Passeios aleatórios: teoria, propriedades e aplicações

Abstract

The present Undergraduate Thesis focuses on the study of the random walk, a fundamental probabilistic model in the theory of stochastic processes and applied statistics. The study adopts a theoretical approach with illustrative examples, exploring the properties, classifications, and applications of random walks in different contexts, ranging from the simple one-dimensional case to versions with barriers and bidimensional and tridimensional generalizations. Based on an axiomatic construction, the work presents theorems and propositions that formalize the behavior of the process, with emphasis on classic results such as the Law of Large Numbers and the Central Limit Theorem. These theoretical foundations are illustrated through computational simulations carried out in the RStudio environment, which allow for the visualization of trajectory evolution and for understanding the role of randomness and variance in the walker’s behavior over time. The work also presents applications of random walks in gambling problems, queueing processes, and birth–death models, highlighting their practical relevance. From an applied perspective, we emphasize the importance of this topic as an essential tool in statistics, since simulation and graphical visualization enhance the development of probabilistic reasoning and conceptual understanding. We aim to convey the view that the random walk constitutes a scientific model of great value, as it combines mathematical rigor, computational experimentation, and practical applicability, serving as a point of convergence between theory, practice, and statistical training.