<s>
In	O
probability	O
theory	O
,	O
there	B-Algorithm
exist	I-Algorithm
several	O
different	O
notions	O
of	O
convergence	B-Algorithm
of	I-Algorithm
random	I-Algorithm
variables	I-Algorithm
.	O
</s>
<s>
The	O
convergence	B-Algorithm
of	O
sequences	O
of	O
random	O
variables	O
to	O
some	O
limit	B-Algorithm
random	O
variable	O
is	O
an	O
important	O
concept	O
in	O
probability	O
theory	O
,	O
and	O
its	O
applications	O
to	O
statistics	O
and	O
stochastic	O
processes	O
.	O
</s>
<s>
The	O
same	O
concepts	O
are	O
known	O
in	O
more	O
general	O
mathematics	O
as	O
stochastic	B-Algorithm
convergence	I-Algorithm
and	O
they	O
formalize	O
the	O
idea	O
that	O
a	O
sequence	O
of	O
essentially	O
random	O
or	O
unpredictable	O
events	O
can	O
sometimes	O
be	O
expected	O
to	O
settle	O
down	O
into	O
a	O
behavior	O
that	O
is	O
essentially	O
unchanging	O
when	O
items	O
far	O
enough	O
into	O
the	O
sequence	O
are	O
studied	O
.	O
</s>
<s>
The	O
different	O
possible	O
notions	O
of	O
convergence	B-Algorithm
relate	O
to	O
how	O
such	O
a	O
behavior	O
can	O
be	O
characterized	O
:	O
two	O
readily	O
understood	O
behaviors	O
are	O
that	O
the	O
sequence	O
eventually	O
takes	O
a	O
constant	O
value	O
,	O
and	O
that	O
values	O
in	O
the	O
sequence	O
continue	O
to	O
change	O
but	O
can	O
be	O
described	O
by	O
an	O
unchanging	O
probability	O
distribution	O
.	O
</s>
<s>
"	O
Stochastic	B-Algorithm
convergence	I-Algorithm
"	O
formalizes	O
the	O
idea	O
that	O
a	O
sequence	O
of	O
essentially	O
random	O
or	O
unpredictable	O
events	O
can	O
sometimes	O
be	O
expected	O
to	O
settle	O
into	O
a	O
pattern	O
.	O
</s>
<s>
These	O
other	O
types	O
of	O
patterns	O
that	O
may	O
arise	O
are	O
reflected	O
in	O
the	O
different	O
types	O
of	O
stochastic	B-Algorithm
convergence	I-Algorithm
that	O
have	O
been	O
studied	O
.	O
</s>
<s>
While	O
the	O
above	O
discussion	O
has	O
related	O
to	O
the	O
convergence	B-Algorithm
of	O
a	O
single	O
series	O
to	O
a	O
limiting	O
value	O
,	O
the	O
notion	O
of	O
the	O
convergence	B-Algorithm
of	O
two	O
series	O
towards	O
each	O
other	O
is	O
also	O
important	O
,	O
but	O
this	O
is	O
easily	O
handled	O
by	O
studying	O
the	O
sequence	O
defined	O
as	O
either	O
the	O
difference	O
or	O
the	O
ratio	O
of	O
the	O
two	O
series	O
.	O
</s>
<s>
Other	O
forms	O
of	O
convergence	B-Algorithm
are	O
important	O
in	O
other	O
useful	O
theorems	O
,	O
including	O
the	O
central	O
limit	B-Algorithm
theorem	O
.	O
</s>
<s>
With	O
this	O
mode	O
of	O
convergence	B-Algorithm
,	O
we	O
increasingly	O
expect	O
to	O
see	O
the	O
next	O
outcome	O
in	O
a	O
sequence	O
of	O
random	O
experiments	O
becoming	O
better	O
and	O
better	O
modeled	O
by	O
a	O
given	O
probability	O
distribution	O
.	O
</s>
<s>
Convergence	B-Algorithm
in	O
distribution	O
is	O
the	O
weakest	O
form	O
of	O
convergence	B-Algorithm
typically	O
discussed	O
,	O
since	O
it	O
is	O
implied	O
by	O
all	O
other	O
types	O
of	O
convergence	B-Algorithm
mentioned	O
in	O
this	O
article	O
.	O
</s>
<s>
However	O
,	O
convergence	B-Algorithm
in	O
distribution	O
is	O
very	O
frequently	O
used	O
in	O
