  Involvement of the environment is indispensable for establishing the
statistical distribution of system. We analyze the statistical distribution of
a quantum system coupled strongly with a heat bath. This distribution is
determined by tracing over the bath's degrees of freedom for the equilibrium
system-plus-bath composite. The stability of system distribution is largely
affected by the system--bath interaction strength. We propose that the quantum
system exhibits a stable distribution only when its system response function in
the frequency domain satisfies $\tilde\chi(\omega = 0+)>0$. We show our results
by investigating the non-interacting bosonic impurity system from both the
thermodynamic and dynamic perspectives. Our study refines the theoretical
framework of canonical statistics, offering insights into thermodynamic
phenomena in small-scale systems.
