  Let $O(\infty)$ and $U(\infty)$ be the inductively compact infinite
orthogonal group and infinite unitary group respectively. The classifications
of ergodic probability measures with respect to the natural group action of
$O(\infty)\times O(m)$ on $\mathrm{Mat}(\mathbb{N}\times m, \mathbb{R})$ and
that of $U(\infty)\times U(m)$ on $\mathrm{Mat}(\mathbb{N}\times m,
\mathbb{C})$ are due to Olshanski. The original proofs for these results are
based on the asymptotic representation theory. In this note, by applying the
Vershik-Kerov method, we propose a simple method for obtaining these two
classifications, making it accessible to pure probabilists.
