Some probability limit theorems
TL;DR: In this article, the authors describe new phenomena, whereas others are refinements of known theorems, which appear to be of a type which cannot be reduced to the study of a functional of the Wiener process.
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Abstract: Some of the present results describe new phenomena, whereas others are refinements of known theorems. While they deal with the limiting behavior of certain functionals of the partial sums, they appear to be of a type which cannot be reduced to the study of a functional of the Wiener process. With the exception of Theorem 6, nothing but (1) will be assumed about the common distribution of the Xit We define the probabilities:
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Citations
On a functional contraction method
Ralph Neininger,Henning Sulzbach +1 more
TL;DR: This approach is an extension of the so-called contraction method to the space C[0,1] of continuous functions endowed with uniform topology and the space D[ 0,1) of cadlag functions with the Skorokhod topology, and develops the use of the Zolotarev metrics on the spaces C and D.
To the Invariance Principle
TL;DR: In this paper, the central limit theorem is applied to sums of random variables on the basis of which the functionals are constructed, and it is shown that the weak convergence of each of these appropriately generalized functionals is equivalent to the applicability of the central-limit theorem to sum-of-random variables.
7
Convergence of two functionals of asymptotically normal sums of independent random variables
TL;DR: In this article, the central limit theorem for the sums of independent random variables that enter the arguments of two functionals is shown to be sufficient and sufficient for one of them to have a weak limit.
References
•Book
Grundzuge Einer Allgemeinen Theorie Der Linearen Integralgleichungen
David Hilbert
- 27 Oct 2009
TL;DR: In this article, a Funktion K(s, t) heise der Kern der Integralgleichungen, i.e., the bestimmende Funktion of a reelle Veranderlichen.
560
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