On properties of a class of strong limits for supercritical superprocesses

Suppose that $X=\{X_t,t\ge 0;{\mathbb{P}}_{\mu }\}$ is a supercritical superprocess in a locally compact separable metric space $E$. Let ${\varphi }_0$ be a positive eigenfunction corresponding to the first eigenvalue ${\lambda }_0$ of the generator of the mean semigroup of $X$. Then $M_t:=e^{-{\lambda }_{0}t}⟨{\varphi }_0,X_t⟩$ is a positive martingale. Let $M_{\mathrm{\infty }}$ be the limit of $M_t$. It is known that $M_{\mathrm{\infty }}$ 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 $[0,\mathrm{\infty })$ and a non-degenerate random variable $W$ such that for any finite nonzero Borel measure $\mu$ on $E$, $$\underset{t\to \mathrm{\infty }}{\lim }{\gamma }_t⟨{\varphi }_0,X_t⟩=W,\text{qquada}\mathrm{.}s\mathrm{.}-{\mathbb{P}}_{\mu }\mathrm{.}$$ In this paper, we mainly investigate properties of $W$. We prove that $W$ has strictly positive density on $(0,\mathrm{\infty })$. We also investigate the small value probability and tail probability problems of $W$.