It is known that in the case of nonidentically distributed random variables certain regularity conditions should be imposed for the moments or for the distributions (e.g., conditions (2.24) and (2.25)).
We next discuss regularity of random variables and processes.
In general, texture synthesis methods in temporal domain are more capable to synthesize dynamic texture that shows repetitive motions, motions with regularity or random motions with fast speed.
Thanks to statistics and statistical probability, the oddity becomes credible that also contingency has its own order, that is to say, that there is some regularity in random distributions and that such regularity can even be calculated.
Using the Lindeberg-Feller central limit theorem for independent random variables under certain regularity conditions (see Appendix D) [24], it can be shown that is distributed under two hypotheses as (17).
Our main result states that, under a suitable assumption on the regularity of the random variables, the integrated density of states of such random operators is locally Hölder continuous at energies below the bottom of the essential spectrum of the background operator for any nonzero disorder, and at energies in the unperturbed spectral gaps, provided the randomness is sufficiently small.
Surprisingly, performance in a random visual search, without any regularity, was better than performance in a mixed design search that contained a beneficial regularity.
Field (1987) as well as Doi and Lewicki (2005) have described how "natural images are not random, instead they exhibit statistical regularities" and have used such statistical regularities for constraining the properties of receptive fields.
Moreover, the accuracy of the representation improves no matter the regularity of the random quantity of interest.
We prove a concentration inequality which asserts that, under some mild regularity conditions, every random variable defined on the product of sufficiently many probability spaces exhibits pseudorandom behavior.
