  The problem of estimating a parameter of a quantum system through a series of
measurements performed sequentially on a quantum probe is analyzed in the
general setting where the underlying statistics is explicitly non-i.i.d. We
present a generalization of the central limit theorem in the present context,
which under fairly general assumptions shows that as the number $N$ of
measurement data increases the probability distribution of functionals of the
data (e.g., the average of the data) through which the target parameter is
estimated becomes asymptotically normal and independent of the initial state of
the probe. At variance with the previous studies [M. Gu\c{t}\u{a}, Phys. Rev. A
83, 062324 (2011); M. van Horssen and M. Gu\c{t}\u{a}, J. Math. Phys. 56,
022109 (2015)] we take a diagrammatic approach, which allows one to compute not
only the leading orders in $N$ of the moments of the average of the data but
also those of the correlations among subsequent measurement outcomes. In
particular our analysis points out that the latter, which are not available in
usual i.i.d. data, can be exploited in order to improve the accuracy of the
parameter estimation. An explicit application of our scheme is discussed by
studying how the temperature of a thermal reservoir can be estimated via
sequential measurements on a quantum probe in contact with the reservoir.
