  The principal aim of this paper is to study the Banach-Saks property when the
speed of convergence of a Cesaro mean sequence can be chosen independtly from
the choice of the initial sequence. We establish links between the uniform
Banach-Saks property, properties $(A_\infty)$ of Partington and the
$p$-Banach-Saks property. We also prove that the $p$-Banach-Saks property and
the strong $p$-Banach-Saks property are equivalent. Many examples are given and
we apply the results to the symmetric Kottman constant.
