  The theory of probability shows that, as the fraction $X_n/Y\to 0$, the
conditional probability for $X_n$, given $X_n+Y \in h_{\delta}:=[h, h+\delta]$,
has a limit law $f_{X_n}(x)e^{-\psi_n(h_\delta)x}$, where $\psi_n(h_\delta) $
equals to $[\partial \ln P(Y \in y_\delta)/\partial y]_{y=h}$ plus an
additional term, contributed from the correlation between $X_n$ and bath $Y$.
By applying this limit law to an isolated composite system consisting of two
strongly coupled parts, a system of interest and a large but finite bath, we
derive the generalized Boltzmann distribution law for the system of interest in
the exponential form of a redefined Hamiltonian and corrected Boltzmann
temperature that reflects the modification due to strong system-bath coupling
and the large but finite bath.
