By Aitken R.J.
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Extra info for A Statistical Study of the Visual Double Stars in the Northern Sky (1915)(en)(5s)
We shall call X the integrand and Z the integrator. Both are now processes. For a guide let us review the construction of the ordinary Lebesgue– Stieltjes integral x dz on the half-line; the stochastic integral X dZ that we are aiming for is but a straightforward generalization of it. The Lebesgue–Stieltjes integral is constructed in two steps. First, it is defined on step functions x. This can be done whatever the integrator z . If, however, the Dominated Convergence Theorem is to hold, even on as small a class as the step functions themselves, restrictions must be placed on the integrator: z must be right-continuous and must have finite variation.
11). There are several ways to attach a numerical size to a function f ∈ L0 , the most common 17 being f 0 = f 0;P def = inf λ : P |f | > λ ≤ λ . It measures convergence in probability, also called convergence in measure; namely, fn → f in probability if dist0 (fn , f ) def = fn − f −−−→ 0 n→∞ 0. 1). There is also a whole slew of absolute-homogeneous but non-subadditive functionals, one for every α ∈ R, that can be used to describe the topology of L0 (P): f [α] = f = inf λ > 0 : P[|f | > λ] ≤ α .
When we write X = Y for two processes X, Y we mean generally that X and Y are indistinguishable. When the probability P ∈ P must be specified, then we talk about P-nearly empty sets or P-nearly vanishing random variables, properties holding P-nearly, processes indistinguishable with P or P-indistinguishable, and P-evanescent processes. A set N is nearly empty if someone with a finite if possibly very long life span t can measure it (N ∈ Ft ) and find it to be negligible, or if it is the countable union of such sets.
A Statistical Study of the Visual Double Stars in the Northern Sky (1915)(en)(5s) by Aitken R.J.