Necessary and Sufficient Condition for Stability of Generalized Expectation Value

Given a probability distribution {pi}i 1,2,...,W , that is, 0 ≤ pi ≤ 1 i 1, 2, . . . ,W and ∑W i 1 pi 1, the ordinary expectation value of a quantity Q of a system under consideration is defined by ∑W i 1 piQi, where W is the total number of accessible states and is enormously large in statistical mechanics, typically being 210 23 . In the field of generalized statistical mechanics for complex systems, on the other hand, discussions are oftenmade about altering this definition. Among others, the so-called “escort average” is widely employed in the field of generalized statistical mechanics 1–3 . It is defined as follows: