In this section, we introduce and demonstrate our main results within a randomized framework, offering insights into long-term behavior. We will employ the symbols a.s -→ and D -→ to denote convergence in almost surely and distribution, respectively. Here, N µ, σ 2 represents a random variable following a normal distribution with mean µ and variance σ 2 , and M ult(n, p) represents a random variable following a multinomial distribution with parameters n and p.