  We consider a Brownian particle moving on a ring. We study the probability
distributions of the total number of turns and the net number of
counter-clockwise turns the particle makes till time t. Using a method based on
the renewal properties of Brownian walker, we find exact analytical expressions
of these distributions. This method serves as an alternative to the standard
path integral techniques which are not always easily adaptable for certain
observables. For large t, we show that these distributions have Gaussian
scaling forms. We also compute large deviation functions associated to these
distributions characterizing atypically large fluctuations. We provide
numerical simulations in support of our analytical results.
