In this paper a theorem on gambling systems for Bernoulli sequences is extended to arbitrary binary random sequences by using the notion of the likelihood ratio, and an analytical technique for the study of a.e. convergence is presented.

A simultaneous confidence band provides useful information on the plausible range of the unknown regression model, and different confidence bands can often be constructed for the same regression model. For a simple regression line, Liu and Hayter [W. Liu, A.J. Hayter, Minimum area confidence set...