On a generalization of the Bernoulli scheme

The paper considers a generalization of Bernoulli's scheme. We consider a sequence of independent identically distributed random variables (r.v.) $X_1, X_2,\ldots,$ taking values $-1, 0, 1$ with probabilities
$$ \mathbf{P} \{X_n=-1\}=p_1, \mathbf{P} \{X_n=0\}=p_2, \mathbf{P} \{X_n=1\}=p_3, $$
where
$$ 0<p_1<1, 0<p_2<1, 0<p_3<1 \text{ and } p_1+p_2+p_3=1. $$
If we are only interested in the number of $-1$ values in a set of $n$ r.v. $X_1, X_2,\ldots,X_n$, then the formulas used for Bernoulli schemes with success probabilities $p_1$ can be applied to such events. Similarly, occurrences of values $+1$ can be treated as occurrences of successes in a Bernoulli scheme with probability of success $p_3$. If you are interested in the appearance of only zero values of $X$, then their number in $n$ trials has a binomial $B(n, p_2)$ distribution, and the mathematical expectation of the number of such appearances is equal to $np_2$.
But in a scheme with three possible variants of values of random variables, a number of new problems appear, in comparison with the Bernoulli scheme. The paper examines some of them, limiting ourselves to situations associated with the appearance of zero values of random variables in a given scheme. Similar results for values $-$1 or $+1$ can be obtained simply by replacing the probability of $p_2$ with $p_1$ or $p_3$ in the resulting formulas. The article examines the relationship of such three-point distributions with a number of other probability laws. A short review of previously obtained results in this area is given and several new ones are added. The research begun in the previous works of the authors was continued.