# Download An introduction to probability theory and its applications by William Feller PDF By William Feller

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According to the central limit theorem, this state must be described, from probabilistic viewpoint, by the Gaussian distribution. Depending on the tensorial properties of the quantity under investigation, one would consider the Gaussian random quantity, the random scalar and vector fields, etc. , ergodicity. , over many realizations of a random quantity. , are of a statistical ensemble, it can be delegated to many considered THE CHANCE ON STAGE 35 space-time points. Indeed, values of a random field and distant points are practically independent and reproduce the result of independent realiza­ tions.

Thus, (v) = 0 and (v2) differs from zero. In the following the velocity v = (v 2 ) l/2 is considered as a "characteristic" one. The characteristic velocity of the motion along the x-axis is given by" * The inelegant factor 3"* arises due to the fact that the average velocity is defined, irrespective of the direction, (v2)=(v2x)+(v2y)+(v2I). This gives (v2x) = -(v 2 ). For 0 = (t>2)'/2 and vx, = (vl)"2 the equality in the text is recovered. 51 DIFFUSION Let us now presume that there exists a certain average time period T over which this velocity remains constant.

However, an intermediate asymptotic behaviour may be, or even typically is, the formation of cellular or network structure: thin channels of high intensity ("enriched phase") isolate, one from another, islands of "poor phase". An example of such intermediate-stage intermittency is provided by the structure of the universe (Shandarin and Zeldovich, 1984). In optics, the problem of structures arises when one considers the passage of light through a plate with a random profile. This situation is described formally as an evolution of a random initial distribution in a deterministic medium.