Property of sums of independent and identically distributed random variables

Question

Let $X_i(i=1,...,n)$ be independent and identically distributed random variables whose probability density functions (PDF) are $f(x)$. Known that $f(x)$ is a continuous even function defined within $\mathbb{R}$, and increases within $\mathbb{R^-}$ (for example, $X_i$ follows standard normal distribution).

My question is, for any given real number $a$, does the equality $\lim_{n\to\infty} \mathbb{P}\{\sum_{i=1}^{n}X_i>a\}=\frac{1}{2}$ always hold?

For example, suppose $X_i$ follows standard normal distribution, then $\sum_{i=1}^{n}X_i$ follows normal distribution with mean $0$ and standard deviation $\sqrt{n}$, and $\mathbb{P}\{\sum_{i=1}^{n}X_i>a\}=1-\Phi(\frac{a}{\sqrt{n}})$, where $\Phi$ is the cumulative distribution function (CDF). Then

$\lim_{n\to\infty} \mathbb{P}\{\sum_{i=1}^{n}X_i>a\}=\lim_{n\to\infty}[1-\Phi(\frac{a}{\sqrt{n}})]=1-\Phi(0)=\frac{1}{2}$

holds.