Via Phi-Sobolev inequalities, we give some sharp integrability conditions on $F$ for the large deviation principle of the empirical mean $\frac{1}{T}{\int_{0}^{T}{F(X_{s})}ds}$ for large time $T$, where $F$ is unbounded with values in some separable Banach space. Several examples are provided.

We prove an almost sure limit theorem on the exact convergence rate of the maximum of standardized gaussian random walk increments. This gives a more precise version of Shao's theorem ( Shao, Q.-M., 1995. On a conjecture of R&eacutev&eacutesz. Proc.…