The innovations algorithm is a classical recursive forecasting algorithm used in time series analysis. We develop the innovations algorithm for a class of nonnegative regularly varying time series models constructed via transformed-linear arithmetic. In addition to providing the best linear predictor, the algorithm also enables us to estimate parameters of transformed-linear regularly-varying moving average (MA) models, thus providing a tool for modeling. We first construct an inner product space of transformed-linear combinations of nonnegative regularly-varying random variables and prove its link to a Hilbert space which allows us to employ the projection theorem, from which we develop the transformed-linear innovations algorithm. Turning our attention to the class of transformed linear MA($\infty$) models, we give results on parameter estimation and also show that this class of models is dense in the class of possible tail pairwise dependence functions (TPDFs). We also develop an extremes analogue of the classical Wold decomposition. Simulation study shows that our class of models captures tail dependence for the GARCH(1,1) model and a Markov time series model, both of which are outside our class of models.

翻译：创新算法是时间序列分析中经典的递归预测方法。本文针对一类通过变换线性算术构建的非负正则变化时间序列模型，发展了创新算法。该算法不仅能提供最优线性预测，还可用于估计变换线性正则变化滑动平均（MA）模型的参数，从而为建模提供工具。我们首先构建了非负正则变化随机变量变换线性组合的内积空间，并证明其与希尔伯特空间的关联，进而可运用投影定理发展出变换线性创新算法。针对变换线性MA(∞)模型类，我们给出了参数估计结果，并证明该类模型在尾部成对依赖函数（TPDF）空间中是稠密的。同时，我们还发展了经典沃尔德分解的极值对应形式。仿真研究表明：尽管GARCH(1,1)模型及马尔可夫时间序列模型不属于本文模型类，但该类模型仍能有效捕捉其尾部依赖特征。