Why is $Y_n$ not a bernoulli process?

Question

for $n \in \mathbb{N}$, let $X_n$ be a Bernoulli process with parameter $p = \frac12$,

let $N = \min \{n \geq 2: X_1 \neq X_n \}$

for $n \in \mathbb{N}$, let $Y_n = X_{N +n -2}$. in a question it says to prove that it's not a Bernoulli process but I can't see why.

first of all, $\mathbb{P}[Y_n = 1] = \mathbb{P}[X_{N +n -2}= 1] = \mathbb{P}[X_{m}= 1] = \frac{1}{2}$ since $m \in \mathbb{N}$

and since for $n \in \mathbb{N}$, $X_n = 0 \text{ or } 1 $ and $m \in \mathbb{N}$ then shouldn't we have $Y_n = 0 \text{ or } 1 $ as well ?

Answer 1

By definition of $N$, we have that $\mathbb P(X_{N-1}=X_N)=0$. Thus, $\mathbb P(Y_1=Y_2)=0$, whereupon $(Y_1,Y_2)$ is not a Bernoulli process (since that would require $\mathbb P(Y_1=Y_2)=\tfrac12$).