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Lindvall, T. , Weak Convergence of Probability Measures and Random Functions in the Function Space D [0, ∞), Journal of Applied Probability , 10 , No. 1, 109–121, 1973. Lindvall T

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Abstract  

First, sufficient conditions are given for a triangular array of random vectors such that the sequence of related random step functions converges towards a (not necessarily time homogeneous) diffusion process. These conditions are weaker and easier to check than the existing ones in the literature, and they are derived from a very general semimartingale convergence theorem due to Jacod and Shiryaev, which is hard to use directly. Next, sufficient conditions are given for the convergence of stochastic integrals of random step functions, where the integrands are functionals of the integrators. This result covers situations which cannot be handled by existing ones.

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of universality theorems are based on Proposition 8, the Mergelyan theorem [ 16 ] on the approximation of analytic functions by polynomials, and equivalents of weak convergence of probability measures. sc>Proof of Theorem 5. By the Mergelyan theorem

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Abstract  

Let M n denote the n-th moment space of the set of all probability measures on the interval [0, 1], P n the uniform distribution on the set M n and r n + 1 the maximal range of the (n + 1)-th moments corresponding to a random moment point C n with distribution P n on M n. We study several asymptotic properties of the stochastic process (r nt⌋+1)t∈[0,T] if n → ∞. In particular weak convergence to a Gaussian process and a large deviation principle are established.

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Summary  

Consider the boundary case in a one-dimensional super-critical branching random walk. It is known that upon the survival of the system, the minimal position after n steps behaves in probability like
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\frac{3} {2}$$ \end{document}
log n when n → ∞. We give a simple and self-contained proof of this result, based exclusively on elementary properties of sums of i.i.d. real-valued random variables.
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Abstract  

Let {X t : 0 ≦ t ≦ 1} be a centered stationary Gaussian process, with correlation function satisfying the condition ρ(t) = 1 − t β L(t), 0 < β < 2, and let L be a slowly varying function at zero. Observing the process at points i/N, i = 0,1,..., N and considering ¦X i/NX (i-1)/N¦p with p > 0, we study the properties of the Donsker line associated with p-th order variations
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\sum\limits_{i = 1}^{[N{\text{ }}t]} {|X_{i/N} } - {\text{ }}X_{(i - 1)/N} |^p$$ \end{document}
. We also study the relationship between the number of crossings of a regularization of the initial process and the local time of the initial process. The results depend on the values of β.
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Abstract  

This note concerns the asymptotic behavior of a Markov process obtained from normalized products of independent and identically distributed random matrices. The weak convergence of this process is proved, as well as the law of large numbers and the central limit theorem.

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