The problem of optimal estimation of functionals $A\xi =\sum\nolimits_{k=0}^{\infty }{}a(k)\xi (k)$ and ${{A}_{N}}\xi =\sum\nolimits_{k=0}^{N}{}a(k)\xi (k)$ which depend on the unknown values of stochastic sequence $\xi (k)$ with stationary $n$th increments is considered. Estimates are based on observations of the sequence $\xi (m)$ at points of time $m=-1,-2,\ldots$. Formulas for calculating the value of the mean square error and the spectral characteristic of the optimal linear estimates of the functionals are derived in the case where spectral density of the sequence is exactly known. Formulas that determine the least favorable spectral densities and minimax (robust) spectral characteristic of the optimal linear estimates of the functionals are proposed in the case where the spectral density of the sequence is not known but a set of admissible spectral densities is given.

翻译：本文研究了依赖于具有平稳 $n$ 阶增量的随机序列 $\xi (k)$ 未知值的泛函 $A\xi =\sum\nolimits_{k=0}^{\infty }{}a(k)\xi (k)$ 与 ${{A}_{N}}\xi =\sum\nolimits_{k=0}^{N}{}a(k)\xi (k)$ 的最优估计问题。估计基于在时间点 $m=-1,-2,\ldots$ 处对序列 $\xi (m)$ 的观测。在序列的谱密度精确已知的情况下，推导了用于计算最优线性估计的均方误差值及谱特征的公式。在序列的谱密度未知但给定了容许谱密度集合的情况下，提出了用于确定最优线性估计的最不利谱密度及极小极大（稳健）谱特征的公式。