Asymptotics of the $k$th record times

Let $\eta_{0,n}\le\eta_{1,n}\le\dots\le\eta_{n,n}$ be the set of ordered statistics constructed by a sequence of independent identically distributed random variables $\eta_0,\eta_1,\dots,\eta_n$ and let $\nu^{(k)}(0)=k-1$,
$$ \nu^{(k)}(n+1)=\min\{j>\nu^{(k)}(n):\eta_j>\eta_{j-k,j-1}\}, \qquad n=0,1,2,\dots, $$
be the $k$th record times. For fixed $k$ and $n$, the asymptotic behavior of the probability $\mathbf P\{\nu^{(k)}(n)>t\}$ is studied as $t\to\infty$.