Martingales in Sequential Analysis and Time Series , 1945 – 1985 ∗

This paper reviews the history of martingales in sequential analysis, beginning with Wald’s ground-breaking paper in 1945 that laid the foundations for the subject, and ending in the decade 1975–1985 when the usefulness of martingale theory was also recognized in time series analysis. The important roles played by martingale inequalities, convergence theory, strong laws, functional central limit theorems, laws of the iterated logarithm, and the optional stopping theorem in developments of the theory of sequential analysis and time series from 1945 to 1985 are also discussed. ∗Research supported by the National Science Foundation grant DMS 0805879. †Department of Statistics, Stanford University, Stanford, CA 94305-4065, USA [email protected]