Limit theorems for the estimation of L1 integrals using
the Brownian motion
Krishna B. Athreyaa , Raoul Normandb , Vivekananda Roya , Sheng-Jhih Wuc
a Iowa State University, Ames, Iowa 50011, USA
of Mathematics, Academia Sinica, 6F, Astronomy-Mathematics Building, No. 1,
Sec. 4, Roosevelt Road, Taipei 10617, Taiwan
c Center for Advanced Statistics and Econometrics Research, School of Mathematical
Sciences, Soochow University, Suzhou 215006, China
b Institute
Abstract
We provide a point estimate for integrals on R, based on the standard Brownian
motion. We prove the consistency of the estimator and limit theorems for the
fluctuations. The proof relies on computing the distribution of the local time of
a Brownian motion at a specific stopping time.
Keywords: Brownian motion, local time, point estimate, Ray-Knight
theorem, regenerative process
2010 MSC: 62F10, 60J65, 60F05
1. Introduction
Let
R f : R → R be integrable with respect to the Lebesgue measure and
λ = R f (x) dx. Consider the problem of obtaining a point estimate and a
confidence interval for λ based on a suitable statistical data.
This problem was solved on Rd in [3] via simple symmetric random walks
on Z and an appropriate randomization around the observed values of the walk.
In that paper, the authors introduced a new Monte Carlo procedure called
R Regenerative Sequence Monte Carlo (RSMC), which works for estimating S f dπ
for an integrable function f on a measure space (S, S, π). While the classical
i.i.d. Monte Carlo method and the currently popular Markov Chain Monte
Carlo method require π(S) < ∞, the RSMC allows π(S) to be finite or infinite.
The key requirement imposed in [3] is that the regenerative sequence used be
such that its occupation measure coincides with a constant multiple of the given
measure π. The regenerative sequence need not even be Markovian.
Here, we produce a solution similar to that of [3], but using the regenerative
property of standard Brownian motion. This property was already exploited by
Email addresses: kbathreya@gmail.com (Krishna B. Athreya),
rnormand@math.sinica.edu.tw (Raoul Normand), vroy@iastate.edu (Vivekananda Roy),
szwu@suda.edu.cn (Sheng-Jhih Wu)
Preprint submitted to Elsevier
January 19, 2015
[1] in proving many results similar to those of Darling and Kac [5], in particular
giving an alternative proof of a well-known result of Kallianpur and Robbins
[6].
The paper is organized as follows. In Section 2, we present our main result.
The main ingredient of the proof is computing the distribution of the local
time of a standard Brownian motion stopped at some specific stopping times,
what is carried out in Section 3. The proof of the main result is then an easy
consequence, or more precisely an easy modification, of the results of [3], and is
sketched in Section 4
2. Main results
Let B = {B(t), t ≥ 0} be the standard Brownian motion (SBM) on R and
T = inf{t > 0, B(t) = 0 & ∃s ∈ (0, t), B(s) = 1}
(1)
be the first hitting time of 0 after hitting 1. By the recurrence property of
SBM, this stopping time is a.s. finite, and by the strong Markov property
{B(T + u), u ≥ 0} is independent of {B(u), 0 ≤ u < T } and has the same
distribution as the SBM. In other words, the process regenerates at time T .
In particular, we may recursively define the regeneration times T0 = 0, T1 = T
and for i ≥ 1,
Ti+1 = inf{t > Ti , B(t) = 0 & ∃s ∈ (Ti , Ti+1 ), B(s) = 1}.
The sequence (Ti+1 − Ti )i≥0 is then i.i.d., and more generally, the variables
{(B(t), Ti ≤ t ≤ Ti+1 ), Ti+1 − Ti }i≥0 are i.i.d. Let also {N (t), t ≥ 0} be the
renewal process generated by the random walk {Ti , i ≥ 0}, i.e. N (t) = k for
Tk ≤ t < Tk+1 .
Consider
R now f : R → R integrable with respect to the Lebesgue measure
and λ = R f (x) dx. The point estimate for λ that we will use is
1
λ (t) =
N (t)
∗
Z
t
f (B(s)) ds.
(2)
0
Our main result is the following.
Theorem 2.1. Let f : R → R be integrable with respect to Rthe Lebesgue
p measure, and define λ and λ∗ as above. Moreover, assume that R |f (x)| |x| dx <
+∞. Then the following hold.
(i) The quantity
2
Z
σ =E
!2
T
f (B(s)) ds
0
is positive and finite.
(ii) λ∗ (t) → λ a.s. as t → +∞.
2
− λ2
(iii) The convergence in distribution
(λ∗ (t) − λ) p
d
N (t) → N(0, 1)
σ
holds as t → +∞.
(iv) The convergence in distribution
(λ∗ (t) − λ) 1/4 d
t
→ Q,
σ
holds as t → +∞, where Q is a random variable having the same distribution as B(V ), where V is a random variable independent of√B and has a
distribution of a stable law of order 1/2, with E(e−sV ) = e− s for s ≥ 0.
