Suppose, $\Xi$ is a collection of random variables. We call $\Xi$ $k$-independent, iff any $k$ distinct elements of $\Xi$ are mutually independent. For example, $2$-independence is pairwise independence and $|\Xi|$-independence is the full mutual independence of random variables from $\Xi$.

Let's define independence depth of $\Xi$ as the maximal number $k$, such that $\Xi$ is $k$-independent.

Suppose $X_1, ... , X_n$ are non-constant real random variables, such that $X_1 + ... + X_n = 0$. What is the largest possible independence depth of $\{X_1, ... , X_n\}$?

I only know the following fact:

Independence depth of $\{X_1, ... , X_n\}$ is strictly less than $n - 1$.

Suppose $\{X_1, ... , X_{n}\}$ is $(n-1)$-independent. Then suppose $Y = -X_n$. Thus we have $Y = \sum{i = 1}^{n - 1} X_k$.

Let's define $\chi_X$ as a characteristic function of a random variable $X$. Then we have $\forall x, y \in \mathbb{R}, k \leq n -1 $:

$$\chi_{X_k}(x)\Pi_{i = 1}^{n-1} \chi_{X_i}(y) = \chi_{X_k}(x)\chi_{Y}(y) = Ee^{i(xX_k + yY)} = Ee^{i((x + y)X_k + \sum_{i = 1}^{k - 1}yX_i + \sum_{i = k+1}^{n-1} yX_i)} = (\Pi_{i = 1}^{k - 1}\chi_{X_i}(y))\chi_{X_k}(\Pi_{i = k + 1}^{n-1}\chi_{X_i}(y))$$

From that and the facts, that characteristic functions are continuous and $\chi_{X_1}(0) = ... = \chi_{X_{n-1}}(0) = 1$ it follows, that $\exists \epsilon > 0$, such that $\forall x \in \mathbb{R}, |y| < \epsilon, k < n - 1$ we have $\chi_{X_k}(x + y) = \chi_{X_k}(x)\chi_{X_k}(Y)$. From that and the fact, that $\mathbb{R}$ is an Archimedean field, we can conclude, that $\forall x, y \in \mathbb{R}, k < n - 1$ we have $\chi_{X_k}(x + y) = \chi_{X_k}(x)\chi_{X_k}(Y)$. And we know, that all non-zero functions with this property are of the form $x \mapsto e^{cx}$. Thus we can conclude, that $\forall k < n - 1$ we have $\chi_{X_k}(x) = e^{ic_kx}$ and thus $X_k = c_k$ almost surely. Thus all $X_k$ and $Y$ (as the sum of them) are constants.