Chebyshev type exponential inequalities for sums of random vectors and random walk trajectories

We obtain analogues of the well-known Chebyshev's exponential inequality $\mathbf P(\xi \ge x)\le e^{-\Lambda^{(\xi)}(x)}$, $x>\mathbf E\,\xi$, for the distribution of a random variable $\xi$, where $\Lambda^{(\xi)}(x):=\sup_\lambda\{\lambda x- \log \mathbf E\,e^{\lambda \xi}\}$ is the large deviation rate function for $\xi$. Generalizations of this relation are established for multivariate random vectors $\xi$, for sums of the vectors, and for trajectories of random processes associated with such sums.