If
X
0
,
X
1
,
…
{X_{0}},{X_1},\ldots
is an arbitrarily-dependent sequence of random variables taking values in
[
0
,
1
]
[0,1]
and if
V
(
X
0
,
X
1
,
…
)
V({X_0},{X_1},\ldots )
is the supremum, over stop rules
t
t
, of
E
X
t
E{X_t}
, then the set of ordered pairs
{
(
x
,
y
)
:
x
=
V
(
X
0
,
X
1
,
…
,
X
n
)
\{ (x,y):x = V({X_0},{X_1},\ldots ,{X_n})
and
y
=
E
(
max
j
⩽
n
X
j
)
y = E({\max _{j\, \leqslant \,n}}{X_j})
for some
X
0
,
…
,
X
n
}
{X_0},\ldots ,{X_n}\}
is precisely the set
\[
C
n
=
{
(
x
,
y
)
:
x
⩽
y
⩽
x
(
1
+
n
(
1
−
x
1
/
n
)
)
;
0
⩽
x
⩽
1
}
;
{C_n} = \{ (x,y):x \leqslant y \leqslant x\,( {1 + n\,(1 - {x^{1/n}})} );0 \leqslant x \leqslant 1\} ;
\]
and the set of ordered pairs
{
(
x
,
y
)
:
x
=
V
(
X
0
,
X
1
,
…
)
\{ (x,y):x = V({X_{0}},{X_1},\ldots )
and
y
=
E
(
sup
n
X
n
)
y = E({\sup _n}\;{X_n})
for some
X
0
,
X
1
,
…
}
{X_0},{X_1},\ldots \}
is precisely the set
\[
C
=
⋃
n
=
1
∞
C
n
.
C = \bigcup \limits _{n = 1}^\infty {{C_n}} .
\]
As a special case, if
X
0
,
X
1
,
…
{X_0},{X_1},\ldots
is a martingale with
E
X
0
=
x
E{X_0} = x
, then
E
(
max
j
⩽
n
X
)
⩽
x
+
n
x
(
1
−
x
1
/
n
)
E({\max _{j \leqslant n}} X) \leqslant x + nx(1 - {x^{1\,/\,n}})
and
E
(
sup
n
X
n
)
⩽
x
−
x
ln
x
E({\sup _n}\;{X_n}) \leqslant x - x\ln \;x
, and both inequalities are sharp.