In other words, λ∗ (t) is a consistent estimator for λ, and the fluctuations
around the limit are p
obtained. Note
√ that, in order to obtain these fluctuations,
one has to divide by N (t), not t as√in the usual Monte Carlo methods. Since
(N (t)) itself turns out to behave like t, the convergence occurs at speed t1/4 ,
as in the examples of [3].
As mentioned above, most of this stems from the results of [3] and [4], so the
bulk of the proof consists in checking that their conditions hold. To this end,
the central point will be proving the next result, which might have an interest
on its own.
In the following, write {L(t, x), t ≥ 0, x ∈ R} for the local time process of
(B(t)). That is, with probability 1, for any Borel set A ⊂ R and t ≥ 0,
Z t
Z
1A (B(s)) ds =
L(t, x) dx.
0
A
It is well-known that such a process exists (see e.g. Chapter VI in [7]). The
next result, proved in Section 3.1 is concerned with the marginal distribution of
L(T, x) for x ∈ R, where T is as in (1). We denote, for x ∈ R,
τx = inf{t ≥ 0, B(t) = x}
to be the hitting time of x for the SBM.
Theorem 2.2. Let Y1 and Y2 be two independent chi-squared random variables
(d)
(d)
with two degrees of freedom, i.e. Y1 = Y2 = Z12 + Z22 for independent standard
normal Z1 and Z2 . Assume moreover that Y1 and Y2 are independent from B,
and let T be as in (1). Then, the following hold:
(d)
(i) for x > 1, L(T, x) = 1{τx ≤T } xY1 ,
(d)
(ii) for x ∈ [0, 1], L(T, x) = xY1 + (1 − x)Y2 ,
(d)
(iii) for x < 0, L(T, x) = 1{τx ≤τ1 } (1 − x)Y1 .
Corollaries of this result are moreover given in Section 3.2, and used afterwards in Section 4 to prove Theorem 2.1.
3
3. Distribution of the local time
3.1. Proof of Theorem 2.2
In the following, we will also consider Brownian motions B a ≡ {B a (t) ≡
B(t) + a, t ≥ 0} where {B(t), t ≥ 0} is SBM. We define τxa as the hitting time
of x by B a . Let {La (t, x), t ≥ 0, x ∈ R} be its local time process.
Let x > 1. On the event {τx > T }, L(T, x) = 0. By the strong Markov
property of B, on the event {τx ≤ T },
(d)
{B(τx + t), t ∈ [0, T − τx ]} = {B x (t), t ∈ [0, τ0x )]},
and further, the left side is independent of {B(u), 0 ≤ u ≤ τx }. Since B does
not accumulate any local time at x before τx , i.e. L(τx , x) = 0, we have
(d)
L(T, x) = Lx (τ0x , x).
By translation invariance of SBM, the right side above has the same distribution as L(τ−x , 0). This, in turn by symmetry of SBM about 0, has the same
distribution as L(τx , 0). Next, the scaling property of local times (see exercise
2.11 Chapter VI and Theorem 2.2 Chapter XI in [7]) ensures that
(d)
L(τx , 0) = xL(τ1 , 0).
Further, the first Ray-Knight theorem (p. 454 in [7]) asserts that the process
{L(τ1 , 1 − a), 0 ≤ a ≤ 1} is a squared Bessel 2 process, i.e. has the distribution
of the squared Euclidean norm of a 2-dimension Brownaian motion:
{H(a), 0 ≤ a ≤ 1} ≡ B1 (a)2 + B2 (a)2 , a ∈ [0, 1] ,
(3)
where B1 and B2 are independent standard Brownian motions. In particular,
(d)
(d)
L(τ1 , 0) = B1 (1)2 + B2 (1)2 = Y1 .
We can thus conclude that, on {τx ≤ T } ,
(d)
L(T, x) = xY1 .
This proves (i).
Next, let x ∈ [0, 1]. The continuity of Brownian trajectories implies that
τx ≤ τ1 ≤ T . Thus the local time at x accumulated on [0, T ] is the sum of
• the local time L(τ1 , x) at x accumulated on [0, τ1 ],
• the local time accumulated at x on [τ1 , T ].
4
But by the strong Markov property again, the second one is independent of
L(τ1 , x) and has the distribution of L1 (τ01 , x). By translation invariance of
SBM, this is distributed as L(τ−1 , x − 1), which by symmetry of SBM has same
distribution as L(τ1 , 1 − x). So,
(d)
L(T, x) = L1 (τ1 , x) + L2 (τ1 , 1 − x)
(4)
for L1 , L2 two independent copies of the process L. By the first Ray-Knight
theorem again, we conclude
(d)
L(T, x) = (1 − x)Y1 + xY2 .
This proves (ii).
Finally, let x < 0. On the event {τx > τ1 }, we have τx > T by continuity of
Brownian trajectories, and thus L(T, x) = 0. On the other hand, on {τx ≤ τ1 },
we have, as before,
(d)
(d)
(d)
(d)
L(T, x) = L(τ1 , x) = Lx (τ1x , x) = L(τ1−x , 0) = (1 − x)L(τ1 , 0) = (1 − x)Y1 ,
where the second equality is by the strong Markov property, the third by the
translation invariance of Brownian motion, the fourth by the scaling property
and the fifth by the Ray-Knight theorem. This proves (iii).
