Abstract. Recent advances in time series analysis provide alternative models for river flows in which the innovations have heavy tails, so that some of the moments do not exist. The probability of large fluctuations is much larger than for standard models. We survey some recent theoretical developments for heavy tail time series models and illustrate their practical application to river flow data from the Salt River near Roosevelt, Arizona. We also include some simple diagnostics that the practitioner can use to identify when the methods of this paper may be useful.

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The innovations algorithm can be used to obtain parameter estimates for periodically stationary time series models. In this paper, we compute the asymptotic distribution for these estimates in the case, where the innovations have a finite fourth moment. These asymptotic results are useful to determine which model parameters are significant. In the process, we also develop asymptotics for the Yule-Walker estimates.

[1] The generation of synthetic river flow samples that can reproduce the essential statistical features of historical river flows is useful for the planning, design, and operation of water resource systems. Most river flow series are periodically stationary; that is, their mean and covariance functions are periodic with respect to time. This article develops model identification and simulation techniques based on a periodic autoregressive moving average (PARMA) model to capture the seasonal variations in river flow statistics. The innovations algorithm is used to obtain parameter estimates. An application to monthly flow data for the Fraser River in British Columbia is included. A careful statistical analysis of the PARMA model residuals, including a truncated Pareto model for the extreme tails, produces a realistic simulation of these river flows.

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