1. Introduction
The Laplace integrals find applications in numerous problems of mathematics and applied science, and the literature on these integrals is abundant. For example, let us mention the applications in statistical physics, see e.g., [
1] or Lecture 5 in [
2], in the pattern analysis [
3], in the large deviation theory  [
4,
5,
6], where it is sometimes referred to as the Laplace–Varadhan method, in the analysis of Weibullian chaos [
7], in the asymptotic methods for large excursion probabilities [
8], in the asymptotic analysis of stochastic processes [
9], and in the calculation of the tunneling effects in quantum mechanics and quantum fields, see [
10,
11]. It can be used to essentially simplify Maslov’s type derivation of the Gibbs, Bose–Einstein and Pareto distribution [
12]. An infinite-dimensional version and a non-commutative versions of the Laplace approximations were developed recently in [
13,
14], respectively.
The majority of research on this topic is devoted to the asymptotic expansions, or even, following the general approach to large deviation of Varadhan, just to the logarithmic asymptotics, see also [
15]. In the present paper, following the recent trend for the searching of the best constants for the error term in the central-limit-type results, see [
16] and references therein, we are interested in exact estimates for the main error term of the Laplace approximation. This approach to Laplace integrals was initiated by the author in book [
9] (Appendix B), where the stress was on the integrals with complex phase. Here we aimed at making these asymptotic more precise for real phase including the most general case of both exponent and the pre-exponential term in the integral depending on the parameter (which is crucial for the applications to the classical conditional large numbers (LLN) that we have in mind here), and stressing two new applications, to the sums instead of integrals (Laplace–Varadhan asymptotics) and to the conditional law of large numbers (LLN) and central limit theorems (CLT) of large deviations.
The content of the paper is as follows. In 
Section 2 we obtained the estimates for the error term in Laplace approximation with minimum of the phase in the interior of the domain of integration improving slightly on estimates from [
9], and in 
Section 3 we derived the resulting LLN and CLT results. In 
Section 4 and 
Section 5 the same program was carried out for the case of phase minima occurring in the border of the domain. In 
Section 6 we derived the analogous results for the case of sums, rather than integrals. In 
Section 7 we show how our results can be applied to the conditional LLN and CLT of large deviations.
  2. Phase Minimum Inside the Domain of Integration
Here we present the estimates of the remainder in the asymptotic formula for the Laplace integrals with the critical point of the phase lying in the interior of the domain of integration, adapting and streamlining the arguments of [
9].
Consider the integral
      
      where 
 is an open bounded subset of the Euclidean space 
, equipped with the Euclidean norm 
, with Euclidean volume 
, the amplitude 
f and the phase 
S are continuous real functions of 
.
Remark 1. The assumption that Ω is bounded is not essential, but simplifies explicit estimates for the error terms. One should think of Ω as a bounded subset of the full domain of integration containing all minimum points of . If f is integrable outside Ω, the integral of  over  will be exponentially small as compared with Equation (
1).
  Recall that the 
kth order derivative
      
      of a real function 
 on 
 can be viewed as the multi-linear map
      
The second derivative will be written as usual in the matrix form
      
We shall denote by 
 the corresponding norm defined as the lowest constant for which the estimate
      
      holds for all 
v.
Remark 2. It is a standard way to define norms of multi-linear mappings, see e.g., [17]. However, as all norms on finite-dimensional spaces are equivalent, the choice of a norm is not very essential here.  Let us make now the following assumptions on the functions f and S:
(C1) 
 is a Lipshitz continuous function of 
x with
      
(C2) 
 is a thrice continuously differentiable function in 
x such that
      
      and
      
      for all 
, 
, 
, with positive constants 
; the latter condition can be concisely written as
      
      where the usual ordering on symmetric matrices is used;
(C3) For any 
 there exists a unique point 
 of global minimum of 
 in 
, and the ball
      
      is contained in 
. Let us denote by 
 the matrix of the second derivatives of 
S at 
, that is
      
