The structure of a stationary Gaussian process near a local maximum with a prescribed height u has been explored in several papers by the present author, see [5]–[7], which include results for moderate u as well as for u→±∞. In this paper we generalize these results to a homogeneous Gaussian field {ξ(t) t ∈ Rⁿ}, with mean zero and the covariance function r(t). The local structure of a Gaussian field near a high maximum has also been studied by Nosko, [8], [9], who obtains results of a slightly different type.

In Section 1 it is shown that if ξ has a local maximum with height u at 0 then ξ(t) can be expressed as $\xi _u (t) = uA(t) - \xi _u^\prime b(t) + \Delta (t),$

WhereA(t) and b(t) are certain functions, θ_u is a random vector, and Δ(t) is a non-homogeneous Gaussian field. Actually ξ_u(t) is the old process ξ(t) conditioned in the horizontal window sense to have a local maximum with height u for t=0; see [4] for terminology.

In Section 2 we examine the process ξ_u(t) as u→−∞, and show that, after suitable normalizations, it tends to a fourth degree polynomial in t₁…, t_n with random coefficients. This result is quite analogous with the one-dimensional case.

In Section 3 we study the locations of the local minima of ξ_u(t) as u → ∞. In the non-isotropic case r(t) may have a local minimum at some point t⁰. Then it is shown in 3.2 that ξ_u(t) will have a local minimum at some point τ^u near t⁰, and that τ^u-t⁰ after a normalization is asymptotically n-variate normal as u→∞. This is in accordance with the one-dimensional case.

In this paper we continue our study of hopficity begun in [1], [2], [3], [4] and [5]. Let A be hopfian and let B have a cyclic center of prime power order. We improve Theorem 4 of [2] by showing that if B has finitely many normal subgroups which form a chain (we say B is n-normal), then AxB is hopfian. We then consider the case when B is a p-group of nilpotency class 2 and show that in certain cases AxB is hopfian.