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We prove that a planar random walk with bounded increments and mean zero which is conditioned to stay in a cone converges weakly to the corresponding Brownian meander if and only if the tail distribution of the exit time from the cone is regularly varying.
A Tauberian theorem and its probability interpretation
Spitzer, F. (1960) · 1960
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Convergence of Probability Measures
Billingsley, P. (1968) · 1968
Earlier work this paper cites.
A unified theory of regularly varying sequences
Bojanic, R. and Seneta, E. (1973) · 1973
Earlier work this paper cites.
Functional central limit theorems for random walks conditioned to stay positive
Iglehart, D. L. (1974) · 1974
Earlier work this paper cites.
On a functional central limit theorem for random walks conditioned to stay positive
Bolthausen, E. (1976) · 1976
Earlier work this paper cites.
A note on Bojanic-Seneta theory of regularly varying sequences
Weissman, I. (1976) · 1976
Cited alongside, same era.
Weak convergence to Brownian meander and Brownian excursion
Durrett, R. T., Iglehart, D. L. and Miller, D. R. (1977) · 1977
Cited alongside, same era.
Excursions in a cone for two-dimensional Brownian motion
Shimura, M. (1985) · 1985
Cited alongside, same era.
A note on the measurability of convex sets
Lang, R. (1986) · 1986
Cited alongside, same era.
A limit theorem for two-dimensional random walk conditioned to stay in a cone
Shimura, M. (1991) · 1991
Later among the works it cites.
Potential theory in conical domains
Varopoulos, N. Th. (1999) · 1999
Later among the works it cites.
Contributions à l’étude d’une marche aléatoire centrifuge et théorèmes limites pour des processus aléatoires conditionnés
Garbit, R. (2008) · 2008
Later among the works it cites.
Brownian motion conditioned to stay in a cone
Garbit, R. (2009) · 2009
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