Let {${xi}_j;j;{geq};1$} be a centered strictly stationary random sequence defined by $S_0;=;0$, $S_n;=;Sigma^n_{j=1};{xi}_j$ and $sigma(n);=;33sqrt {ES^2_n}$ where $sigma(t),;t;>;0$, is a nondecreasing continuous regularly varying function. Suppose that there exists $n_0;{geq};1$ such that, for any $n;{geq};n_0$ and $0;{leq};{varepsilon};<;1$, there exist positive constants $c_1$ and $c_2$ such that $c_1e^{-(1+{varepsilon})x^2/2};{leq};P{frac{{mid}S_n{mid}}{sigma(n)};{geq};x};{leq};c_2e^{-(1-{varepsilon})x^2/2$, $x;{geq};1$ Under some additional conditions, we investigate some limsup results for the increments of partial sum processes of the sequence {${xi}_j;j;{geq};1$}.