# On the asymptotic behaviour of the empirical random field of the Brownian motion

Bolthausen, E (1984). On the asymptotic behaviour of the empirical random field of the Brownian motion. Stochastic Processes and their Applications, 16(2):199-204.

## Abstract

Let ξt, t greater-or-equal, slanted 0, be a d-dimensional Brownian motion. The asymptotic behaviour of the random field ƒmaps to∫t0ƒ(ξs) ds is investigated, where ƒ belongs to a Sobolev space of periodic functions. Particularly a central limit theorem and a law of iterated logarithm are proved leading to a so-called universal law of iterated logarithm.

## Abstract

Let ξt, t greater-or-equal, slanted 0, be a d-dimensional Brownian motion. The asymptotic behaviour of the random field ƒmaps to∫t0ƒ(ξs) ds is investigated, where ƒ belongs to a Sobolev space of periodic functions. Particularly a central limit theorem and a law of iterated logarithm are proved leading to a so-called universal law of iterated logarithm.

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