On the Kolmogorov–Hajek–Rényi inequality for normed integrals of weakly dependent processes

We consider a process of the form $\zeta_\varepsilon(t)=\sqrt{\varepsilon}\int_0^{t/\varepsilon}\eta(s)\,ds$, $t\in [0,1]$, where $\eta(t)$, $t\ge0$, is a strictly stationary process with zero mean satisfying either the uniform strong mixing condition or the absolute regularity condition and find an estimate from below for the probability of the event that $|\zeta_{\varepsilon}(t)|$, $t\in [0,1]$, lies within a domain with growing curved boundaries.