  Let $p_n$ denote the $n$-th prime, and for any $k \geq 1$ and sufficiently
large $X$, define the quantity $$ G_k(X) := \max_{p_{n+k} \leq X} \min(
p_{n+1}-p_n, \dots, p_{n+k}-p_{n+k-1} ),$$ which measures the occurrence of
chains of $k$ consecutive large gaps of primes. Recently, with Green and
Konyagin, the authors showed that \[ G_1(X) \gg \frac{\log X \log \log
X\log\log\log\log X}{\log \log \log X}\] for sufficiently large $X$. In this
note, we combine the arguments in that paper with the Maier matrix method to
show that \[ G_k(X) \gg \frac{1}{k^2} \frac{\log X \log \log X\log\log\log\log
X}{\log \log \log X}\] for any fixed $k$ and sufficiently large $X$. The
implied constant is effective and independent of $k$.