Notice that from convexity of 
S in 
 and Assumption (C3) it follows that
      
Our approach to the study of the Laplace integral 
 is based on its decomposition
      
      with
      
Remark 3. In the proof below one can use  instead of Equations (
2) 
with , the lower bounds coming from the estimate of  below, and the upper bound from the estimate of  below.  Proposition 1. Under Assumptions (C1)–(C3),where  is a bounded function depending on , and  is exponentially small, compared to the main term. Explicitly  Proof.  From the Taylor formula for functions on 
        it follows that
        
Consequently, for 
 we have by Assumption (C2) that
        
It follows then from Equation (
4) that
        
        so that
        
To go further we shall need the Taylor expansion of 
S up to the third order. Namely, from Equation (
9) we deduce the expansion
        
        where, due to the equation 
,
        
Turning to 
 we further decompose it into the four integrals
        
        with
        
It follows from Equation (
14) that, for 
, 
. Using Equation (
14) again and the trivial estimate 
, we conclude that, for 
,
        
From the standard integral
        
        we deduce that
        
Next,
        
        or, using Equation (
17) with 
,
        
Next,
        
        where
        
        is the area of the unit sphere in 
. Changing 
r to 
z so that
        
        and thus 
, the last integral rewrites as
        
        so that, using the inequality 
,
        
Remark 4. For  we get simplyand for  the same with  instead of 2.
  Finally 
 is calculated explicitly giving the main term of asymptotics:
        
Summarizing the estimates for all integrals involved and performing elementary simplifications, in particular using 
 and 
, yields estimate Equation (
7). □
 Proposition 2. Under (C1)–(C3) assume additionally that S is four times differentiable and f has a Lipschitz continuous first derivatives with respect to x with Thenwhere the exponentially small term  has exactly the same estimate as in the previous Proposition and  is a bounded function depending on . Explicitly,  Remark 5. The key difference in the error term here is the denominator N instead of  in Equation (
6).
  Proof.  We again decompose 
 in the sum 
 with 
 given by Equation (
5) and estimate 
 by Equation (
12). Estimation of 
 needs more careful analysis using further terms of the Taylor expansion of 
S and 
f. Namely we decompose it first as
        
        with
        
From Equation (
14) we get
        
From Equation (
17) with 
 we deduce that
        
To evaluate 
 we use the Taylor expansion of 
S to the fourth order yielding
        
        with
        
Consequently, 
 can be represented as 
 with
        
Using the estimate for 
 we obtain
        
To evaluate 
 we expand 
f in Taylor series writing
        
Substituting this in 
 and using the fact that the integral of an odd function over a ball centered at the origin vanishes, we get
        
        with
        
The first two integrals are estimated as above, that is
        
        and
        
Finally, 
 was estimated in Proposition 1 by representing it as the difference between the integral over the whole space 
 and the integral over 
, the first term yielding the main term of the asymptotics and the second one being exponentially small. Exponentially small terms are exactly the same as in the previous Proposition. Summarizing the estimates obtained and slightly simplifying, yields Equation (
22). □
   3. LLN and CLT for Internal Minima of the Phase
Theorem 1. Let Ω be a bounded open subset of  and ,  be continuous functions on  satisfying conditions of Proposition 1. Assume that  is strictly positive and the sequence of global minima  converges, as , to a point  belonging to the interior of Ω.
Let  denote a Ω-valued random variable having density  that is proportional to , that is (i) Then  weakly converge to . More explicitly, for a smooth g, one haswith a constant  depending on  and , which can be explicitly derived from Equations (
7) 
and (
8).
 (ii) If additionally S satisfies the conditions of Proposition 2, thenwith a constant  depending on  and .  Proof.  From Propositions 1 and 2 we conclude that
        
        and
        
        in cases (i) and (ii) respectively. The estimates of Equations (
25) and (
26) are then obtained from the triangle inequality. □
 Next we were interested in the convergence of the normalized fluctuations of 
 around 
, namely, of the random variables
      
