  Statistical fluctuation problems are faced by all quantum key distribution
(QKD) protocols under finite-key condition. Most of the current statistical
fluctuation analysis methods work based on independent random samples, however,
the precondition cannot be always satisfied on account of different choice of
samples and actual parameters. As a result, proper statistical fluctuation
methods are required to figure out this problem. Taking the after-pulse
contributions into consideration, we give the expression of secure key rate and
the mathematical model for statistical fluctuations, focusing on a decoy-state
QKD protocol (Sci Rep. 3, 2453, 2013) with biased basis choice. On this basis,
a classified analysis of statistical fluctuation is represented according to
the mutual relationship between random samples. First for independent identical
relations, we make a deviation comparison between law of large numbers and
standard error analysis. Secondly, we give a sufficient condition that Chernoff
bound achieves a better result than Hoeffding's inequality based on independent
relations only. Thirdly, by constructing the proper martingale, for the first
time we represent a stringent way to deal with statistical fluctuation issues
upon dependent ones through making use of Azuma's inequality. In numerical
optimization, we show the impact on secure key rate, the ones and respective
deviations under various kinds of statistical fluctuation analyses.
