# Random variables whose expectations are cumulants

In my research I stumbled about the following class of random variables: Let $X_0,X_1,\dots$ be random variables on a common probability space with finite moments of all orders. We then define

\begin{align}K(X_0)&:=X_0,\\ K_{t_1}(X_0,X_1)&:=X_1\big(\chi_{t_1\le 1}-\mathbb E\big)X_0,\\ K_{t_1,\dots,t_k}(X_0,\dots,X_k)&:=X_k \big(\chi_{t_k\le t_{k-1}}-\mathbb E\big)X_{k-1}\big(\chi_{t_{k-1}\le t_{k-2}}-\mathbb E\big)X_{k-2}\dots X_1\big(\chi_{t_1\le 1}-\mathbb E\big)X_0,\end{align} where $\chi_{t_k\le t_{k-1}}$ are characteristic functions and the $\mathbb E$ are expectation operators which take expectations of the random variables to the right of $\mathbb E$. For example \begin{align}K_{t_1}(X_0,X_1)&=X_1X_0 \chi_{t_1\le 1} - X_1(\mathbb E X_0),\\ K_{t_1,t_2}(X_0,X_1,X_2)&= X_2 X_1 X_0 \chi_{t_2\le t_1\le 1} - X_2 (\mathbb E X_1 X_0) \chi_{t_1\le 1} - X_2X_1(\mathbb E X_0)\chi_{t_2\le t_1}+X_2(\mathbb E X_1) (\mathbb E X_0).\end{align}

It turns out that

$$\sum_{\sigma\in S_k} \mathbb E\int_0^1 \dots \int_{0}^1 K_{t_1,\dots,t_k}(X_0,X_{\sigma(1)},\dots,X_{\sigma(k)})d t_1\dots d t_k = \text{cum}\big(X_0,\dots,X_{k}\big),$$

where cum is the joint cumulant of a collection of random variables and $S_k$ is the group of permutations on $\{1,\dots, k\}$. For example

\begin{align}\mathbb E \int_{0}^1 \int_0^1 K_{t_1,t_2}(X_0,X_{\sigma(1)}X_{\sigma(2)})d t_1dt_2 &= \mathbb E \int_0^1\int_0^1 X_{\sigma(2)}(\chi_{t_2\le t_1}-\mathbb E)X_{\sigma(1)}(\chi_{t_1\le 1}-\mathbb E)X_0 dt_1 d t_2\\ &= \mathbb E X_{\sigma(2)}\big(\frac{1}{2}-\mathbb E\big)X_{\sigma(1)}(1-\mathbb E)X_0\\ &= \frac{1}{2}\mathbb E X_{\sigma(2)}X_{\sigma(1)}X_0-\mathbb E X_{\sigma(2)}\mathbb EX_{\sigma(1)}X_0 - \frac{1}{2}\mathbb E X_{\sigma(2)}X_{\sigma(1)}\mathbb EX_0\\ &\qquad +\mathbb EX_{\sigma(2)}\mathbb EX_{\sigma(1)}\mathbb E X_0, \end{align} which after the sum over the two permutations is $$\mathbb E X_0 X_1 X_2 -\mathbb E X_1 \mathbb E X_0 X_2-\mathbb E X_2 \mathbb E X_0 X_1 - \mathbb E X_0 \mathbb E X_1 X_2+2\mathbb E X_0\mathbb E X_1\mathbb E X_2=\text{cum}(X_0,X_1,X_2).$$

I could very well imagine that these $K_{t_1,\dots,t_k}$, which I would be inclined to call 'pre-cumulants', are known objects. Has anyone ever seen random variables giving rise to cumulants in this way? If yes, I would like to use the established name for them.

If not I would also be interested whether one can define similar random variables recursively, for which the expectations are also cumulants, but do not involve integrations.

• It seems to me that your symbols mess up? – Henry.L May 11 '17 at 17:25
• I tried to add a few more details.. which symbols mess up? – Julian May 11 '17 at 19:38
• The definitions of K's? – Henry.L May 11 '17 at 19:41
• I made the definition more direct now instead of recursive.. Does this clarify the confusion? – Julian May 11 '17 at 19:44
• What are the $\mathbb{E}$ inside brackets, are they expectations? w.r.t. what? – Henry.L May 11 '17 at 19:45