  Using a generalised Bochner type representation for Olshanski spherical
pairs, we establish a L\'evy-Khinchin formula for the continuous functions of
negative type on the space $V_\infty= M(\infty, \mathbb C)$ of infinite
dimensional square complex matrices relatively to the action of the product
group $K_\infty=U(\infty)\times U(\infty)$. The space $V_\infty$ is the
inductive limit of the spaces $V_n=M(n, \mathbb C)$, and the group $K_\infty$
is the inductive limit of the product groups $K_n=U(n)\times U(n)$, where
$U(n)$ is the unitary group.
