  We give a concentration inequality based on the premise that random variables
take values within a particular region. The concentration inequality guarantees
that, for any sequence of correlated random variables, the difference between
the sum of conditional expectations and that of the observed values takes a
small value with high probability when the expected values are evaluated under
the condition that the past values are known. Our inequality outperforms other
well-known inequalities, e.g. the Azuma-Hoeffding inequality, especially in
terms of the convergence speed when the random variables are highly biased.
This high performance of our inequality is provided by the key idea in which we
predict some parameters and adopt the predicted values in the inequality.
