Another approach to Brownian motion

For the partial sums Sn = Y1 + . . . + Yn of a centered stationary strongly mixing sequence {Yi} with finite second moment, the well-known sufficient conditions for the central limit theorem are that Var(Sn)/n is slowly varying as n → ∞ and the sequence {S2 n/σn} is uniformly integrable (σn = Var(Sn)). The conditions are checkable under various mixing conditions and they lead to the central limit theorem under the normalization σn(see, Denker (1986) and Peligrad (1986) for a survey). Dehling, Denker and Philipp (1986) proved an interesting central limit theorem using the nontraditional normalization ρn = E|Sn|. One of their results, Theorem 3, roughly states that if both sequences σn/n and ρn/ √ n are slowly varying as n → ∞, then the central limit theorem holds. On the other hand, Braverman, Mallows and Shepp (1995) showed that if the absolute moments of partial sums of i.i.d. symmetric variables are equal to those of normal variables, then the marginals have normal distribution. This fact suggested the conjecture that probably the absolute moments alone characterize the homogeneous process with independent increments (see Bryc (2002) for a discussion on this topic and related conjectures). Our main interest in this topic is to prove some of these conjectures and to apply them to understand the nature of the intricate normalization in Dehling, Denker and Philipp (1986).