On probabilities of expectations

Question

Question

Let $X_1 , X_2, \cdots , X_n$ be independent and identically distributed with $P(X_i =1)=P(X_i=-1)=p$ and $P(X_i=0)=1-2p$ for all $i=1,2,\cdots,n$. Define $$ a_n=P(\prod_{i=1}^{n}X_i=1), b_n=P(\prod_{i=1}^{n}X_i=-1), c_n=P(\prod_{i=1}^{n}X_i=0)$$

Find $\lim_{n\to\infty} a_n,\lim_{n\to\infty} b_n,\lim_{n\to\infty} c_n$.

My Approach

Since $(\prod_{i=1}^n X_i=1)=1$ hence $a_n=P(\prod_{i=1}^{n}X_i=1)=P(X=1)=p$ and so $\lim_{n\to\infty}a_n=p$.

Similarly for others. Is this approach correct?

Answer 1

Since $1-2p$ is a probability, we can assume that $p \leq 1/2$. If $p<1/2$, then $c_n=1-\text{Pr[No $X_i$ is zero]=}1-(2p)^n$. Thus the third limit is 1. The sum of the three quantities is 1, so the first two limits are zero.