Eccentric in act, yet to show persistence (8)
I believe the answer is:
tenacity
'persistence' is the definition.
('tenacity' can be a synonym of 'persistence')
'eccentric in act yet' is the wordplay.
'eccentric' indicates anagramming the letters.
'in'+'act'+'yet'='inactyet'
'inactyet' with letters rearranged gives 'TENACITY'.
'to show' acts as a link.
(Other definitions for tenacity that I've seen before include "Dogged resolve" , "Persistence in holding on" , "Perseverance, doggedness" , "Persistent determination, doggedness" , "Persistence, resolution" .)