The General Moment Problem, A Geometric Approach

Let g₁, , gₙ and h be given real-valued Borel measurable functions on a fixed measurable space T = (T, A). We shall be interested in methods for determining the best upper and lower bound on the integral (h) = Th (t) (dt), given that is a probability measure on T with known moments (gⱼ) = yⱼ, j = 1, , n. More precisely, denote by M^+ = M^+ (T) the collection of all probability measures on T such that (|gⱼ|) < (j = 1, , n) and (|h|) <. For each y = (y₁, , yₙ) Rⁿ, consider the bounds L (y) = L (y | h) = (h), U (y) = U (y | h) = (h), where is restricted by M^+ (T) ; (g₁) = y₁, , (gₙ) = yₙ. If there is no such measure we put L (y) = +, U (y) = -. In many applications, h is the characteristic function (indicator function) h = Iₛ of a given measurable subset S of T. In that case we usually write instead L (y | Iₛ) = Lₛ (y), U (y | Iₛ) = Uₛ (y). Thus, Lₛ (y) (S) Uₛ (y) are the best possible bounds on the probability mass (S) contained in S, given that M^+ and that (g) = y. Here, g denotes the mapping g: T Rⁿ defined by g (t) = (g₁ (t), , gₙ (t) ). By g₀ we shall denote the function on T with g₀ (t) = 1 for all t T. The following tentative method for finding L (y h) may be said to go back to Markov 8 and Riesz 13, see 7. Choose an (n + 1) -tuple d^ = (d₀, d₁, , dₙ) of real numbers such that d₀ + d₁g₁ (t) + + dₙgₙ (t) h (t) for all t T, and define B (d^) = \z Rⁿ: z = g (t) for some t T with ⁿ₉=₀ dⱼgⱼ (t) = h (t) \. Then L (y h) = d₀ + ⁿ₉=₁ dⱼyⱼ for each y conv B (dᵃst), (conv = convex hull). The main purpose of the present paper is to investigate the merits of this method and certain more general methods. It turns out (Theorem 5) that for almost all y Rⁿ there exists at most one admissible d^ with y conv B (d^). Moreover, provided y (V) where V = conv g (T), there exists at least one such d^ if and only if there exists a measure M^+ with (g) = y and (h) = L (y h). A sufficient condition for the latter would be that T has a compact topology with respect to which g is continuous and h is lower semi-continuous. More interesting is a related method for finding L (y h), see Theorem 6, which will work for each y (V) as soon as g is bounded. The situation where y (V) is discussed in Section 4. It appears that the assumption y (V) is a rather natural one. We have chosen to develop the important special case h = Iₛ in a partly independent manner, see the Sections 5, 6, and 7. In this case, the (n + 1) -tuple d^ must satisfy align*d₀ + ⁿ₉=₁ dⱼzⱼ 1 for all z g (T), \\ 0 for all z g (S'). align* Here, S' denotes the complement of S in T. Assuming that d₁, , dₙ are not all zero, let us associate to d^ the pair of hyperplanes H and H' with equations ⁿ₉=₁ ₃䲛ₙ䲛 = 1 - d₀ and ⁿ₉=₁ dⱼzⱼ = -d₀, respectively. This pair is such that H, H' are distinct parallel hyperplanes with g (S') and H on opposite sides of H' and g (T) and H' on the same side of H; such a pair H, H' will be said to be admissible. Observe that B (d^) = (g (S) cap H) cup (g (S') cap H'), with H, H' as the admissible pair determined by d^. The present (n + 1) -tuple d^ is useful, for determining Lₛ (y) = L (y Iₛ) for at least some points y, only when both g (S) cap H 0 and g (S') cap H' 0. That is, H' should not only support the set g (S') but even "intersect" it; similarly, H and g (S). Fortunately, one can usually replace "intersect" by "touch". More precisely (Corollary 13), if H and H' form an admissible pair as above then Lₛ (y) = d₀ + ⁿ₉=₁ dⱼyⱼ for each point y such that both y (V), y conv\H cap {conv g (S) \} cup \H' cap {conv g (S') \}, a bar denoting closure. Provided g is bounded the latter generalization will yield the value Lₛ (y) for all relevant y, see Theorem 7. Whether or not g is bounded, we have for almost all y that there can be at most one admissible pair of hyperplanes H and H' yielding Lₛ (y) in the above manner. A detailed discussion of the method on hand may be found in Section 6. The present method is geometrical in the following sense: (i) one only needs to know the sets g (S) and g (S') in Rⁿ; (ii) afterwards, one considers all the pairs H and H' of parallel hyperplanes touching g (S) and g (S') in the above manner. Each such pair yields Lₛ (y) for certain values y; varying the pair H, H' one often obtains the value Lₛ (y) for all relevant y Rⁿ. Usually, there are many different regions in y-space, each with its own analytic formula for Lₛ (y). Nevertheless, all these different formulae are derived from one and the same geometrical principle. A number of specific illustrations, all with n = 2, are presented in Section 7. They indicate that it is often quite easy to solve the following problem in a geometric manner. Let X be a random variable taking its values in a measurable space T, such that E (g₁ (X) ) = y₁, E (g₂ (X) ) = y₂, with g₁ and g₂ as known real-valued Borel measurable functions on T. The problem is to determine the best possible lower bound Lₛ (y) on Pr (X S) where S is a given Borel measurable subset of T.