Limit theorems for random walks
TLDR
In this article, a random walk S τ which is obtained from a simple random walk by a discrete time version of Bochner's subordination was considered and it was shown that S τ converges in the Skorohod space to the symmetric α-stable process B α.About:
This article is published in Stochastic Processes and their Applications.The article was published on 2017-10-01 and is currently open access. It has received 26 citations till now. The article focuses on the topics: Random walk & Subordinator.read more
Citations
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Critical Exponents for Long-Range $${O(n)}$$ O ( n ) Models Below the Upper Critical Dimension
TL;DR: In this article, the authors considered the critical behavior of long-range (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 28, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48
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Critical exponents for long-range O(n) models below the upper critical dimension
TL;DR: The proof adapts and applies a rigorous renormalisation group method developed in previous papers with Bauerschmidt and Brydges for the nearest-neighbour models in the critical dimension, and is based on the construction of a non-Gaussian renormisation group fixed point.
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Critical two-point function for long-range $O(n)$ models below the upper critical dimension
TL;DR: In this article, the authors considered the long-range spin-spin interactions and weakly self-avoiding walk on a 4-dimensional lattice spin model and showed that the two-point function at the critical point decays with distance.
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Invariance principles for random walks in random environment on trees
TL;DR: In this article, the scaling limit of the edge-reinforced random walk on a size-conditioned Galton-Watson tree with finite variance as a Brownian motion in a random Gaussian potential on the CRT with a drift proportional to the distance to the root.
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Approximation of Fractional Local Times: Zero Energy and Derivatives
TL;DR: In this paper, Jeganathan et al. derived conditions under which these processes verify a (possibly uniform) law of large numbers, as well as a second order limit theorem.
References
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Book
Convergence of Probability Measures
TL;DR: Weak Convergence in Metric Spaces as discussed by the authors is one of the most common modes of convergence in metric spaces, and it can be seen as a form of weak convergence in metric space.
Book
Foundations of modern probability
TL;DR: In this article, the authors discuss the relationship between Markov Processes and Ergodic properties of Markov processes and their relation with PDEs and potential theory. But their main focus is on the convergence of random processes, measures, and sets.
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Some theorems on stable processes
R. M. Blumenthal,R. K. Getoor +1 more
TL;DR: In this paper, the authors extend the Fourier inversion theorem for radial functions and obtain the asymptotic distribution of the eigenvalues for certain operators that are naturally associated with the symmetric stable processes.
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Some Useful Functions for Functional Limit Theorems
TL;DR: This paper facilitates applications of the continuous mapping theorem by determining when several important functions and sequences of functions preserve convergence.