To simplify the formulas below we shall assume that , but everything remains valid under general f satisfying the assumptions above,
To analyze the fluctuations, we use their moment generating functions
      
      for 
.
The numerator in Equation (
30) can be written in the form of Equation (
1) as
      
      where the new phase is
      
To shorten the notations, we shall denote by primes the derivatives of 
S or 
 with respect to the variable 
x. 
 is also convex, as 
S is, and has the same derivatives of order 2 and higher as 
S. To apply the Laplace method we need to find its point of global minimum, which coincides with its (unique) critical point, that we denote by 
 and that solves the equation
      
As a preliminary step to proving our CLT let us perform some elementary analysis of this equation proving its well posedness and finding its dependence on N in the first approximation. We shall need the following elementary result.
Lemma 1. Let  be a smooth convex function in  s.t.  everywhere and . Then for any K the mapping  is a diffeomorphism of the ball  on its image and this image contains the ball .
 Proof.  Injectivity is straightforward from convexity. Let us prove the last statement, that is, that for any 
 there exists 
 such that 
. For any 
, this claim is equivalent to the existence of a fixed point for a mapping
        
        in 
. By the famous fixed point principle, to show the existence of a fixed point, it is sufficient to show that 
 maps 
 to itself, that is, 
 whenever 
. Let
        
        and take 
. Then the symmetric matrix 
 is such that 
 for all 
. Hence, if 
 we have
        
Hence, the inequality  is fulfilled whenever , as was claimed. □
 Thus the image of the set  contains the ball of radius , so that for every  there exists a unique  such that .
On the other hand, for any 
K we can take 
, which is such that
      
      for all 
 and 
. Consequently, by Lemma 1, for such 
p and 
N, there exists a unique solution 
 of Equation (
31) in 
, and 
, i.e.,
      
Next, expanding 
 in the Taylor series around 
 (where 
), we find from Equation (
31) that
      
      and thus
      
	  (recall that we denote 
).
This allows us to improve the preliminary estimate of Equation (
32) and to obtain
      
Hence from Equation (
33) we get
      
Finally we conclude that
      
      with
      
We can now prove a convergence result that can be called the CLT for Laplace integrals.
Theorem 2. Under the assumption of Theorem 1 (i), assume additionally that  converges to  quickly enough, that iswith positive constants . Then the fluctuations  converge weakly to a centered Gaussian random variable with the moment generating function  Proof.  We show that the moment generating functions of the fluctuations 
 given by Equation (
30) converge, as 
, to the function 
, the convergence being uniform on bounded subsets of 
p. By the well known characterization of weak convergence this will apply the weak convergence of the random fluctuations 
.
Applying Proposition 1 to the numerator and denominator of the r.h.s. of Equation (
30) we get, for 
,
        
        where 
 is a bounded function, with a bound, depending on 
, that can be found explicitly from Equation (
7).
We have
        
        with
        
        and consequently
        
Using Equation (63) we conclude that
        
        where the constant 
c depends on 
.
Next, from Equation (
35) we get
        
        so that
        
        with another constant 
c depending on 
. Consequently, we deduce from Equation (
41) that
        
        with some functions 
, which are bounded on bounded subsets of 
p, implying the required convergence of the functions 
. □
   4. Phase Minimum on the Border of the Domain of Integration
Here we present the estimates of the remainder in the asymptotic formula for the Laplace integrals with the critical point of the phase lying on the boundary of the domain of integration.
Let us start with a simple one-dimensional result, which is version of the well known Watson lemma. The proof can be performed as above by decomposing the domain of integration  into the two intervals:  and . We omit the detail of the proof.
Lemma 2. Let  and  be two continuous functions on the domain  with . Let f be continuously differentiable and S be twice continuously differentiable with respect to x, with the uniform boundsand the lower boundwith some strictly positive constants , where by primes we denote derivatives with respect to x. Then, for the Laplace integralwe have the asymptotic expressionwhere  Remark 6. One can obtain similar result by decomposing  for any , in which case the exponentially small term will get the estimate This also shows that Lemma 2 remains essentially valid for small a of order , , which is used in the proof of the next result.
 Let us turn to the general case. Namely, assume 
 is a bounded open set in 
. The coordinates in 
 will be denoted 
 with 
. Let
      
      with some smooth function 
. It will be convenient to introduce the sections of 
 as the sets
      
