What is the probability distribution of the Sum of continuous Unimodal RVs if nothing is known about its individual distributions?

Question

A) What is the probability distribution of the Sum of continuous Unimodal RVs if nothing is known about its individual distributions?

I want to know if is possible to make some insight about the distribution of the Sum of continuous unimodal random variables if nothing is known about its individual distributions (finitely many), as it is a sum of "don´t know if identically distributed" and "don't know if independent" random variables..

B)There exists a worst case scenario?

What I know is going to be true:

1. Because of linearity of the Expected Value I understand that, if each variable is label as $X_i$, the sum $S_i$ will have expected value $\mu_S = E[\Sigma X_i] = \Sigma E[X_i] = \Sigma \mu_i$.
2. Also because of concentration inequalities, that the mode $\theta$ of each r.v. $X_i$ will behave as $ \frac{|\mu_i - \theta_i|}{\sigma_i}\leq \sqrt{3}$ [1], and also, if and only if the distribution of the Sum results to be unimodal, then $ \frac{|\mu_S - \theta_S|}{\sigma_S}\leq \sqrt{3}$, but I don't know if it is going to have such a kind of distribution.
3. Also I know that if the Law of the big numbers applies, here meaning that each $X_i$ are independent, the resultant distribution will be Gaussian.
4. And finally, if each $X_i$ distribute Gaussian, but are somehow correlated, then the sum will distribute similarly to a "wider" Gaussian [2], but with heavy tails (similar to a T-Student distribution). So my intuition is that in a general case, something between a heavy-tailed Gaussian or a skewed alike distribution will result (like a log-normal).

At first I think about a worst-case for continuous R.V.s for a random series, like an Uniform distribution "extended" somehow to infinity, like a "lying down" Dirac delta distribution: instant changes between $X_i(t)$ and $X_i(t+\epsilon)$ will be infinite apart each other even if $\lim_{\epsilon \rightarrow 0}$ is happening, but its not possible to generate this kind of distribution. Then I found "concentration inequalities" (Markov's, Gauss', Chevyshev's, etc.), so, if a point $X_i(t)$ happen, the next one will be "in probability" at finite distance of there, even if their distribution could achieve infinite values and its power is unbounded (like in Cauchy's distribution), but it don´t really says if the point can "jumps" to an infinite distance or not. And finally I found "martingales" where by definition an "infinite jump" is forbidden, even when its total variation is infinite so its derivatives are undefined (but I don't really understand them, are deep-mathematical objects for me).

If the questions is too "general", I am thinking that $X_i$ are prices (like stocks, or commodities), that kind of somehow limited variables $0 \leq X_i < \infty$, but if is possible to be more general, the best ($-\infty < X_i < \infty$). Also, if needed, assume that its variance is finite $\sigma^2 < \infty$.

I am thinking how will distribute a portfolio, which is a weighted sum of random series, where some movements of each series $X_i(t)$ are random, but some random movements could be "correlated" (market shocks, as example), so I won't make any assumption about it. Neither about its "Stationarity".

My doubt born because of this: if I have two Gaussian R.V.s with same mean and different variance, lets say $\mu_1=\mu_2$ and $\sigma_1^2 < \sigma_2^2$, and I make them compete, in average neither will have advantage since $X_1 - X_2 \sim N(0,\sigma_1^2+\sigma_2^2)$, but if I have more than three variables it stops to be true and the variable $X_i$ with higher variance $\sigma_i$ will have advantage, this because $(X_1 - X_2 > 0)\cap(X_1 - X_3 > 0)$ (X1 wins) is not independent of $X_1 > X_2$, $X_1 > X_3$ and $X_2 > X_3$, as can be seen in the distribution of $\max{\{X_i\}}$). I was shocked at first, and I don't know if this is properly "included" in the Markowitz portfolio theory - also neither if Simpson's Paradox is avoided (if someone can explain this too, it will be awesome!), that is why I want to know how will distribute a general case sum of random variables.

C) More restricted scenario

Also, since under some assumptions, prices tend to behave like Log-normal distributed, like a Law of big numbers applied to its log values: if $X_i(t) = X_0 \cdot (1+\tau_1)\cdot(1+\tau_2)\cdots (1+\tau_n)$ and I call $1+\tau_i = e^{z_i}$ then $\log(\frac{X_i(t)}{X_0}) = \Sigma z_i$ with $z_i$ random variables where the Law of big numbers could be applied (some assumptions needed)...

So, if the questions still being too general, I also would like to know this more restricted case: what will be the distribution of $S = \Sigma X_i$ if every $X_i$ is distributed continuous with $0 \leq X_i < \infty$, its distribution have finite mean and variance $0< \mu_i < \infty$ and $\sigma_i^2 < \infty$, its distribution is unimodal and skewed to zero so its mode is that $\theta_i < \nu_i < \mu_i$ with $\nu_i$ its median, but not necessarily distributed Log-normaly. Always without knowing its individual distributions, or if they are independent or not.

Here my intuition says that S will distributed also positively skewed, but it will depends in the weight of each $X_i$ in the sum. Maybe like a Johnson's SU distribution [3], but with a parameter "managed" by the correlation.

Since in positive skewed distributions $\theta < \mu$ will happens, I believe using the word "expected value" make some misunderstands since the value that I will see happening the most is the mode $\theta$ which will be commonly under the "expected average" $\mu$, which could tell me to change my strategy because I am not achieving the mean value I am looking for when my portfolio is optimal, in other words, it is "normal" to have less return that the theoretical average.

D) I don´t know if there is a portfolio optimization than instead of maximizing the expected value where designed to maximize the mode, if someone knows, please let me know any reference.