We consider a sequence of random variables $(X_n : \Omega \longrightarrow \{-1,1\})_{n \in \mathbf{N}}$ defined on the same probability space $(\Omega, \mathscr{A}, P)$, taking values in $\{-1,1\}$, mutually independent and centered. The random variable $N_n$ counts the number of equality indices of the path $(X_1(\omega), \cdots, X_{2n}(\omega))$, and it has been shown that: $$\mathbb{E}(N_n) = \sum_{i=1}^n \frac{\binom{2i}{i}}{4^i}.$$ Deduce the equivalent: $$\mathbb{E}(N_n) \underset{n \to +\infty}{\sim} \frac{2}{\sqrt{\pi}} \sqrt{n}.$$
We consider a sequence of random variables $(X_n : \Omega \longrightarrow \{-1,1\})_{n \in \mathbf{N}}$ defined on the same probability space $(\Omega, \mathscr{A}, P)$, taking values in $\{-1,1\}$, mutually independent and centered. The random variable $N_n$ counts the number of equality indices of the path $(X_1(\omega), \cdots, X_{2n}(\omega))$, and it has been shown that:
$$\mathbb{E}(N_n) = \sum_{i=1}^n \frac{\binom{2i}{i}}{4^i}.$$
Deduce the equivalent:
$$\mathbb{E}(N_n) \underset{n \to +\infty}{\sim} \frac{2}{\sqrt{\pi}} \sqrt{n}.$$