Distributions Determined by Cutting a Simplex with Hyperplanes

Suppose that X₁, X₂, …, Xₙ are random variables uniformly distributed over the simplex of points x₁, x₂, ⋯, xₙ such that xᵢ 0 for i = 1, 2, ⋯, n and ∑ⁿ₁ xᵢ 1. The distribution of X = ∑ⁿ₁ cᵢXᵢ for constants cᵢ satisfying c₁ > c₂ > ⋯ > cₙ > 0 is easily seen to be given by {equation*}{align*}{1.1}P(X x) &= 1 - ∑^rj = 1 (c_j - x)^n c_j∏i ≠ j (c_j - c_i)⁻¹ \\ &= x^n∏^ni = 1 c_i⁻¹ - ∑^nj = r + 1 (x - c_j)^n c_j∏i ≠ j (c_i - c_j)^ {-1}{align*}{equation*} for 0 x c₁, where r is the largest positive integer such that x cᵣ. A geometric derivation is given in Section 2 which relies on a principle of inclusion and exclusion and identifies the terms in (1.1) as volumes of various simplices. It is remarked in Section 3 that the argument of Section 2 extends in principle to give the joint distribution of a set of different linear combinations of X₁, X₂, ⋯, Xₙ. Finally, the relations to the theory of order statistics from a uniform distribution and to the theory of serial correlation are noted. A simple derivation of (1.1) will now be sketched. Suppose that Z₁, Z₂, ⋯, Zn + 1 are $n + 1$ independent random variables each with the standard exponential density function exp (-z) for 0 z < ∞. By a well-known argument involving the partial fraction expansion of the characteristic function (cf, Box (1954) Theorem 2.4), the density of Z = ∑ⁿ₁ cⱼZⱼ is found to be ∑ⁿ₁ (wⱼ/cⱼ) exp (-z/cⱼ) where {equation*}{1.2}w_j = cⁿ⁻¹_j∏i ≠ j (c_j - c_i)⁻¹.{equation*} Now a set of random variables X₁, X₂, ⋯ Xₙ uniformly distributed over the simplex as described above may be represented as Xⱼ = Zⱼ/Y for j = 1, 2, ⋯, n where Y = ∑ⁿ⁺¹₁ Zⱼ, and the set X₁, X₂, ⋯, Xₙ thus created is distributed independently of Y. Thus X = ∑ⁿ₁ cⱼXⱼ may be represented as {equation*}{1.3}X = Z/Y{equation*} where X and Y are independent. In a relation like (1.3) the marginal distributions of Z and Y together with the fact of independence of X and Y are sufficient to determine uniquely the distribution of X. To see this, note that cf (log X) (log Y) = cf (log Z), where cf denotes the characteristic function. Thus, the characteristic function of log X is determined by those of log Y and log Z. Now if Z were known to have the scaled exponential density c⁻¹ exp (-z/c) for z 0 while Y had the gamma density yⁿ exp (-y)/n! for y 0, it would follow that X had the scaled beta density nc⁻¹(1 - x/c)ⁿ⁻¹ for 0 x c. But the density of Z is, in fact, a mixture of scaled exponentials. Thus, the density of X is the same mixture of scaled betas and is given by ∑ⁿ₁ wⱼfⱼ(x), where the wⱼ's are defined by (1.2) and {equation*}{align*}{1.4}f_j(x) &= nc⁻¹_j (1 - x/c_j)ⁿ⁻¹ 0 x c_j \\ &= 0{align*}.{equation*} The result just given yields immediately the first line of (1.1). To derive the second line of (1.1), set Xn + 1 = 1 - ∑ⁿ₁ Xⱼ and note that c₁ - X = c₁ Xn + 1 + (c₁ - cₙ)Xₙ + ⋯ + (c₁ - c₂)X₂ has a distribution of the same type as X. Applying the first line of (1.1) to c₁ - X yields the second line of (1.1). The distribution of ∑n + 1₁dᵢXᵢ is also given by (1.1), where X₁, X₂, ⋯, Xn + 1 are uniformly distributed over the simplex xᵢ 0 for i = 1, 2, ⋯, n + 1 and ∑n + 1₁ xᵢ = 1, and where constants dᵢ satisfy the condition {equation*}{1.5}d_1 > d_2 > ⋯ > d_r x > dr + 1 > ⋯ > dn + 1.{equation*} To see this, let cᵢ = dᵢ - dn + 1 and notice that ∑n + 1₁ dᵢXᵢ = X + dn + 1.