Kac-Stroock type approximations for the Brownian motion

2025/11/21 by Bardina, Xavier, Boukfal, Salim
#60F05 #60F17 #60G50 #60K05 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2511.17281

Abstract

In the present paper we show that the processes X_n = \X_n(t) \colon t ∈ [0,1]\, n ∈ ℕ, defined by X_n(t) = √(n)C∫₀^t (-1)^L(nu) du, where L = \L(t) \colon t ≥ 0\ is a renewal processes whose inter-arrival times satisfy some integrability conditions and C > 0 is some normalizing constant, weakly converge, in the space of continuous functions over [0,1], C([0,1]), to the Brownian motion as n approaches infinity. Thus, generalizing the result of D. W. Stroock (1982), where L is taken to be a standard Poisson process. In particular, we see that these results are a mere consequence of Donsker's invariance principle.

Citations

• Strong limit of processes constructed from a renewal proces
• Weak approximation of the complex Brownian sheet from a L 'evy sheet and\n applications to SPDEs

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