Download Representing a distribution by stopping a Brownian Motion: Root`s

BULL. AUSTRAL. MATH. SOC.
VOL.
60C40
34 (1986) 427-431.
REPRESENTING A DISTRIBUTION BY STOPPING A
BROWNIAN MOTION: ROOT'S CONSTRUCTION
SHEY SHIUNG SHEU
A closed subset
(i)
C
of
[O,°°] x [-">,«>]
(«°,x) e C ,vx
(ii)
(t,±»)
(t,x)
(iii)
6 C , V t ,
e C
C , let
variable
X
implies
(s,x) e C , ¥ S > t .
x (C)
be
C
starting at the origin and a
inf{i : (t,B(t)) S C} . A random
(or a distribution
exists a barrier
F ) is called achievable if there
B(T(C))
so that
In this paper we shall show that if
with finite mean or if
then
X
if
,
Given a Brownian motion (B(t))
barrier
i s c a l l e d a barrier
is achievable.
X
is distributed
X
as
X(F) .
is bounded above or below
has zero mean and
£"( |X|
log \x\) < °>
This result gives a partial answer to a
problem raised by Loynes [7].
1. Introduction
In dealing with various limit theorems for sums of independent
random variables, Skorohod (see [9], page 163) introduced a method to
imbed a mean-zero random variable
X
into a Brownian motion
starting at the origin; that is, he found a stopping time
x
B(t) , t > 0,
(relative
to a filtration generally larger than the Brownian filtration) so that
B
has the same distribution as
E(X ) = E{T) .
If one requires
Brownian filtration ( x
X
x
(denoted by
B
~ X ) and, furthermore,
to be a stopping time relative to the
depends only on Brownian paths), whether such
x
can be still constructed has been a research problem for many authors
Received 28 January 1986.
Copyright Clearance Centre, Inc. Serial-fee code:
$A2.00 + 0.00.
427
000-9727/86
Downloaded from https:/www.cambridge.org/core. IP address: 88.99.165.207, on 16 Jun 2017 at 11:34:15, subject to the Cambridge Core terms of use,
available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700010315
428
Shey Shiung Sheu
(see Root [8], Dubins [6], Chacon and Walsh [4], Azema and Yor [I], Bass
[2], Vallois [10], etc.).
most intuitive.
Among these constructions, Root's seems
His stopping time is the hitting time of a certain set
in the compactified time-state space
C
in
H
(i)
(ii)
(iii)
H = [0,°°] x [-«>,<»] . A closed set
will be called a barrier if
(»,a;) G C
(t,+«) G C
(t,x) G C
for all a; G C ,
for all
£,
implies
(s,x) G C
for all s > £ .
The space of barriers will be compact under the Hausdorff metric. For
a barrier
C , let
T (C) = inf{£ , B(T)) G C] . Root proved that if X
has zero mean and finite variance, then there exists a barrier
that
C so
2
S ( T ( C ) ) ~ X , and E{i{C)) = E(X ) . Loynes [7] defines a random
variable
X
to be achievable if there exists a barrier
C
so that
B ( T ( C ) ) ~ X . He posed the problem of finding conditions for X
achievable.
In this respect, any X
to be
with zero mean and finite variance
is achievable; any degenerate random variable is achievable and,
therefore, being zero-mean is not a necessary condition.
Loynes [7] showed that if X
E{X) > 0 , or on [a,°°)
and E(X) < 0 , then
also pointed out that if X
In fact,
is concentrated on a half line (-°°,b] and
X
is achievable. He
is achievable, then
1
E( |X|")
< °° for all
p , 0 < p < 1 . Unfortunately, Loynes
results do not cover important
cases such as Poisson distributions ( X
concentrated on the positive
half line but E(X) > 0 ) .
In this paper, we shall improve his results.
2. Main results
Call a sequence of random variables
if
vc > 0, 3A > 0
such that
{X } stochastically bounded
P{\X \ >A) < e
for all n .
00
LEMMA 2.1. Let
C
-*• Cm .
probability.
then
{C } _
be a sequence of barriers such that
Then the corresponding hitting times
In particular, if
Ca
x (C ) •*• T (CJ
consists of points at «
in
only 3
i{C ) is not stochastically bounded.
Proof.
This is just a rephrase of Lemma 1 in Loynes [7].
D
Downloaded from https:/www.cambridge.org/core. IP address: 88.99.165.207, on 16 Jun 2017 at 11:34:15, subject to the Cambridge Core terms of use,
available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700010315
Brownian motion
LEMMA 2.2. Let {X}
429
be a sequence of random variables such
CO
that X
converges to X^ in distribution. Let {C }
be a
sequence of barriers such that C •* C^ and C^ consists of at least
one finite point. If B.n~X,\<.n<<»3
Proof.
then B „
By Lemma 2.1, t (C ) ->• i{C ) in probability.
By assumption,
PiTiC^) < =>) = 1 . Therefore, there exists a subsequence
almost surely.
