  In this paper, we investigate the almost surely pointwise convergence problem
of free KdV equation, free wave equation, free elliptic and non-elliptic
Schr\"odinger equation respectively. We firstly establish some estimates
related to the Wiener decomposition of frequency spaces which are just Lemmas
2.1-2.6 in this paper. Secondly, by using Lemmas 2.1-2.6, 3.1, we establish the
probabilistic estimates of some random series which are just Lemmas 3.2-3.11 in
this paper. Finally, combining the density theorem in L$^{2}$ with Lemmas
3.2-3.11, we obtain almost surely pointwise convergence of the solutions to
corresponding equations with randomized initial data in $L^{2}$, which require
much less regularity of the initial data than the rough data case. At the same
time, we present the probabilistic density theorem, which is Lemma 3.11 in this
paper.
