AK-type stability theorems on cross t-intersecting families

Two families, $\mathcal{A}$ and $\mathcal{B}$, of subsets of $[n]$ are cross $t$-intersecting if for every $A\in \mathcal{A}$ and $B \in {\mathcal B}$,$A$and$B$intersectinatleast$t$elements\mathrm{.}Forarealnumber$p$ and a family $\mathcal{A}$ the product measure ${\mu }_p(\mathcal{A})$ is defined as the sum of $p^{\mathrm{\mid }A\mathrm{\mid }}(1-p{)}^{n-\mathrm{\mid }A\mathrm{\mid }}$ over all $A\in \mathcal{A}$. For every non-negative integer $r$, and for large enough $t$, we determine, for any $p$ satisfying $\frac{r}{t+2r-1}\le p\le \frac{r+1}{t+2r+1}$, the maximum possible value of ${\mu }_p(\mathcal{A}){\mu }_p(\mathcal{B})$ for cross $t$-intersecting families $\mathcal{A}$ and $\mathcal{B}$. In this paper we prove a stronger stability result which yields the above result.