~X .
x (C ,) -»• T (CJ)
By the continuity of Brownian paths, we conclude
THEOREM 2.3. Any random variable X bounded below or above with
finite mean is achievable. In particular, the Poisson random variable
is achievable.
Proof.
Without loss of generality, we may assume that
By Loynes1 results, we may also assume
f kM
with probability y d
)
b
with probability —(1
)
-y(n-l) (kM+b)
where
k
X > b > -» .
M = E(X)> 0 . Let
is chosen so that
with probability - ,
k > 0 , (k - 2)M +b > 0 . Since
Y
has_
mean zero and finite variance, it is achievable and the barrier can be
expressed as
{(t,x) : t > 0 , x = kM or
{(t,X) : t > t
J? = n
n
, X = b]
for some
if X > n , and let M
I
x
n
=x
if
^ < « ;
n
with probability 1 - —
n
-{n-l)M
Z
> 0 . Let
= £(X ) . Let
n
X
t
- j(n - 1){kM +b)} U
n
!
with probability —
has mean zero and finite variance and hence, is achievable. The
corresponding barrier can be expressed as
C
- {(t,x) : b < x < n ,
t > tn(x)} u {(t,x) -. t > 0 , x - -(n-l)Mn) . For n > kM , let
t'-= inf{t (x) :fc< x < kM} . since n > kM , -(rc-l)M > —(n-1
K
n
n
2
Downloaded from https:/www.cambridge.org/core. IP address: 88.99.165.207, on 16 Jun 2017 at 11:34:15, subject to the Cambridge Core terms of use,
available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700010315
430
Shey Shiung Sheu
we have
t' < t
.
But
Y
converges in distribution to
P(Y = kM) = P(Y = b) = — .
Therefore,
number and consequently,
C
converges to
X
Lemma 2.2.
t
will converge to a finite
will not diverge to infinity.
in distribution, X
We may assume that
X
< X < b
Of course,
such that
X
B
; X
= 0 , when
is achievable.
~ X
x
and
a
-*• -°° , b
X < a
Let
T
or
-> °°
such that if
X > bn , and
X
= X,
E(.X ) = 0 .
= T (C' ) be the stopping time
2
£ ( T ) = E[X ) .
n
n
E(X) = 0 ,
is neither bounded above nor bounded
Then there exist sequences
a
Z
°
Proof.
when
Since
is achievable by Lemma 2.1 and
THEOREM 2.4. If X is a random variable satisfying
£"(|^| log \X\) < °° , then X is aeh.ieva.ble.
below.
Y , where
By the famous Burkholder-Gundy' s
n
inequality (see Theorem 6.1 in Burkholder [3]), we have
E[/r~) < a E ( sup
\B(t) |) .
By Doob's inequality (see Doob [5], page 317) and the fact that
sup
\B(t)|
is bounded, we have
n
Hence,
bounded.
{E(/x~) }
Since
is bounded, which implies
X
converges to
by Lemma 2.1 and Lemma 2.2.
X
{ T } is stochastically
in distribution,
X
is achievable
D
References
[7]
J. Az&na and M. Yor, "Une solution simple au proble'me de Skorokhod",
Sent. Prob. XIII, Lecture Notes in Math. 721 (1979), Springer,
90-115.
Downloaded from https:/www.cambridge.org/core. IP address: 88.99.165.207, on 16 Jun 2017 at 11:34:15, subject to the Cambridge Core terms of use,
available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700010315
Brownian motion
[2]
431
R.F. Bass, "Skorohod imbedding via stochastic integrals", Sem. Prob.
XVII, Lecture notes in Math., 986 (1983), Springer, 221-224.
[3]
D.L. Burkholder, "Distribution function inequalities for martingales",
Ann. Prob., (1) 1 (1973), 19-42.
[4]
R.V. Chacon and J.B. Walsh, "One-dimensional potential embedding",
Sem. Prob. X, Lecture Notes in Math., 511 (1976), Springer,
19-23.
[5]
J.L. Doob, Stochastic Process (Wiley, New York, 1953).
[6]
L.E. Dubins,
"On a theorem of Skorohod", Ann. Math. Stat., 39 (1968),
2094-2097.
[7]
R.M. Loynes, "Stopping times on Brownian motion:
some properties
of Root's construction", Z. Wahr. verm Gebiete, 16 (1970),
211-218.
[8] D.H. Root, "The existence of certain stopping times on Brownian
motion", Ann. Math. Stat., (2) 40 (1967), 715-718.
[9] A.V. Skorokhod, Studies in the Theory of Random Processes (AddisonWesley, Reading, 1965).
[7 0] P. Vallois, "Le probleme de Skorokhod sur R: une approche avec le
temps local", Sem. Prob. XVII, Lecture Notes in Math., 986
(1983), Springer, 227-239.
Institute of Applied Mathematics,
National Tsing Hua University,
Hsinchu, Taiwan 30043,
Republic of China.
Downloaded from https:/www.cambridge.org/core. IP address: 88.99.165.207, on 16 Jun 2017 at 11:34:15, subject to the Cambridge Core terms of use,
available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700010315