The maximum probability method in problems of sequential analysis

The author considers the following probability maximizing approach in optimal stopping. As an example of the problem considered the author solves a disorder problem for a sequence of independent random variables. By analogous methods, a version of the general problem is solved for independent random variables appearing according to a renewal process. The paper contains a collection of results obtained by the author in earlier papers [Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 25 (1977), no. 8, 833–837; MR0468065; Stochastic Process. Appl. 6 (1977/78), no. 2, 153–163; MR0468066; Bol. Soc. Mat. Mexicana (2) 22 (1977), no. 1, 35–40; MR0651552; Stochastics 3 (1979), no. 1, 61–71; MR0546700].