Randomness and Hyper-randomness by Igor I. Gorban

Author:Igor I. Gorban
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(6.2)

where is the sample mean of the average fluctuations; and to define the concept of statistical stability of the process with respect to the standard deviation, we can put the analysis of changes in the sample variance

(6.3)

of the fluctuations in the sample standard deviation

(6.4)

where is the average of the sample standard deviations.

For a random sample the simplest variants in the formulation may be the following (Gorban 2011).

A random sample (sequence of random variables) X 1 , X 2 , … is statistically stable with respect to the average if, when the sample size N goes to infinity, the expectation of the sample variance (6.1) goes to zero.

A random sample (sequence of random variables) X 1 , X 2 , … is statistically stable with respect to the standard deviation if, when the sample size N goes to infinity, the expectation of the sample variance (6.3) goes to zero.

Random samples (sequences of random variables) that do not satisfy these definitions are considered to be statistically unstable with respect to the average and standard deviation, respectively.

The type of convergence in this case is not important. However, to make the definitions with the necessary mathematical rigor, we shall refer to convergence in probability here.

A reduction of the variance of the sample mean with increasing data volume can be caused, not only by stabilization of the average, but also by decreasing dispersion in the initial process. To mitigate this effect, it seems reasonable to redefine the concept of statistical stability as follows.

A random sample (sequence of random variables) X 1 , X 2 , … is statistically stable with respect to the average if, when the sample size N goes to infinity, the parameter of statistical instability with respect to the average , viz.,

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