In this thesis, we study the Hardy-Littlewood majorant problem randomized via stochastic processes. Stationary processes, random walks and the Poisson processes are used for randomization, and we show the Hardy-Littlewood majorant property holds almost surely for deterministic sets perturbed by these processes. We also perturb a very large class of sparse sets, including the Green-Ruzsa set by Poisson processes and demonstrate that the Hardy-Littlewood majorant property remains valid up to a negligible probability. Additionally, we investigate how randomization affects the expected values of L2-norm and L4-norm of an exponential sum whose frequencies constitute an arithmetic progression of larger step size. Furthermore, we estimate the expected value of the Ln-norms, n ∈ 2N of exponential sums whose frequencies are randomized via Poisson processes, and these norms can be interpreted as lattice points in regions or solutions of diophantine equations in an average sense.