resultative state of 死ぬ (死んでいる)

The verb 死ぬ ('to die') is a change-of-state verb, so its ~ている form 死んでいる means 'is in the state of having died' — i.e. 'is dead' — never the progressive 'is dying.' English speakers often want to read it as the latter; that's a hallmark mistake to avoid. Japanese treats many verbs of change this way (結婚している = 'is married', 死んでいる = 'is dead').