Decomposition of Recurrent Choices into Stochastically Independent Counts

Consider a fixed set of alternatives {1, …, k } available at each of a random number N of choice opportunities, exactly one alternative from {1, …, k } being selected at each such choice opportunity. Let the distribution of the conditional random vector {X₁, …, X_k | Σ X_i = N} be known, X_i being the number of times the i th alternative is chosen. What is the class of all possible (k + 1)-vectors of probability mass functions {R(n), R₁(x₁), …, R_k(x_k)} such that if N is distributed according to R(n), the components of the unconditional random vector {X₁, …, X_k} are mutually independent random variables distributed according to R₁(x₁), …, R_k(z_k), respectively? This paper presents a complete and constructive solution of this problem for a broad class of conditional random vectors {X₁, …, X_k | Σ X_i - N}. In particular, the solution applies to all situations where the sequence of potentially observable values of X_i (for any i = 1, …, k) forms an interval of consecutive integers, finite or infinite. When, for some i = 1, …, k, this sequence contains finite gaps, the solution may or may not apply in its entirety. it is suggested, however, that in many, if not all, such situations the representation of recurrent choices by conditional vectors {X₁, …, X_k | Σ X_i = N} may not be optimal in the first place. A more natural representation, to which the solution proposed applies universally, is provided by {M₁, …, M_k | Σ M_i = M}, where M_i is the ordinal position of an observable value of X_i in the sequence of all such values.