The displacements of a particle from classical Brownian motion form a Gaussian distribution. However, the distribution of variables drawn from a distribution with infinite variance as opposed to a finite variance in classical Brownian motion is not Gaussian. We show that the sum of a large number of lognormal variables tends to a Levy subordinator distribution when \(0 < \alpha < 1\). This distribution is surprisingly common in real-life stochastic systems such as the stock market and relativistic Brownian motion.