Partial sum processes and continued fractions

Given{Xi}i=1∞, a sequence of real valued random variables, we define S0=0, Sn=∑i=1nXi, and define the normalized partial sum process {Yn(t):0≤t≤1} by linear interpolation of Ynin=SiSn (assuming P(Sn=0)=0 for all n≥1). In this note the convergence of Yn(⋅) in [0,1] is investigated under various assumptions on {Xi}i=1∞. Of particular interest is the special case where the Xi are the coefficients in the continued fraction expansion of a point x∈[0,1] chosen according to Gauss measure.