We study the effect of approximation errors in assessing the extreme behaviour of univariate functionals of random objects. We build our framework into a general setting where estimation of the extreme value index and extreme quantiles of the functional is based on some approximated value instead of the true one. As an example, we consider the effect of discretisation errors in computation of the norms of paths of stochastic processes. In particular, we quantify connections between the sample size $n$ (the number of observed paths), the number of the discretisation points $m$, and the modulus of continuity function $\phi$ describing the path continuity of the underlying stochastic process. As an interesting example fitting into our framework, we consider processes of form $Y(t) = \mathcal{R}Z(t)$, where $\mathcal{R}$ is a heavy-tailed random variable and the increments of the process $Z$ have lighter tails compared to $\mathcal{R}$.

翻译：本文研究了近似误差对随机对象单变量泛函极端行为评估的影响。我们将研究框架构建为一个通用场景，其中泛函的极端值指数与极端分位数的估计基于近似值而非真实值。以随机过程路径范数计算中的离散化误差为例，我们量化了样本量$n$（观测路径数量）、离散化点数$m$以及描述底层随机过程路径连续性的连续模函数$\phi$之间的关联。作为符合该框架的有趣案例，本文进一步考虑了形如$Y(t) = \mathcal{R}Z(t)$的过程，其中$\mathcal{R}$为厚尾随机变量，且过程$Z$的增量尾部相较于$\mathcal{R}$更轻。