Simple upper and lower bounds for the multivariate Laplace approximation (Q401425)

The authors consider multiple integrals of the form NEWLINE\[NEWLINE J(\lambda)=\int_{R^n}e^{-\lambda f(t)}g(t)d^nt, NEWLINE\]NEWLINE and their Laplace approximation for large positive \(\lambda\). Under similar assumptions to those of the classical theory for the functions \(f\) and \(g\), the authors propose a new proof for the Laplace approximation, NEWLINE\[NEWLINE J(\lambda)={e^{-\lambda f(t_0)}g(t_0)\over\sqrt{H_f(t_0)}}\left({2\pi\over\lambda}\right)^{n/2}(1+E(\lambda)), NEWLINE\]NEWLINE where \(t_0\) is the unique absolute minimal point of \(f(t)\) and \(H_f(t)\) is the Hessian matrix of \(f\). The authors derive upper and lower bounds for the error term \(E(\lambda)\). Without being very specific, just with the aim of giving a rough idea, we find that they are of the form NEWLINE\[NEWLINE -\sum_{k=1}^3{K_k\over\lambda^{\alpha_k}}\leq E(\lambda)\leq \sum_{k=4}^9{K_k\over\lambda^{\alpha_k}}, NEWLINE\]NEWLINE NEWLINEwhere \(K_k\) and \(\alpha_k\) are positive quantities (independent of \(\lambda\)) depending on the smallest eigenvalue of \(H_f(t_0)\). These quantities are also related to the exponents that define the degree of regularity of the functions \(f\) and \(g\) at the point \(t_0\) in such a way that, the more degree of regularity, the better the bounds are. The paper include examples that show that their error estimates are best possible.NEWLINENEWLINEFinally, the authors consider the interesting example of the approximation of the sum of powers of binomial coefficients NEWLINE\[NEWLINE S(p,N)=\sum_{k=0}^N\left(\begin{matrix} N\\ k \end{matrix}\right)^p,NEWLINE\]NEWLINE for large \(N\). Using an appropriate multi-dimensional integral representation of \(S(p,N)\), the authors apply their theory to derive an asymptotic approximation of \(S(p,N)\) for large \(N\) with error bounds that improve the existing ones in the literature.