This is an announcement for the paper "The Kadec-Pe\l czynski theorem in $L^p$, $1\le p<2$" by Istvan Berkes and Robert Tichy. Abstract: By a classical result of Kadec and Pe\l czynski (1962), every normalized weakly null sequence in $L^p$, $p>2$ contains a subsequence equivalent to the unit vector basis of $\ell^2$ or to the unit vector basis of $\ell^p$. In this paper we investigate the case $1\le p<2$ 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 $\int_{\mathbb{R}} x^2 d\mu (x)\in L^{p/2}$. Archive classification: math.FA Submitted from: berkes@tugraz.at The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1506.07453 or http://arXiv.org/abs/1506.07453