
48 Modeling and In verse Problems in the Presence of Uncertainty
There are another two alternative and equivalent ways to define convergenc e
almost surely; that is, condition (2.117) can be r eplaced by either
lim
m→∞
Prob
\
j≥m
(|X
j
− X| < ǫ)
= 1, for any positive number ǫ,
or
lim
m→∞
Prob
[
j≥m
(|X
j
− X| ≥ ǫ)
= 0, for any positive number ǫ.
By the above equation, we easily see that convergence a.s. implies convergence
in probability, and thus implies convergence in distribution. In gener al, the
conve rse is not true; that is, convergence in probability does not imply conver-
gence almost surely. However, convergence in probability implies c onverg