  The ongoing miniaturization of semiconductor lasers has enabled ultra-low
threshold devices and even provided a path to approach thresholdless lasing
with linear input-output characteristics. Such nanoscale lasers have initiated
a discourse on the origin of the physical mechanisms involved and their
boundaries, such as the required photon number, the importance of optimized
light confinement in a resonator and mode-density enhancement. Here, we
investigate high-$\beta$ metal-clad coaxial nanolasers, which facilitate
thresholdless lasing. We experimentally and theoretically investigate both the
conventional lasing characteristics, as well as the photon statistics of the
emitted light. While the former lacks adequate information to determine the
threshold to coherent radiation, the latter reveals a finite threshold pump
power. Our work clearly highlights an important and often misunderstood aspect
of high-$\beta$ lasers, namely that a thresholdless laser does have a finite
threshold pump power and must not be confused with a hypothetical zero
threshold laser.