practice	O
;	O
most	O
often	O
it	O
arises	O
from	O
application	O
of	O
the	O
central	O
limit	B-Algorithm
theorem	O
.	O
</s>
<s>
For	O
example	O
,	O
if	O
are	O
distributed	O
uniformly	O
on	O
intervals	O
,	O
then	O
this	O
sequence	O
converges	B-Algorithm
in	I-Algorithm
distribution	I-Algorithm
to	O
the	O
degenerate	O
random	O
variable	O
.	O
</s>
<s>
Indeed	O
,	O
for	B-Algorithm
all	I-Algorithm
n	O
when	O
,	O
and	O
for	B-Algorithm
all	I-Algorithm
when	O
.	O
</s>
<s>
However	O
,	O
for	O
this	O
limiting	O
random	O
variable	O
,	O
even	O
though	O
for	B-Algorithm
all	I-Algorithm
.	O
</s>
<s>
Thus	O
the	O
convergence	B-Algorithm
of	O
cdfs	O
fails	O
at	O
the	O
point	O
where	O
is	O
discontinuous	O
.	O
</s>
<s>
For	O
random	B-General_Concept
vectors	I-General_Concept
the	O
convergence	B-Algorithm
in	O
distribution	O
is	O
defined	O
similarly	O
.	O
</s>
<s>
The	O
definition	O
of	O
convergence	B-Algorithm
in	O
distribution	O
may	O
be	O
extended	O
from	O
random	B-General_Concept
vectors	I-General_Concept
to	O
more	O
general	O
random	O
elements	O
in	O
arbitrary	O
metric	O
spaces	O
,	O
and	O
even	O
to	O
the	O
“	O
random	O
variables	O
”	O
which	O
are	O
not	O
measurable	O
—	O
a	O
situation	O
which	O
occurs	O
for	O
example	O
in	O
the	O
study	O
of	O
empirical	B-General_Concept
processes	I-General_Concept
.	O
</s>
<s>
This	O
is	O
the	O
“	O
weak	O
convergence	B-Algorithm
of	O
laws	O
without	O
laws	O
being	O
defined	O
”	O
—	O
except	O
asymptotically	O
.	O
</s>
<s>
for	B-Algorithm
all	I-Algorithm
continuous	O
bounded	O
functions	O
.	O
</s>
<s>
Since	O
,	O
the	O
convergence	B-Algorithm
in	O
distribution	O
means	O
that	O
the	O
probability	O
for	O
to	O
be	O
in	O
a	O
given	O
range	O
is	O
approximately	O
equal	O
to	O
the	O
probability	O
that	O
the	O
value	O
of	O
is	O
in	O
that	O
range	O
,	O
provided	O
is	O
sufficiently	O
large	O
.	O
</s>
<s>
In	O
general	O
,	O
convergence	B-Algorithm
in	O
distribution	O
does	O
not	O
imply	O
that	O
the	O
sequence	O
of	O
corresponding	O
probability	O
density	O
functions	O
will	O
also	O
converge	O
.	O
</s>
<s>
However	O
,	O
according	O
to	O
Scheffé	O
’s	O
theorem	O
,	O
convergence	B-Algorithm
of	O
the	O
probability	O
density	O
functions	O
implies	O
convergence	B-Algorithm
in	O
distribution	O
.	O
</s>
<s>
The	O
portmanteau	B-Algorithm
lemma	I-Algorithm
provides	O
several	O
equivalent	O
definitions	O
of	O
convergence	B-Algorithm
in	O
distribution	O
.	O
</s>
<s>
The	O
lemma	O
states	O
that	O
converges	B-Algorithm
in	I-Algorithm
distribution	I-Algorithm
to	O
if	O
and	O
only	O
if	O
any	O
of	O
the	O
following	O
statements	O
are	O
true	O
:	O
</s>
<s>
for	B-Algorithm
all	I-Algorithm
continuity	O
points	O
of	O
;	O
</s>
<s>
for	B-Algorithm
all	I-Algorithm
bounded	O
,	O
continuous	O
functions	O
(	O
where	O
denotes	O
the	O
expected	O
value	O
operator	O
)	O
;	O
</s>
<s>
for	B-Algorithm
all	I-Algorithm
bounded	O
,	O
Lipschitz	O
functions	O
;	O
</s>
<s>
for	B-Algorithm
all	I-Algorithm
nonnegative	O
,	O
continuous	O
functions	O
;	O
</s>
<s>
for	B-Algorithm
all	I-Algorithm
continuity	O
sets	O
