A note on speed of convergence to the quasi-stationary distribution

In this note we show that for Z being a birth and death process on Z or Brownian motion with drift and (…), the speed of convergence to the quasi-stationary distribution is of order 1/t. The corresponding version that X is the number of calls in M/M/1 queue or the reflected Brownian motion is also considered. The result is obtained by asymptotic expansions of some transition functions. For this we use some new asymptotic expansion of the Bessel function.