We study asymptotic behavior of stochastic approximation procedures with three main characteristics: truncations with random moving bounds, a matrix-valued random step-size sequence, and a dynamically changing random regression function. In particular, we show that under quitemild conditions, stochastic approximation procedures are asymptotically linear in the statistical sense, that is, they can be represented as weighted sums of random variables. Therefore, a suitable formof the central limit theoremcan be applied to derive asymptotic distribution of the corresponding processes. The theory is illustrated by various examples and special cases.

中文翻译：

---

截断随机逼近过程的渐近行为

我们研究具有三个主要特征的随机逼近过程的渐近行为：具有随机移动边界的截断，矩阵值的随机步长序列和动态变化的随机回归函数。特别是，我们表明，在相当温和的条件下，随机逼近过程在统计意义上是渐近线性的，也就是说，它们可以表示为随机变量的加权和。因此，可以应用中心极限定理的合适形式来导出相应过程的渐近分布。各种实例和特殊情况说明了这一理论。