Table 3 Examples of realized cumulants l...

Table 3

Examples of realized cumulants

l	Realized lth cumulant
2	$\sum_{j = 1}^{N} {(Δ S_{t_{j}})}^{2}$
3	$\sum_{j = 1}^{N} ({(Δ S_{t_{j}})}^{3} + 3 Δ M_{t_{j}}^{(2)} Δ S_{t_{j}})$
4	$\sum_{j = 1}^{N} ({(Δ S_{t_{j}})}^{4} + 6 Δ M_{t_{j}}^{(2)} {(Δ S_{t_{j}})}^{2} + 3 {(Δ M_{t_{j}}^{(2)})}^{2} + 4 Δ M_{t_{j}}^{(3)} Δ S_{t_{j}})$
5	$\sum_{j = 1}^{N} ({(Δ S_{t_{j}})}^{5} + 10 Δ M_{t_{j}}^{(2)} {(Δ S_{t_{j}})}^{3} + 15 {(Δ M_{t_{j}}^{(2)})}^{2} Δ S_{t_{j}} + 10 Δ M_{t_{j}}^{(3)} {(Δ S_{t_{j}})}^{2} + 10 Δ M_{t_{j}}^{(3)} Δ M_{t_{j}}^{(2)} + 5 Δ M_{t_{j}}^{(4)} Δ S_{t_{j}})$
6	$\sum_{j = 1}^{N} ({(Δ S_{t_{j}})}^{6} + 15 Δ M_{t_{j}}^{(2)} {(Δ S_{t_{j}})}^{4} + 20 Δ M_{t_{j}}^{(3)} {(Δ S_{t_{j}})}^{3} + 45 {(Δ M_{t_{j}}^{(2)})}^{2} {(Δ S_{t_{j}})}^{2} + 15 {(Δ M_{t_{j}}^{(2)})}^{3} + 60 Δ M_{t_{j}}^{(3)} Δ M_{t_{j}}^{(2)} Δ S_{t_{j}} + 15 Δ M_{t_{j}}^{(4)} {(Δ S_{t_{j}})}^{2} + 10 {(Δ M_{t_{j}}^{(3)})}^{2} + 15 Δ M_{t_{j}}^{(4)} Δ M_{t_{j}}^{(2)} + 6 Δ M_{t_{j}}^{(5)} Δ S_{t_{j}})$

Note(s): This table describes the realized lth cumulants in Bae and Lee (2021) and Fukasawa and Matsushita (2021). $Δ M_{t_{j}}^{(l)}$ in the row 2–6 are $M_{t_{j}}^{(l)} - M_{t_{j - 1}}^{(l)}$ ⁠. Recall that $S_{t} = M_{t}^{(1)}$

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