On the Upper Bound for Large Deviations of Sums of I.I.D. Random Vectors

Let $X_1, X_2, \cdots$ be a sequence of i.i.d. random vectors with values in $\mathbb{R}^d, \mu = \mathscr{L}(X_1)$ and let $\lambda$ be the convex conjugate of $\log \hat{\mu}$, where $\hat{\mu}$ is the Laplace transform of $\mu$. For every $d \geq 2$, a probability measure $\mu$ and an open set $A$ in $\mathbb{R}^d$ are constructed so that $\lim \inf_{n\rightarrow\infty} \frac{1}{n} \log P\big(\frac{S_n}{n} \in A\big) > - \Lambda (A),$ where $S_n = X_1 + \cdots + X_n$ and $\Lambda (A) = \inf_{x \in A} \Lambda (x)$. It is also shown that if $\mu$ satisfies certain regularity conditions, then $\lim \sup_{n\rightarrow \infty} \frac{1}{n} \log P\big(\frac{S_n}{n} \in A\big) \leq - \Lambda (A),$ holds for all Borel sets in $\mathbb{R}^d$.