of	O
random	O
variable	O
;	O
</s>
<s>
The	O
continuous	O
mapping	O
theorem	O
states	O
that	O
for	O
a	O
continuous	O
function	O
,	O
if	O
the	O
sequence	O
converges	B-Algorithm
in	I-Algorithm
distribution	I-Algorithm
to	O
,	O
then	O
converges	B-Algorithm
in	I-Algorithm
distribution	I-Algorithm
to	O
.	O
</s>
<s>
Note	O
however	O
that	O
convergence	B-Algorithm
in	O
distribution	O
of	O
to	O
and	O
to	O
does	O
in	O
general	O
not	O
imply	O
convergence	B-Algorithm
in	O
distribution	O
of	O
to	O
or	O
of	O
to	O
.	O
</s>
<s>
Lévy	O
’s	O
continuity	O
theorem	O
:	O
the	O
sequence	O
converges	B-Algorithm
in	I-Algorithm
distribution	I-Algorithm
to	O
if	O
and	O
only	O
if	O
the	O
sequence	O
of	O
corresponding	O
characteristic	O
functions	O
converges	B-Algorithm
pointwise	I-Algorithm
to	O
the	O
characteristic	O
function	O
of	O
.	O
</s>
<s>
Convergence	B-Algorithm
in	O
distribution	O
is	O
metrizable	O
by	O
the	O
Lévy	O
–	O
Prokhorov	O
metric	O
.	O
</s>
<s>
A	O
natural	O
link	O
to	O
convergence	B-Algorithm
in	O
distribution	O
is	O
the	O
Skorokhod	O
's	O
representation	O
theorem	O
.	O
</s>
<s>
The	O
basic	O
idea	O
behind	O
this	O
type	O
of	O
convergence	B-Algorithm
is	O
that	O
the	O
probability	O
of	O
an	O
“	O
unusual	O
”	O
outcome	O
becomes	O
smaller	O
and	O
smaller	O
as	O
the	O
sequence	O
progresses	O
.	O
</s>
<s>
The	O
concept	O
of	O
convergence	B-Algorithm
in	O
probability	O
is	O
used	O
very	O
often	O
in	O
statistics	O
.	O
</s>
<s>
Convergence	B-Algorithm
in	O
probability	O
is	O
also	O
the	O
type	O
of	O
convergence	B-Algorithm
established	O
by	O
the	O
weak	O
law	O
of	O
large	O
numbers	O
.	O
</s>
<s>
Then	O
is	O
said	O
to	O
converge	O
in	O
probability	O
to	O
X	O
if	O
for	O
any	O
and	O
any	O
δ>0	O
there	B-Algorithm
exists	I-Algorithm
a	O
number	O
N	O
(	O
which	O
may	O
depend	O
on	O
ε	O
and	O
δ	O
)	O
such	O
that	O
for	B-Algorithm
all	I-Algorithm
n≥N	O
,	O
Pn(ε )	O
<	O
δ	O
(	O
the	O
definition	O
of	O
limit	B-Algorithm
)	O
.	O
</s>
<s>
Notice	O
that	O
for	O
the	O
condition	O
to	O
be	O
satisfied	O
,	O
it	O
is	O
not	O
possible	O
that	O
for	O
each	O
n	O
the	O
random	O
variables	O
X	O
and	O
Xn	O
are	O
independent	O
(	O
and	O
thus	O
convergence	B-Algorithm
in	O
probability	O
is	O
a	O
condition	O
on	O
the	O
joint	O
cdf	O
's	O
,	O
as	O
opposed	O
to	O
convergence	B-Algorithm
in	O
distribution	O
,	O
which	O
is	O
a	O
condition	O
on	O
the	O
individual	O
cdf	O
's	O
)	O
,	O
unless	O
X	O
is	O
deterministic	O
like	O
for	O
the	O
weak	O
law	O
of	O
large	O
numbers	O
.	O
</s>
<s>
At	O
the	O
same	O
time	O
,	O
the	O
case	O
of	O
a	O
deterministic	O
X	O
cannot	O
,	O
whenever	O
the	O
deterministic	O
value	O
is	O
a	O
discontinuity	O
point	O
(	O
not	O
isolated	O
)	O
,	O
be	O
handled	O
by	O
convergence	B-Algorithm
in	O
distribution	O
,	O
where	O
discontinuity	O
points	O
have	O
to	O
be	O
explicitly	O
excluded	O
.	O
</s>
<s>
Convergence	B-Algorithm
in	O
probability	O
is	O
denoted	O
by	O
adding	O
the	O
letter	O
p	O
over	O
an	O
arrow	O
indicating	O
convergence	B-Algorithm
,	O
or	O
using	O
the	O
"	O
plim	O
"	O
probability	O
