A Comprehensive Study of the Almost Sure Convergence of Boundary Points in Continuous-Time Random Walks

In this work, we rigorously derive the almost sure limit points for suitably normalized partial sums of continuous-time random walks. These walks are a specialized form of random walks, subordinated to an underlying renewal process. This mathematical framework is widely employed in physics as a robust model for capturing the complexities of anomalous diffusion, which deviates from classical diffusion behaviours observed in standard stochastic processes.