of the sizes of the pieces. Self-similarity means that the mean will increase or decrease as more data are analyzed, depending on the relative contribution of the frequent small values versus the infrequent large values.

		The variance is the second moment. It is the average of the square of the difference between each value and the mean value of the sizes of the pieces. Typically, self-similarity means that the variance will increase as more data are analyzed because there are an ever larger number of self-similar fluctuations.

		We are so used to the properties of Gaussian distributions, where the mean and variance exist, that we assume that these moments must always exist. The existence of the mean and variance depend on the data satisfying the mathematical assumptions that these moments approach finite, nonzero limiting values as more data is analyzed. This fact is not well known. When the data are fractal, it does not satisfy these assumptions, the mean and variance do not exist, and therefore they are not useful to characterize the properties of fractal data.

 

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