Let a Wiener process W be observed on some nonbounded set T ⊂ R + . We show that after reasonable time- and space-rescaling the correspondent interpolated sample paths converge with probability one (in the usual sense of functional laws) to some limit set in C[0, 1]. The complete description of the limit set for arbitrary T is given. Unlike the various known functional laws, this limit set is not necessarily convex. We also supply an invariance principle which permits us to obtain the same results for partially observed processes generated by the sums of i.i.d. random variables.
Scopus subject areas
- Statistics and Probability
- Statistics, Probability and Uncertainty