  Given a strictly convex multiobjective optimization problem with objective
functions $f_1,\dots,f_N$, let us denote by $x_0$ its solution, obtained as
minimum point of the linear scalarized problem, where the objective function is
the convex combination of $f_1,\dots,f_N$ with weights $t_1,\ldots,t_N$. The
main result of this paper gives an estimation of the averaged error that we
make if we approximate $x_0$ with the minimum point of the convex combinations
of $n$ functions, chosen among $f_1,\dots,f_N$, with probabilities
$t_1,\ldots,t_N$, respectively, and weighted with the same coefficient $1/n$.
In particular, we prove that the averaged error considered above converges to 0
as $n$ goes to $\infty$, uniformly w.r.t. the weights $t_1,\ldots,t_N$. The key
tool in the proof of our stochastic approximation theorem is a geometrical
property, called by us small diameter property, ensuring that the minimum point
of a convex combination of the function $f_1,\dots,f_N$ continuously depends on
the coefficients of the convex combination.
