Table 4 The 2–6 realized joint cumulants...

Table 4

The 2–6 realized joint cumulants

l	$c M_{l - 1,1}^{r e a l} (S_{1}, S_{2})$
2	$\sum_{j = 1}^{N} Δ S_{1, t_{j}} Δ S_{2, t_{j}}$
3	$\sum_{j = 1}^{N} (({(Δ S_{1, t_{j}})}^{2} + Δ M_{t_{j}}^{(2,0)}) Δ S_{2, t_{j}} + 2 Δ S_{1, t_{j}} Δ M_{t_{j}}^{(1,1)})$
4	$\sum_{j = 1}^{N} (({(Δ S_{1, t_{j}})}^{3} + 3 Δ M_{t_{j}}^{(2,0)} Δ S_{1, t_{j}} + Δ M_{t_{j}}^{(3,0)}) Δ S_{2, t_{j}} + 3 ({(Δ S_{1, t_{j}})}^{2} + Δ M_{t_{j}}^{(2,0)}) Δ M_{t_{j}}^{(1,1)} + 3 Δ S_{1, t_{j}} Δ M_{t_{j}}^{(2,1)})$
5	$\sum_{j = 1}^{N} (({(Δ S_{1, t_{j}})}^{4} + 6 Δ M_{t_{j}}^{(2,0)} {(Δ S_{1, t_{j}})}^{2} + 3 {(Δ M_{t_{j}}^{(2,0)})}^{2} + 4 Δ M_{t_{j}}^{(3,0)} Δ S_{1, t_{j}} + Δ M_{t_{j}}^{(4,0)}) Δ S_{2, t_{j}} + 4 ({(Δ S_{1, t_{j}})}^{3} + 3 Δ M_{t_{j}}^{(2,0)} Δ S_{1, t_{j}} + Δ M_{t_{j}}^{(3,0)}) Δ M_{t_{j}}^{(1,1)} + 6 ({(Δ S_{1, t_{j}})}^{2} + Δ M_{t_{j}}^{(2,0)}) Δ M_{t_{j}}^{(2,1)} + 4 Δ S_{1, t_{j}} Δ M_{t_{j}}^{(3,1)})$
6	$\sum_{j = 1}^{N} (({(Δ S_{1, t_{j}})}^{5} + 10 Δ M_{t_{j}}^{(2,0)} {(Δ S_{1, t_{j}})}^{3} + 10 Δ M_{t_{j}}^{(3,0)} {(Δ S_{1, t_{j}})}^{2} + 15 {(Δ M_{t_{j}}^{(2,0)})}^{2} Δ S_{1, t_{j}} + 10 Δ M_{t_{j}}^{(3,0)} Δ M_{t_{j}}^{(2,0)} + 5 Δ M_{t_{j}}^{(4,0)} Δ S_{1, t_{j}} + Δ M_{t_{j}}^{(5,0)}) Δ S_{2, t_{j}} + 5 ({(Δ S_{1, t_{j}})}^{4} + 6 Δ M_{t_{j}}^{(2,0)} {(Δ S_{1, t_{j}})}^{2} + 3 {(Δ M_{t_{j}}^{(2,0)})}^{2} + 4 Δ M_{t_{j}}^{(3,0)} Δ S_{1, t_{j}} + Δ M_{t_{j}}^{(4,0)}) Δ M_{t_{j}}^{(1,1)} + 10 ({(Δ S_{1, t_{j}})}^{3} + 3 Δ M_{t_{j}}^{(2,0)} Δ S_{1, t_{j}} + Δ M_{t_{j}}^{(3,0)}) Δ M_{t_{j}}^{(2,1)} + 10 ({(Δ S_{1, t_{j}})}^{2} + Δ M_{t_{j}}^{(2,0)}) Δ M_{t_{j}}^{(3,1)} + 5 Δ S_{1, t_{j}} Δ M_{t_{j}}^{(4,1)})$

Note(s): This table describes detailed forms of the realized lth joint cumulants $c M_{l - 1,1}^{r e a l} (S_{1}, S_{2})$ up to order six. $Δ M_{t_{j}}^{(l - k, k)}$ in the row 2–6 are $M_{t_{j}}^{(l - k, k)} - M_{t_{j - 1}}^{(l - k, k)}$ ⁠. Recall that $S_{1, t} = M_{t}^{(1,0)}$ and $S_{2, t} = M_{t}^{(0,1)}$

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