Probabilities of moderate deviations in the multidimensional case (Q1056981)

Suppose \(X^{(1)},X^{(2)},..\). are independent identically distributed random vectors in \(R^ s\) with mean zero and covariance matrix the identity. Denote the distribution function of \(\sum^{n}X^{(i)}/\sqrt{n}\) by \(F_ n\) and that of the normal distribution with mean zero and covariance matrix the identity by \(\Phi\). The paper's main result is rather too technical to quote in full. For it a condition on the decay of \(P(| X^{(1)}| \geq y)\) is imposed; then for any Borel set D at a distance of less than \(c\sqrt{\ln (n)}\) from the origin and satisfying a condition on \(\Phi\) (D) one has \[ F_ n(D)=[\Phi (D)+\theta L\Phi (\partial D^{\epsilon_ n})](1+o(1))\quad as\quad n\to \infty, \] where \(0\leq \theta \leq 1\), L is a constant, \(\partial D^{\epsilon_ n}\) is the \(\epsilon_ n\)-neighbourhood of the boundary of D and \(\epsilon_ n\to 0\) at a specified rate. The proof is by detailed and fairly intricate analysis.