The Kadec-Peł czynski theorem in $L^p$, $1\le p<2$

Abstract: By a classical result of Kadec and Pe\l czynski (1962), every normalized weakly null sequence in $L<sup>p$, $p&gt;2$ contains a subsequence equivalent to the unit vector basis of $\ell<sup>2$ or to the unit vector basis of $\ell<sup>p$. In this paper we investigate the case $1\le p&lt;2$ and show that a necessary and sufficient condition for the first alternative in the Kadec-Pe\l czynski theorem is that the limit random measure $\mu$ of the sequence satisfies $$\int_{\mathbb{R}} x² d\mu (x)\in L<sup>{p/2}$$.