# Joint probability of two sums of independent variables with common terms

I have $n$ independent and identically distributed random variables $(X_1, \dots, X_n)$. Let $F$ be the CDF of their distribution. I also know the CDF of any sum i.i.d. variables that follow the distribution.

Let $k < l$.

I would like to calculate:

$$\forall t \in \mathbb{R} \qquad \mathbb{P}\left(\sum_{i=1}^lX_i \leq t \text{ and } \sum_{i=k}^nX_i \leq t\right)$$

Since both sums have terms in common, I don't see how to proceed.

If necessary, I could make some additional assumptions on the variables, but I would like to keep the result as general as possible.

Denote for compactness $$S_{1,l} = \sum_{i=1}^lX_i$$ and analogously for any other sum, the first subscript denoting the beginning value of the index, the second, the ending value of the index. Then we have

$$\mathbb{P}\left(\left\{\sum_{i=1}^lX_i \leq t \right\} \land \left\{\sum_{i=k}^nX_i \leq t \right\}\right) = \mathbb{P}\Big(\left\{S_{1,k-1}+S_{k,l}\leq t \right\} \land \left\{S_{k,l}+S_{l+1,n}\leq t \right\}\Big)$$

$$=\mathbb{P}\Big(\left\{S_{1,k-1}\leq t-S_{k,l} \right\} \land \left\{S_{l+1,n}\leq t-S_{k,l} \right\}\Big)$$

Now you have two independent sums being smaller or equal from a (function of a) third independent sum. If $f()$ is the density of a sum, and $F()$ the corresponding cdf, and denoting for further compactness $$S_{1,k-1} = S_1,\; S_{l+1,n} = S_2,\;S_{k,l} =S_3$$

we have using independence

$$\mathbb{P}\Big(\left\{S_1\leq t-S_3 \right\} \land \left\{S_{2}\leq t-S_{3} \right\}\Big) = \int_{-\infty}^{\infty}\int_{-\infty}^{t-s_3}\int_{-\infty}^{t-s_3}f_1(s_1)f_2(s_2)f_3(s_3)ds_1ds_2ds_3$$

$$=\int_{-\infty}^{\infty}f_3(s_3)\int_{-\infty}^{t-s_3}f_2(s_2)\int_{-\infty}^{t-s_3}f_1(s_1)ds_1ds_2ds_3$$ $$= \int_{-\infty}^{\infty}f_3(s_3)F_2(t-s_3)F_1(t-s_3)ds_3$$