limit	B-Algorithm
operator	O
:	O
</s>
<s>
Convergence	B-Algorithm
in	O
probability	O
implies	O
convergence	B-Algorithm
in	O
distribution	O
.	O
</s>
<s>
In	O
the	O
opposite	O
direction	O
,	O
convergence	B-Algorithm
in	O
distribution	O
implies	O
convergence	B-Algorithm
in	O
probability	O
when	O
the	O
limiting	O
random	O
variable	O
X	O
is	O
a	O
constant	O
.	O
</s>
<s>
Convergence	B-Algorithm
in	O
probability	O
does	O
not	O
imply	O
almost	O
sure	O
convergence	B-Algorithm
.	O
</s>
<s>
Convergence	B-Algorithm
in	O
probability	O
defines	O
a	O
topology	B-Architecture
on	O
the	O
space	O
of	O
random	O
variables	O
over	O
a	O
fixed	O
probability	O
space	O
.	O
</s>
<s>
This	O
is	O
the	O
type	O
of	O
stochastic	B-Algorithm
convergence	I-Algorithm
that	O
is	O
most	O
similar	O
to	O
pointwise	B-Algorithm
convergence	I-Algorithm
known	O
from	O
elementary	O
real	O
analysis	O
.	O
</s>
<s>
Using	O
the	O
notion	O
of	O
the	O
limit	B-Algorithm
superior	O
of	O
a	O
sequence	O
of	O
sets	O
,	O
almost	O
sure	O
convergence	B-Algorithm
can	O
also	O
be	O
defined	O
as	O
follows	O
:	O
</s>
<s>
Almost	O
sure	O
convergence	B-Algorithm
is	O
often	O
denoted	O
by	O
adding	O
the	O
letters	O
a.s.	O
over	O
an	O
arrow	O
indicating	O
convergence	B-Algorithm
:	O
</s>
<s>
For	O
generic	O
random	O
elements	O
 { Xn } 	O
on	O
a	O
metric	O
space	O
,	O
convergence	B-Algorithm
almost	O
surely	O
is	O
defined	O
similarly	O
:	O
</s>
<s>
Almost	O
sure	O
convergence	B-Algorithm
implies	O
convergence	B-Algorithm
in	O
probability	O
(	O
by	O
Fatou	O
's	O
lemma	O
)	O
,	O
and	O
hence	O
implies	O
convergence	B-Algorithm
in	O
distribution	O
.	O
</s>
<s>
It	O
is	O
the	O
notion	O
of	O
convergence	B-Algorithm
used	O
in	O
the	O
strong	O
law	O
of	O
large	O
numbers	O
.	O
</s>
<s>
The	O
concept	O
of	O
almost	O
sure	O
convergence	B-Algorithm
does	O
not	O
come	O
from	O
a	O
topology	B-Architecture
on	O
the	O
space	O
of	O
random	O
variables	O
.	O
</s>
<s>
This	O
means	O
there	O
is	O
no	O
topology	B-Architecture
on	O
the	O
space	O
of	O
random	O
variables	O
such	O
that	O
the	O
almost	O
surely	O
convergent	B-Algorithm
sequences	I-Algorithm
are	O
exactly	O
the	O
converging	O
sequences	O
with	O
respect	O
to	O
that	O
topology	B-Architecture
.	O
</s>
<s>
In	O
particular	O
,	O
there	O
is	O
no	O
metric	O
of	O
almost	O
sure	O
convergence	B-Algorithm
.	O
</s>
<s>
This	O
is	O
the	O
notion	O
of	O
pointwise	B-Algorithm
convergence	I-Algorithm
of	O
a	O
sequence	O
of	O
functions	O
extended	O
to	O
a	O
sequence	O
of	O
random	O
variables	O
.	O
</s>
<s>
Sure	O
convergence	B-Algorithm
of	O
a	O
random	O
variable	O
implies	O
all	O
the	O
other	O
kinds	O
of	O
convergence	B-Algorithm
stated	O
above	O
,	O
but	O
there	O
is	O
no	O
payoff	O
in	O
probability	O
theory	O
by	O
using	O
sure	O
convergence	B-Algorithm
compared	O
to	O
using	O
almost	O
sure	O
convergence	B-Algorithm
.	O
</s>
<s>
This	O
is	O
why	O
the	O
concept	O
of	O
sure	O
convergence	B-Algorithm
of	I-Algorithm
random	I-Algorithm
variables	I-Algorithm
is	O
very	O
rarely	O
used	O
.	O
</s>
<s>
Convergence	B-Algorithm
in	O