We are interested in the asymptotics of the Laplace integral
      
      with continuous functions 
f and 
S referred to as the amplitude and phase respectively.
Let us first discuss the case of 
 with a plane boundary, that is with 
, or equivalently with
      
We shall assume the following:
(C1’) 
 is a continuously differentiable function on 
 (up to the border) with
      
(C2’) 
 is thrice continuously differentiable function of 
x and 
y such that
      
	  (where ≥ is the usual order on symmetric matrices) and
      
      with positive constants 
, and
      
Remark 7. As was noted above, the norms of higher derivatives in the estimates that we are using are their norms as multi-linear operators. For instance,  is the minimum of constants α such that  (C3’) For any 
, there exists a unique point of global minimum of 
S in 
, this point lies on the boundary 
, i.e., it has coordinates 
 with some 
, and the box
      
      is contained in 
. We shall also use the sections
      
Let us denote by 
 the matrix of the second derivatives of 
S as a function of 
y at 
, and by 
 the gradient of 
S as a function of 
x at 
, that is
      
The approach of our analysis is to decompose the integral 
 into the sum of two integrals
      
      over the sets 
 and 
, to represent the first integral as the double integral, so that
      
      and to use Proposition 1 for the estimation of 
, 
, and finally Lemma 2 to estimate 
.
Theorem 3. Under the assumptions (C1’)–(C3’), the formulaholds for  from Equation (47) and , where  is an exponentially small term and  Proof.  Integral 
 from Equation (
50) yields clearly an exponentially small contribution, similar to the integral 
 in Proposition 1, so we omit the details here.
To calculate 
 we have to know critical points of the phase 
 as a function of 
y, that is the solutions 
 of the equation
        
As 
S is convex in 
y, the solution is unique, if it exists. Proceeding as in Lemma 1, that is, searching for a fixed point of the mapping
        
        we find that there exists a unique solution 
 of Equation (
54) whenever
        
        such that
        
Next, using the Taylor expansion of 
 around the point 
 we get that
        
        with
        
This implies
        
        so that
        
        which is an essential improvement as compared with the initial estimate of Equation (
56). It ensures that the distance from 
 to the border of 
 is of order 
, so that Proposition 1 can in fact be applied to the integral 
 leading to
        
        where 
 is exponentially small compared to the main term and
        
In order to apply Lemma 2 we need to get lower and upper bounds to the quantities
        
        respectively.
But the second term vanishes. Hence
        
Next, differentiating Equation (
54) with respect to 
y we obtain
        
        implying the estimate
        
Consequently, using the formula for the differentiation of the determinant of invertible symmetric matrices,
        
        we can estimate
        
Hence Lemma 2 can be applied to the calculation of 
 given by Equations (
51) and (
59) yielding Equation (
52). □
 Remark 8. Arguing as in Proposition 2, one can improve the estimate of the remainder term in Equation (
52) 
to be of order , by assuming more regularity on S and f.  The general case of Equation (
45) can be directly reduced to the case of 
 from Equation (
47). In fact, changing coordinates 
 to 
 with 
 we get that 
 turns to 
. Making this change of the variable of integration in 
 yields
      
      with 
, 
. Assuming that these functions satisfy the conditions of Theorem 3 we obtain
      
