Asymptotic and non-asymptotic estimates for multivariate Laplace integrals

We derive bilateral asymptotic as well as non-asymptotic estimates for the multivariate Laplace integrals. Possible applications: Tauberian theorems for random vectors.

Notations. Previous results. Statement of problem. We will impose in the sequel the following condition on the set X.
say for all the values Z ≥ 1 sufficiently large. Denote so that dim(x) = dim(λ) = d.
The one-dimensional case d = 1 was considered in [6,21,22]; a preliminary result may be found in [23].
We will generalize the main results obtained in the articles [21,22,23], where are described also some applications of these estimates, in particular, in the probability theory. The estimates given below may be considered in turn as a generalization of the classical saddle-point method ( [11]). The paper is organized as follows. In section 2 and in section 3 we deduce respectively an upper and a lower direct estimate for the Laplace integral I(λ); section 4 and section 5 contain an investigation of the inverse problem and, respectively, an upper and a lower estimate for the source function through the exponential integral. In section 6 we consider the multidimensional Tauberian theorems for exponential integrals; in section 7 some important examples are described. The last section contains the concluding remarks.
Denote, as usually, Let us mention briefly a possible application. Recall that the so-called (multivariate) moment generating function (MGF) for the random vector (r.v.) ⃗ ξ is defined by the equality where f ξ (x) denotes the density of the r.v. ξ, if there exists. So, the MGF function exp (φ ξ (λ)) is, on the other terms, the multivariate Laplace integral. It will be presumed that the r.v. ξ satisfies the so-called Cramer's condition: and that the density function there exists.
Recall that the well-known Young-Fenchel or Legendre transform for the function ζ ∶ X → R is defined as follows If some function φ = φ(λ) is defined and is finite in a set V , i.e. dom[φ] = V, convex or not, one can define formally This notion plays an important role in the probability theory. Namely, let ξ = ⃗ ξ be a random vector for which where T ξ = T ξ (x) denotes the tail function for the r.v. ξ ∶ the so-called generalized Chernoff's inequality, see e.g. [7,8,22].
Moreover, this assertion may be reversed under some natural conditions (smoothness, convexity etc.) in the following sense. Suppose d = 1 (one-dimensional case) and that the last estimate (1.7) holds true. Then, under appropriate conditions (see [22]), for some finite constant C 1 .
2 Main result. A direct approach. Upper estimate.
Let us introduce some preliminary notations and conditions. Put here and in the sequel ǫ = const ∈ (0, 1).
Let µ be the classical Lebesgue measure and let ζ = ζ(x), x ∈ R d + , be a non-negative strictly convex continuous differentiable function. The function K(ǫ), ǫ > 0, defined by (2.1), satisfies the following estimate Proof. There exist positive constants C 1 , C 2 , . . . , C d and a number C 0 ∈ R such that Indeed, one can apply the well-known Fenchel-Morau theorem so that, for an arbitrary y 0 ∈ R d + , Definition 2.1. Let D ⊂ X be a non-empty subset of the whole set X. We introduce the so-called regional Young-Fenchel transform for the function ζ(⋅) We represent now three methods for an upper estimate of I(λ) for sufficiently large values of the real parameter λ .
It is proved in particular in [22] that Note that in [22] was considered the one-dimensional case d = 1; but the general one may be investigated quite analogously. In detail, let ǫ ∈ (0, 1) be some number for which K(ǫ) ∈ (0, ∞). Consider the following probability measure, more precisely, the family of probability measures or symbolically We have So, the relation (2.4) is proved.
As a slight consequence: C. An opposite method, which was introduced in a particular case in [21], [22]. Define the following integral if, of course, it is finite at least for some value ǫ ∈ (0, 1). .
Let again ǫ = const ∈ (0, 1). Applying the well-known Young inequality we have Of course We conclude Furthermore, we will use the following elementary inequality alike ones in the monograph [25], chapter 3; and suppose that lim Λ(λ)→∞ π(λ) = 0; (2.11) so that the value Λ 0 = Λ(κ) may be chosen such that Let us impose the following condition on the function Define also Choosing ǫ = π(λ) in the domain min i λ(i) ≥ Λ(1) , we have the following Theorem 2.1. If the function φ(⋅) = ζ * (⋅) satisfies the condition (2.12), then and, after the minimization over ǫ , E. Let us consider an arbitrary simple partition X = X 0 ∪ X 1 , X 0 ∩ X 1 = ∅ of the whole set X onto two disjoint measurable subsets. We deduce splitting integral I(λ) into two ones and applying the foregoing estimates: (2.14) We obtained actually the following compound estimate.
Then, ∀λ ∶ λ ≥ c and ∀ǫ ∈ (0, 1), As a slight consequence: Remark 2.1. Introduce the following condition on the function ζ(⋅): This condition is satisfied if, for example, the function ζ = ζ(x), x ∈ X is regular varying: where m = const > 0, ⋅ is the ordinary Euclidean norm (or an arbitrary other non-degenerate vector one) and L = L(r), r ≥ 1, is some positive continuous slowly varying function as r → ∞, and we suppose where, as before, M = M (r), r ≥ 1, is some positive continuous slowly varying function as r → ∞.
One can apply the spherical coordinates: where Γ is the classical Gamma function.
To summarize: as ǫ → 0+ Thus, in this case, the values K = K(ǫ) and R = R(ǫ), ǫ ∈ (0, 1), are finite with concrete estimate following from (2.20): If the condition of Remark 2.1 is satisfied, then Theorem 2.2. Let X = R d + and µ be the ordinary Lebesgue measure. Suppose that the random vector ⃗ ξ , with non-negative entries {ξ(i)}, i = 1, 2, . . . , d , satisfies the Cramer's condition: Then It is sufficient to consider only the two-dimensional case: assume Main result. A direct approach. Lower estimate.
We introduce additional notations.
◻ As a slight consequence we get: and, if we choose ǫ = π κ (λ), Let us define the following function so, by (3.1), we have For instance, it is reasonable to suppose in addition, see e.g. Example 3.1 below, that Let us consider the following important example.
Example 3.1. Suppose that X = R d + , dµ = dx and that the function ζ = ζ(x), x ∈ X = R d + is non-negative, strictly convex, twice continuous and differentiable as well as its conjugate ζ * (λ) and such that its second (matrix) derivative , i, j = 1, 2, . . . , d is a strictly positive definite matrix for all sufficiently large values min i x(i).
Denote also We deduce after simple calculations, using Taylor's formula, that the set X 0 (ǫ, λ) is asymptitical equivalent, as ǫ → 0+ , to the following one (multidimensional ellipsoid) The case when the value ǫ = ǫ(λ) is dependent on λ, but such that can not be excluded. It is no hard to calculate the "volume" of ellipsoidX 0 ∶ If, for instance, d = 1, m = const > 1, and we find, after some calculations, The last estimate is in full accordance, up to a multiplicative constant, with the exact asymptotic estimates for I m (λ), as λ → ∞, which may be find, e.g., in the well-known book [11], sections 1, 2: The upper estimate corresponding to the lower one obtained above, for the integral I m (λ), has the form