-th	O
mean	O
tells	O
us	O
that	O
the	O
expectation	O
of	O
the	O
-th	O
power	O
of	O
the	O
difference	O
between	O
and	O
converges	O
to	O
zero	O
.	O
</s>
<s>
This	O
type	O
of	O
convergence	B-Algorithm
is	O
often	O
denoted	O
by	O
adding	O
the	O
letter	O
Lr	O
over	O
an	O
arrow	O
indicating	O
convergence	B-Algorithm
:	O
</s>
<s>
The	O
most	O
important	O
cases	O
of	O
convergence	B-Algorithm
in	O
r-th	O
mean	O
are	O
:	O
</s>
<s>
Convergence	B-Algorithm
in	O
the	O
r-th	O
mean	O
,	O
for	O
r	O
≥	O
1	O
,	O
implies	O
convergence	B-Algorithm
in	O
probability	O
(	O
by	O
Markov	O
's	O
inequality	O
)	O
.	O
</s>
<s>
Furthermore	O
,	O
if	O
r	O
>	O
s	O
≥	O
1	O
,	O
convergence	B-Algorithm
in	O
r-th	O
mean	O
implies	O
convergence	B-Algorithm
in	O
s-th	O
mean	O
.	O
</s>
<s>
Hence	O
,	O
convergence	B-Algorithm
in	O
mean	O
square	O
implies	O
convergence	B-Algorithm
in	O
mean	O
.	O
</s>
<s>
None	O
of	O
the	O
above	O
statements	O
are	O
true	O
for	O
convergence	B-Algorithm
in	O
distribution	O
.	O
</s>
<s>
The	O
chain	O
of	O
implications	O
between	O
the	O
various	O
notions	O
of	O
convergence	B-Algorithm
are	O
noted	O
in	O
their	O
respective	O
sections	O
.	O
</s>
<s>
Convergence	B-Algorithm
in	O
probability	O
implies	O
there	B-Algorithm
exists	I-Algorithm
a	O
sub-sequence	O
which	O
almost	O
surely	O
converges	O
:	O
</s>
<s>
Convergence	B-Algorithm
in	O
r-th	O
order	O
mean	O
implies	O
convergence	B-Algorithm
in	O
probability	O
:	O
</s>
<s>
Convergence	B-Algorithm
in	O
r-th	O
order	O
mean	O
implies	O
convergence	B-Algorithm
in	O
lower	O
order	O
mean	O
,	O
assuming	O
that	O
both	O
orders	O
are	O
greater	O
than	O
or	O
equal	O
to	O
one	O
:	O
</s>
<s>
If	O
converges	O
in	O
probability	O
to	O
X	O
,	O
and	O
if	O
for	B-Algorithm
all	I-Algorithm
n	O
and	O
some	O
b	O
,	O
then	O
converges	O
in	O
rth	O
mean	O
to	O
X	O
for	B-Algorithm
all	I-Algorithm
.	O
</s>
<s>
Usually	O
,	O
convergence	B-Algorithm
in	O
distribution	O
does	O
not	O
imply	O
convergence	B-Algorithm
almost	O
surely	O
.	O
</s>
<s>
However	O
,	O
for	O
a	O
given	O
sequence	O
 { Xn } 	O
which	O
converges	B-Algorithm
in	I-Algorithm
distribution	I-Algorithm
to	O
X0	O
it	O
is	O
always	O
possible	O
to	O
find	O
a	O
new	O
probability	O
space	O
( Ω	O
,	O
F	O
,	O
P	O
)	O
and	O
random	O
variables	O
{	O
Yn	O
,	O
n	O
=	O
0	O
,	O
1	O
,	O
...	O
}	O
defined	O
on	O
it	O
such	O
that	O
Yn	O
is	O
equal	O
in	O
distribution	O
to	O
for	O
each	O
,	O
and	O
Yn	O
converges	O
to	O
Y0	O
almost	O
surely	O
.	O
</s>
<s>
If	O
for	B-Algorithm
all	I-Algorithm
ε	O
>	O
0	O
,	O
</s>
<s>
the	O
above	O
sequence	O
of	O
tail	O
probabilities	O
is	O
summable	O
for	B-Algorithm
all	I-Algorithm
)	O
,	O
then	O
also	O
converges	O
almost	O
surely	O
to	O
X	O
.	O
</s>
<s>
The	O
dominated	O
convergence	B-Algorithm
theorem	O
gives	O
sufficient	O
conditions	O
for	O
almost	O
sure	O
convergence	B-Algorithm
to	O
imply	O
L1-convergence	O
:	O
</s>
<s>
A	O
necessary	O
and	O
sufficient	O
condition	O
for	O
L1	O
convergence	B-Algorithm
is	O
and	O
the	O
sequence	O
(	O
Xn	O
)	O
is	O
uniformly	O
integrable	O
.	O
</s>