      where
      
      and with similar change in the constants appearing in 
 and 
.
  5. LLN and CLT for Minima on the Boundary
The results on weak convergence of random variables with exponential densities given above for the case of the phase having minimum in the interior of the domain can be now recast for the case of the phase having minimum on the boundary of the domain of integration. The following statements are proved by literally the same argument as Theorems 1 and 2. We omit details.
Theorem 4. Let Ω be a bounded open set in  with coordinates , , and let Let the functions ,  be a continuous functions on  satisfying condition (C1’)- (C3’) from Theorem 3. Assume moreover that f is bounded below by a positive constants and that the sequence of global minima  converges, as , to a point  belonging to the interior of Ω.
Let  denote a -valued random variable having density  that is proportional to , that is Then  weakly converge to a constant . More explicitly, for a smooth g, one haswith a constant c depending only on S (actually on the bounds for the derivatives of S up to the third order).  Theorem 5. Under the assumptions of Theorem 4 assume additionally that Then the fluctuations  converge weakly to a -dimensional random vector such that its last coordinates form a centered Gaussian random vector with the moment generating functionand the first coordinate is independent and represents a - exponential random variable. The rates of convergence with all explicit constants are obtained directly from Theorem 3.    6. Laplace Sums with Error Estimates
It is more or less straightforward to modify the above results to the of sums rather than integrals. Namely, instead of the integral 
 from Equation (
1) let us consider the sum
      
      where 
 is an open polyhedron of the Euclidean space 
, with Euclidean volume 
, the amplitude 
f and the phase 
S are continuous real functions.
Theorem 6. Under the assumptions of Proposition 1,whereand where  and  are the same as in Proposition 1 and  is yet another constant depending on .  Proof.  We use the well known (and easy to prove) fact (a simplified version of the Euler–Maclorin formula) that
        
Consequently,
        
        where 
 is from Equation (
1). The first integral on the r.h.s. of Equation (
68) is clearly of order 
, as compared with the main term of 
 given in Proposition 1. The pre-exponential term in the second integral vanishes at the critical point 
 of 
. Hence the required estimate for the second integral is obtained directly from Proposition 1. □
 Now all LLN and CLT results obtained above for continuous distributions can be reformulated and proved straightforwardly for the case of discrete random variables taking values in the lattice  with probabilities proportional to .
  7. Application to LLN and CLT of Large Deviations
Conditional LLN (conditioned on the sums of the corresponding random variables to stay in a certain prescribed domain, usually some linear subspace or a convex set) are well developed in probability (see e.g., [
2,
18] for two different contexts). The results above can be used to supply exact estimates for the error terms in these approximations. To illustrate this statement in the most transparent way let us start with the classical multidimensional local theorem of large deviations as given in [
4] (that extends earlier results of [
6]). Namely, let 
 be a sequence of independent identically distributed 
-valued random vectors. Assume that the set 
O of vectors 
 such that the moment generating function 
 is well defined has a nonempty interior 
. It is well known (and easy to see) that the functions 
v and 
 are convex and the sets 
 and its closure 
 are convex. The function 
 is called the entropy and it is concave. Moreover, the infimum in its definition is attained, so that there exists 
 such that
      
      and the function 
 is a diffeomorphism of 
 onto some open domain 
 in 
. Assume that the random variable 
 has a bounded probability density 
, and define the family of distributions 
 with the densities
      
Let  be the density of the averaged sum .
Theorem 1 of [
4] states (though we formulate it equivalently in terms of the density of 
, rather than 
 as is done in [
4]) that if 
 is any compact set in 
, then
      
      where 
s is arbitrary, the estimate is uniform for 
, 
 is the matrix of the second moments of the distributions 
, the coefficients 
 depend only on 
 moments of 
 and are uniformly bounded in 
.
The densities of Equation (
69) are exactly of the type dealt with in our Theorems 1, 2, and 4, and Equation (
5). Thus, these theorems are applied directly for finding the rates of convergence for LLN and CLT for the sums of independent variables when 
 is reduced to some convex bounded set with smooth boundary or a linear subspace. These conditional versions of LLN may be applied even if 
 is not defined, so that the usual LLN does not hold.
When the random variable  has values in a lattice, a version with sums, that is Theorem 6, should be applied to get the rates of convergence in the corresponding laws of large numbers.