Mathematical Foundations of Time Series Analysis: A Concise Introduction

The goal of this book is to provide a concise summary of the foundational principles of time series analysis with special focus on the underlying mathematics. It is based on the author’s lectures to students of mathematics, financial mathematics, physics and economics.

The author implicitly expects readers to have a mathematical background that includes at least the equivalent of advanced undergraduate courses in probability and statistics. Moreover, much of the book would not make sense to readers who have not previously studied time series and seen both examples and applications of the various techniques described there.

The author’s approach is to provide definitions and then to state and prove theorems. He does this with as few words as possible. Very little narrative connects individual sections. The book’s subtitle says “concise”, and here this means extremely terse. It is likely that there are more equations than sentences in the text.

The author begins by identifying and analyzing typical assumptions that one can make in order to estimate the probability distributions underlying time series. He focuses first on univariate time series with terms equally spaced in time. The general assumptions he discusses include strong or weak stationarity and various forms of ergodicity. More specific assumptions such as normality or linearity of underlying distributions are also considered.

Spectral analysis of time series gets a good deal of attention. The author first considers univariate time series for harmonic processes. Then he extends his analysis to general processes and considers the effect of linear filters on spectral representation. This part concludes with a discussion of spectral analysis for real-valued vector time series.

Much of the rest of the book considers modeling time series with autoregressive processes. Once again the emphasis is on univariate processes. Most attention is given to autoregressive moving average (ARMA) methods and their variations (ARIMA, VARMA, ARCH and GARCH). The latter two variations are commonly used in modeling financial time series with significant time-varying volatility.

Three concluding sections address predictability, prediction methods, as well as estimation methods for mean, autocovariance, and spectral density functions.

One should not expect to learn time series analysis from this text. It seems best suited as a reference, especially for those with more than modest expertise in the subject. Even those with plenty of experience might have trouble tracking down results given the extreme terseness of the text.

Bill Satzer (bsatzer@gmail.com) was a senior intellectual property scientist at 3M Company. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.