  Suppose that $X=\{X_t, t\ge 0; \mathbb{P}_{\mu}\}$ is a supercritical
superprocess in a locally compact separable metric space $E$. Let $\phi_0$ be a
positive
  eigenfunction corresponding to the first eigenvalue $\lambda_0$ of the
generator of the mean semigroup of $X$. Then
$M_t:=e^{-\lambda_0t}\langle\phi_0, X_t\rangle$ is a positive martingale. Let
$M_\infty$ be the limit of $M_t$. It is known that $M_\infty$ is non-degenerate
iff the $L\log L$ condition is satisfied. When the $L\log L$ condition may not
be satisfied, we recently proved in (arXiv:1708.04422) that there exist a
non-negative function $\gamma_t$ on $[0, \infty)$ and a non-degenerate random
variable $W$ such that for any finite nonzero Borel measure $\mu$ on $E$, $$
\lim_{t\to\infty}\gamma_t\langle \phi_0,X_t\rangle
=W,\qquad\mbox{a.s.-}\mathbb{P}_{\mu}. $$ In this paper, we mainly investigate
properties of $W$. We prove that $W$ has strictly positive density on
$(0,\infty)$. We also investigate the small value probability and tail
probability problems of $W$.
