  We define a block persistence probability $p_l(t)$ as the probability that
the order parameter integrated on a block of linear size $l$ has never changed
sign since the initial time in a phase ordering process at finite temperature
T<T_c.
  We argue that p_l(t)\sim l^{-z\theta_0}f(t/l^z) in the scaling limit of large
blocks, where \theta_0 is the global (magnetization) persistence exponent and
f(x) decays with the local (single spin) exponent \theta for large x. This
scaling is demonstrated at zero temperature for the diffusion equation and the
large n model, and generically it can be used to determine easily \theta_0 from
simulations of coarsening models. We also argue that \theta_0 and the scaling
function do not depend on temperature, leading to a definition of \theta at
finite temperature, whereas the local persistence probability decays
exponentially due to thermal fluctuations. We also discuss conserved models for
which different scaling are shown to arise depending on the value of the
autocorrelation exponent \lambda. We illustrate our discussion by extensive
numerical results. We also comment on the relation between this method and an
alternative definition of \theta at finite temperature recently introduced by
Derrida [Phys. Rev. E 55, 3705 (1997)].
