Non-Gaussian stable stochastic models have attracted growing interest in recent years, due to their connections to limit theorems and due to empirical evidence pointing to heavier-than-Gaussian probability tails in many natural situations. We study the structure of two broad classes of stable stochastic processes through some convergence results. In the first half of the thesis, we study the integrated periodogram for discretetime infinite moving average processes with i.i.d. stable noise. We show that for such processes, a collection of weighted integrals of the periodogram, considered as a function-indexed stochastic process, converges weakly to a limit which can be represented as an infinite Fourier series with i.i.d. stable coefficients. The convergence works under certain assumptions on the Fourier coefficients of the index functions. We also extend the weak convergence results to stochastic volatility processes with stable noise, which are of interest in financial time series analysis. In the second half, we describe a family of continuous-time stable processes with stationary increments that are asymptotically or exactly self-similar. We show that they arise naturally as a large time scale limit in a situation where many users perform independent random walks and collect heavy-tailed random rewards depending on their position on the integer line. We study various properties of the limiting process. This work generalizes an earlier construction by Cohen and Samorodnitsky (2006).