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∙
When
W
n
:
n
1,2,...,
n
converges in probability to
c
we also say
that
c
is the
probability limit
of
W
n
. Sometimes we get sloppy and
refer to
W
n
, but convergence in probability is about
sequences
of
random variables.
∙
More notation. We write
plim
n
→
W
n
c
(and often drop “
n
→
”) or
W
n
p
→
c
15
∙
There is an important notation for the special case
W
n
p
→
0. As we
will see later, to facilitate the alebra of asymptotic theory, it is handy to
write
W
n
o
p
1
(read “little oh pee one”) when
W
n
p
→
0. We will see
in Section 4 where the “1” in parentheses comes from.
∙
The definition extends immediately to the idea of convergence in
probability to a random variable rather than a constant (although it
comes up less often). We require, for all
0,
P

W
n
−
W

≥
→
0 as
n
→
.
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