ARCH（$\infty$）模型自协方差函数的长期行为
Long run behaviour of the autocovariance function of ARCH($\infty$) models
摘要：
研究了非线性（$\infty$）方程记忆结构的渐近性质.这种渐近分析是通过将非线性（$\infty$）方程的自协方差函数表示为线性Volterra求和方程的解，并通过线性Volterra算子的容许性理论分析相关的预解方程的性质来实现的。证明了自协方差函数是次指数（或几何）衰减的当且仅当预解方程的核具有相同的衰减性质。它还表明，上亚指数界的自协方差函数的结果，当且仅当类似的界限适用于内核。
英文摘要：
The asymptotic properties of the memory structure of ARCH($\infty$) equations are investigated. This asymptotic analysis is achieved by expressing the autocovariance function of ARCH($\infty$) equations as the solution of a linear Volterra summation equation and analysing the properties of an associated resolvent equation via the admissibility theory of linear Volterra operators. It is shown that the autocovariance function decays subexponentially (or geometrically) if and only if the kernel of the resolvent equation has the same decay property. It is also shown that upper subexponential bounds on the autocovariance function result if and only if similar bounds apply to the kernel.
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