On the lengths of quotients of ideals and depths of fiber cones

Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring, $I$ an $\mathfrak{m}$-primary ideal of $R$ and $J$ its minimal reduction. We study the depths of $F(I)$ under certain depth assumptions on $G(I)$ and length condition on quotients of powers of $I$ and $J$, namely ${\sum }_{n\ge 0}\lambda (\mathfrak{m}{I}^{n+1}\mathrm{/}\mathfrak{m}J{I}^n)$ and ${\sum }_{n\ge 0}\lambda (\mathfrak{m}{I}^{n+1}\cap J\mathrm{/}\mathfrak{m}J{I}^n)$.