There are applications, however, in which sensitivity to relative displacement or refractive index differences between reference and sample paths is not desirable; alternatively, one may be interested in the measurement of some other property.
22.
When the number of components \ bar { k } goes to infinity, continuous-time MSM converges to a multifractal diffusion, whose sample paths take a continuum of local H�lder exponents on any finite time interval.
23.
As in most of the definitions of Brownian motion it is required that the sample paths are continuous almost surely, one then uses kolmogorov continuity theorem to construct a continuous modification of the process constructed by Kolmogorov extension theorem.
24.
The advantage of this technique is twofold : it reduces the number of normal samples to be taken to generate " N " paths, and it reduces the variance of the sample paths, improving the accuracy.
25.
Instead of generating sample paths randomly, it is possible to systematically ( and in fact completely deterministically, despite the " quasi-random " in the name ) select points in a probability spaces so as to optimally " fill up " the space.
26.
Almost surely, a sample path " X " of Brownian motion in the Euclidean plane has Hausdorff dimension equal to 2, but the 2-dimensional Hausdorff measure " ? " 2 ( " X " ) is zero.
27.
If a continuous-time real-valued stochastic process meets certain moment conditions on its increments, then the Kolmogorov continuity theorem says that there exists a modification of this process that has continuous sample paths with probability one, so the stochastic process has a continuous modification or version.
28.
Some authors define a " continuous ( stochastic ) process " as only requiring that the index variable be continuous, without continuity of sample paths : in some terminology, this would be a continuous-time stochastic process, in parallel to a " discrete-time process ".
29.
Keith, Timothy Z ., Sheila M . Pottebaum, and Steve Eberhart . 1986 . " Effects of Self-concept and Locus of Control on Academic Achievement : A Large-sample Path Analysis . " Journal of Psychoeducational Assessment 4 ( 1 ) : 61-72.
30.
The article Dirichlet process begins " In probability theory, a Dirichlet process over a set S equipped with a suitable sigma-algebra is a stochastic process whose sample path is a probability distribution on S . " I think I sort of understand this, but to be sure, what is a sample path?
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