Kac–Stroock type approximations for the Brownian motion from renewal processes

In the present paper we show that the processes X n = { X n ( t ) : t ∈ [ 0 , 1 ] } , n ∈ N , defined by X n ( t ) = n C ∫ 0 t ( − 1 ) L ( n u ) d u , 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.