Weak convergence of a time and space scaled semi-Markov process to a limiting diffusion process

Key Points

• This research aims to establish weak convergence of a specific semi-Markov process to a limiting diffusion process.
• Consider a semi-Markov process with values in R^d.
• Develop a scheme of series using time and space scaling.
• Prove the weak convergence in a Skorokhod space under specific conditions.
• Weak convergence was successfully proved in the Skorokhod space.
• Convergence holds under defined conditions on the distribution of jumps.
• The limiting behavior aligns with predictions of the diffusion process.