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She's big in the satin set, improbably (8) T???N???
Ultimately yearns to live in coastal regions (7) ?E?C?E? looks like BEACHES, but why?
Behold a huge initial outlay of capital (4) ?S?O looks like OSLO, but why?

She's big in the satin set, improbably (8)

titaness

Ultimately yearns to live in coastal regions (7)

to live - BE
yearns - ACHES

Titaness
anagram of "satin set"

Beaches
to live=BE+ACHES (yearns)

Oslo
OS=huge +LO (behold)

Behold a huge initial outlay of capital (4)

behold - LO
huge - OS (outsize)

Thanks ixion and mattrom, appreciated.