This work continues the research started by Rudzkis (Soviet Math Dokl, 45(1), 226–228, 1992), Rudzkis and Bakshaev (Lithuanian Mathematical Journal, 52(2), 196–213, 2012) and extends it to the case of random fields close to Gaussian ones. Let \( \{\xi (t), t \in \mathbb {R}^{m} \}\) be a differentiable (in the mean square sense) random field with \(\mathbb {E} \xi (t)\equiv 0, \mathbb {D} \xi (t)\equiv 1\) and continuous trajectories. The paper is devoted to the problem of large excursions of the random field ξ. Let T be an m-dimensional interval and u(t) be a continuously differentiable function. We investigate the asymptotic properties of the probability \(P=\mathbb {P}\{ \xi (t)< u(t), t \in T \}\) as \(\inf _{t \in \mathbb {R}^{m}} u(t) \rightarrow \infty \) and the mixed cumulants of the random field ξ and its partial derivatives tend to zero, i.e. the scheme of series is considered. It is shown that if the random field ξ satisfies certain smoothness and regularity conditions, then \(\frac {1-P}{1-G}=1+o(1)\), where G is a constructive functional depending on u, T and a matrix function \(R(t) = cov(\xi ^{\prime }(t),\xi ^{\prime }(t))\), \(\xi ^{\prime }(t) = \left (\frac {\partial \xi (t)}{\partial t_{1}},...,\frac {\partial \xi (t)}{\partial t_{m}} \right )\).

这项工作延续了Rudzkis（苏联数学Dokl，45（1），226-228，1992），Rudzkis和Bakshaev（立陶宛数学杂志，52（2），196-213，2012）开始的研究并将其扩展到案例接近高斯场的随机场。令\（\ {\ xi（t），t \ in \ mathbb {R} ^ {m} \} \）是具有\（\ mathbb {E} \ xi（ t）\ equiv 0，\ mathbb {D} \ xi（t）\ equiv 1 \）和连续轨迹。本文致力于解决随机场ξ的大偏移问题。设T为m维区间，u（t）是一个连续可微的函数。我们研究概率\（P = \ mathbb {P} \ {\ xi（t）<u（t），t \ in T \} \）作为\（\ inf _ {t \ in \ mathbb {R} ^ {m}} u（t）\ rightarrow \ infty \）和随机场ξ及其偏导数的混合累积量趋于零，即考虑级数方案。结果表明，如果随机场ξ满足一定的光滑度和规则性条件，则\（\ frac {1-P} {1-G} = 1 + o（1）\），其中G是取决于u的构造函数，T和矩阵函数\（R（t）= cov（\ xi ^ {\ prime}（t），\ xi ^ {\ prime}（t））\），\（\ xi ^ {\ prime}（t）= \ left（\ frac {\ partial \ xi（t）} {\ partial t_ {1}}，...，\ frac {\ partial \ xi（t） } {\ partial t_ {m}} \ right）\）。