Inverse approach. Upper estimation.
Let now the representation (1.2) be given on the form of an inequality for a certain non-negative continuous function J = J(λ). Here we derive the upper bound for the source function ζ = ζ(x) for all the sufficiently large values Λ(x) = min i x(i), i = 1, 2, . . . , d, of course, under appropriate conditions. Let us impose the following condition on our datum. Namely, assume that for some finite constant C 12 Suppose also that the function ζ(⋅) is non-negative, continuous and convex. We have, by virtue of Theorem 2.1, Under the above conditions and by virtue of Fenchel-Moreau Theorem, we have

Inverse approach. Lower estimation.
Let now the representation (1.2) be given on the form of an inequality for a certain non-negative continuous function K = K(λ). Here we derive the lower bound for the source function ζ = ζ(x). Let us impose the following condition on our datum. Namely, assume that there exists C 13 = C 13 [ζ] = const ∈ (0, 1) such that Suppose, as above, that the function ζ(⋅) is non-negative, continuous and convex. We have, by virtue of Theorem 3.1 and its consequences, 6 Multivariate Tauberian theorems.
Preface. Tauberian theorems are named the relations between asymptotical or not-asymptotical behavior of some function (sequence) and correspondent behavior of its certain integral transform, for example, Laplace, Fourier or power series transform, see [32,19]. They play a very important role, for example, in the probability theory (see [3]), to establish the connection between the behavior of tail of distribution for a random variable and the asymptotic one of its Moment Generation Function (MGF). There are many results in this direction for one-dimensional case, as well as asymptotical ones, see e.g. in [1,4,9,10,12,13,16,17,20,24,29,33].
Direct approach. Proof. Choosing ǫ = ǫ(λ) = π(λ) = π 1 (λ) we have, for sufficiently large values Λ(λ) = min i λ(i), The term on the left hand side tends to zero as Λ → ∞, the limit of the quantity on the right hand side is equal to one. In detail, On the other hand, from the condition (2.12) it follows therefore, by virtue of condition (6.1), This completes the proof. ◻ Theorem 6.2. (Lower limit). Under the same assumptions of Theorem 3.1 for the function φ(λ) = ζ * (λ), if in addition suppose that Proof. The proof is completely alike to the one based on Theorem 6.1 and may be omitted.
In this section we consider X = R d as well as λ ∈ R d .
Definition 7.1. Recall that the function g = g(x) ∶ R d → R is said to be radial, or equally spherical invariant, iff it depends only on the Euclidean norm x of the vector x = ⃗ x, namely there exists g 0 ∶ R → R such that g(x) = g 0 ( x ).
Lemma 7.1. Suppose that the function g ∶ R d → R is radial and such that its Young-Fenchel transformation g * (y) there exists. Then it is again a radial function, namely there is a function g 0 ∶ R → R for which g * (y) = g * 0 ( y ) = sup z∈R ( y z − g 0 (z)). (7.1) As a consequence, it is an even function. Moreover, the optimal value in the definition of the Young-Fenchel transformation, i.e. the variable x(y) = x[g](y) ∶= argmax x∈R d ((x, y) − g(x)), so that g * (y) = (y, x[g](y) − g(x[g])(y)), is also a radial function if, of course, there exists and is uniquely determined.

Concluding remarks.
A. It is interesting, by our opinion, to generalize the estimates obtained in the second section to the case of infinite-dimensional linear spaces, as well as to generalize our estimates for the more general integrals of the form B. One can consider also the applications of the obtained results in the Probability theory, namely, in the theory of great deviation, asymptotical or not.