<s>
In	O
probability	O
theory	O
,	O
an	O
empirical	B-General_Concept
process	I-General_Concept
is	O
a	O
stochastic	O
process	O
that	O
describes	O
the	O
proportion	O
of	O
objects	O
in	O
a	O
system	O
in	O
a	O
given	O
state	O
.	O
</s>
<s>
For	O
a	O
process	O
in	O
a	O
discrete	O
state	O
space	O
a	O
population	B-General_Concept
continuous	I-General_Concept
time	I-General_Concept
Markov	I-General_Concept
chain	I-General_Concept
or	O
Markov	B-General_Concept
population	I-General_Concept
model	I-General_Concept
is	O
a	O
process	O
which	O
counts	O
the	O
number	O
of	O
objects	O
in	O
a	O
given	O
state	O
(	O
without	O
rescaling	O
)	O
.	O
</s>
<s>
Applications	O
of	O
the	O
theory	O
of	O
empirical	B-General_Concept
processes	I-General_Concept
arise	O
in	O
non-parametric	B-General_Concept
statistics	I-General_Concept
.	O
</s>
<s>
That	O
is	O
,	O
Fn	O
converges	O
to	O
F	O
pointwise	B-Algorithm
.	O
</s>
<s>
Glivenko	O
and	O
Cantelli	O
strengthened	O
this	O
result	O
by	O
proving	O
uniform	B-Algorithm
convergence	I-Algorithm
of	O
Fn	O
to	O
F	O
by	O
the	O
Glivenko	O
–	O
Cantelli	O
theorem	O
.	O
</s>
<s>
By	O
the	O
central	O
limit	O
theorem	O
,	O
converges	B-Algorithm
in	I-Algorithm
distribution	I-Algorithm
to	O
a	O
normal	O
random	O
variable	O
N( 0	O
,	O
P(A )	O
(	O
1−	O
P(A )	O
)	O
)	O
for	O
fixed	O
measurable	O
set	O
A	O
.	O
</s>
<s>
Similarly	O
,	O
for	O
a	O
fixed	O
function	O
f	O
,	O
converges	B-Algorithm
in	I-Algorithm
distribution	I-Algorithm
to	O
a	O
normal	O
random	O
variable	O
,	O
provided	O
that	O
and	O
exist	O
.	O
</s>
<s>
is	O
called	O
an	O
empirical	B-General_Concept
process	I-General_Concept
indexed	O
by	O
,	O
a	O
collection	O
of	O
measurable	O
subsets	O
of	O
S	O
.	O
</s>
<s>
is	O
called	O
an	O
empirical	B-General_Concept
process	I-General_Concept
indexed	O
by	O
,	O
a	O
collection	O
of	O
measurable	O
functions	O
from	O
S	O
to	O
.	O
</s>
<s>
A	O
significant	O
result	O
in	O
the	O
area	O
of	O
empirical	B-General_Concept
processes	I-General_Concept
is	O
Donsker	O
's	O
theorem	O
.	O
</s>
<s>
It	O
has	O
led	O
to	O
a	O
study	O
of	O
Donsker	O
classes	O
:	O
sets	O
of	O
functions	O
with	O
the	O
useful	O
property	O
that	O
empirical	B-General_Concept
processes	I-General_Concept
indexed	O
by	O
these	O
classes	O
converge	B-Algorithm
weakly	I-Algorithm
to	O
a	O
certain	O
Gaussian	B-General_Concept
process	I-General_Concept
.	O
</s>
<s>
As	O
an	O
example	O
,	O
consider	O
empirical	B-General_Concept
distribution	I-General_Concept
functions	I-General_Concept
.	O
</s>
<s>
In	O
this	O
case	O
,	O
empirical	B-General_Concept
processes	I-General_Concept
are	O
indexed	O
by	O
a	O
class	O
It	O
has	O
been	O
shown	O
that	O
is	O
a	O
Donsker	O
class	O
,	O
in	O
particular	O
,	O
</s>
