Assume the following set up:
Let $Z_i = \min\{k_i, X_i\}, i=1,...,n$. Also $X_i \sim U[a_i, b_i], \; a_i, b_i >0$. Moreover $k_i = ca_i + (1-c)b_i,\;\; 0<c<1$ i.e. $k_i$ is a convex combination of the boundaries of the respective supports. $c$ is common for all $i$.
I think I have the distribution of $Z_i$ right: it is a mixed distribution.
It has a continuous part,
$$X_i \in [a_i, k_i), Z_i=X_i \Rightarrow \Pr(Z_i \le z_i) = \frac {z_i-a_i}{b_i-a_i}$$
and then a discontinuity and a discrete part where probability mass concentrates:
$$\Pr(Z_i=k_i) = \Pr(X_i > k_i) = 1- \Pr(X_i \le k_i)$$
$$= 1- \frac {k_i - a_i}{b_i-a_i} = 1-\frac {(1-c)(b_i-a_i)}{b_i-a_i} =c$$
So in all $$F_{Z_i}(z_i) = \begin{cases} 0\qquad z_i<a_i\\ \\ \frac {z_i-a_i}{b_i-a_i}\qquad a_i\le z_i<k_i \\ \\1\qquad k_i\le z_i\end{cases}$$
while for the mixed "discrete/continuous" mass/density function, it is $0$ outside the interval $[a_i, k_i]$, it has a continuous part that is the density of a uniform $U(a_i, b_i)$,$\frac {1}{b_i-a_i}$ but for $a_i\le z_i<k_i$, and it concentrates positive probability mass $c >0$ at $z_i = k_i$.
In all, it sums up to unity over the reals.
I would like to be able to derive, or say something about, the distribution and /or moments of the random variable $S_n \equiv \sum_{i=1}^n Z_i$, as $n\rightarrow \infty$.
Say, if the $X_i$'s are independent, it looks like $\Pr(S_n = \sum_i^nk_i) = c^n \rightarrow 0$ as $n\rightarrow \infty$. Can I "ignore" that part, even as an approximation? Then I would be left with a random variable that ranges in the interval $[\sum_{i=1}^na_i,\; \sum_{i=1}^nk_i)$, looking like the sum of censored uniforms, on their way to become "un-censored", and so maybe some central limit theorem... but I am probably diverging rather than converging here, so, any suggestions?
PS: This question is relevant, Deriving the distribution of the sum of censored variables, but @Glen_b 's answer is not what I need -I have to work this thing analytically, even using approximations. This is research, so please treat it like homework -general suggestions or references to literature are good enough.