Kac-Stroock type approximations for the Brownian motion from renewal processes
In the present paper we show that the processes X={X(t):t∈[0,1]}, n∈N, defined by X(t)=nC∫ (-1)du, where L={L(t):t≥0} is a renewal process whose inter-arrival times satisfy some integrability conditions and C>0 is some normalizing constant, weakly converge, in the space of continuous functions ov...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2026 |
| País: | España |
| Institución: | Universitat Autònoma de Barcelona |
| Repositorio: | Dipòsit Digital de Documents de la UAB |
| Idioma: | inglés |
| OAI Identifier: | oai:dnet:uabarcelona_::b1c583cfc91e830cf92d65d9531718a7 |
| Acceso en línea: | https://ddd.uab.cat/record/327372 https://dx.doi.org/urn:doi:10.1016/j.spl.2026.110714 |
| Access Level: | acceso abierto |
| Palabra clave: | Brownian motion Renewal process Weak convergence Kac-Stroock approximations |
| Sumario: | In the present paper we show that the processes X={X(t):t∈[0,1]}, n∈N, defined by X(t)=nC∫ (-1)du, where L={L(t):t≥0} is a renewal process 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. This result thus generalizes the well-known 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. |
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