3.2. Corollaries
From Theorem 2.2, we successively deduce the following two corollaries. The
second one will be the main ingredient in the proof of Theorem 2.1.
Corollary 3.1. For all x ∈ R, E(L(T, x)) = 2,

8x
 2
2
E L(T, x) = 8(x − x + 1)


8(1 − x)
and
for x > 1,
for x ∈ [0, 1],
for x < 0.
(d)
Proof. In the following, recall that E(Y1 ) = 2 and E(Y12 ) = 8, since Y1 =
Z12 + Z22 for Z1 , Z2 independent N (0, 1) random variables.
Now, for x > 1, by Theorem 2.2 (i), and independence of 1{τx ≤T } and Y1 ,
E(L(T, x)) = P(τx ≤ T )xE(Y1 ).
But τx ≤ T if and only if the Brownian motion hits 1, then hits x before hitting
0, and thus, by the strong Markov property of SBM,
P(τx ≤ T ) = P(τx1 < τ01 ) =
1
,
x
where the last equality is the usual gambler’s ruin estimate (see Proposition 2.8
Chapter II in [7]). Thus,
E(L(T, x)) =
1
× x × 2 = 2.
x
5
Next, for x ∈ [0, 1], by Theorem 2.2 (ii),
E(L(T, x)) = xE(Y1 ) + (1 − x)E(Y2 ) = 2x + 2(1 − x) = 2.
Finally, for x < 0, by gambler’s ruin estimates again, we have
P(τx ≤ τ1 ) =
1
1−x
and thus by Theorem 2.2 (iii) and independence of 1{τx ≤τ1 } and Y1 , we have
E(L(T, x)) =
1
× (1 − x) × 2 = 2.
1−x
For the second moment, note that, by independence as above we have by
Theorem 2.2 (i), for x > 1,
E(L(T, x)2 ) = E(12{τx ≤T } )x2 E(Y12 ) = P(τx ≤ T )x2 E(Y12 ) =
1
× x2 × 8 = 8x.
x
The same works for x < 0. Finally, for x ∈ [0, 1], by independence,
E(L(T, x)2 ) = x2 E(Y12 ) + (1 − x)2 E(Y22 ) + 2x(1 − x)E(Y1 )E(Y2 )
= 8x2 + 8(1 − x)2 + 8x(1 − x)
= 8(x2 − x + 1)
and the proof is complete.
Corollary
R p 3.2. Let f : R → R be Lebesgue integrable, and assume moreover
that R |x||f (x)| dx < ∞. Then
Z
!2
T
|f (B(s))| ds
E
< ∞,
0
where T is as in (1) and {B(s), s ≥ 0} is SBM.
Proof. Since f is integrable, by the occupation times formula (see Corollary
1.6 Chapter VI in [7]),
! Z
Z T
Z
E
|f (Bs )| ds =
|f (x)|E(L(T, x)) dx = 2 |f (x)| dx < ∞,
0
R
R
RT
RT
so that 0 |f (Bs )| ds < ∞ a.s. and V := 0 f (Bs ) ds is a well-defined random
variable. Next, by Minkowski inequality [2], for any p ∈ [1, ∞),
Z
p 1/p
(E|V | )
≤
|f (x)|(EL(T, x)p )1/p dx.
R
6
But, by Corollary 3.1 (ii), there is a C ∈ (0, ∞) such that for all x in R,
p
(EL(T, x)2 )1/2 ≤ C |x|.
This yields
(E|V |2 )1/2 ≤ C
Z
p
|f (x)| |x| dx,
R
which is finite by hypothesis.
Remark 3.3. By the same proof, and with an easy extension of Corollary 3.1,
it holds that E(|V |p ) < ∞ when
Z
p−1
|f (x)||x| p dx < ∞.
R
In particular, V has moments of all order when
Z
|f (x)||x| dx < ∞.
R
3.3. A related result and an open problem
From the proof of Theorem 2.2 (ii), more precisely Equation (4), we have
the following result.
Proposition 3.4. Let H1 (·) and H2 (·) be two independent Bessel 2 processes
as defined in (3). Then
(d)
{L(T, x), x ∈ [0, 1]} = {H1 (x) + H2 (1 − x), x ∈ [0, 1]}.
This gives the distribution of the process (L(T, x), x ∈ [0, 1]). A natural open
problem is determining the distribution of the whole process (L(T, x), x ∈ R).
The techniques used above do not readily apply to this question.
4. Proof of Theorem 2.1
We recall the main result of [4]. Assume that f ∈ L1 (R) and
!2
Z
T
< ∞.
f (B(s)) ds
E
(5)
0
Then
p
Z t
N (t)
1
d
f (B(s)) ds − λ → N(0, 1)
σ
N (t) 0
where
2
Z
σ =E
!2
T
f (B(s)) ds
− λ2 .
0
p
R
By Corollary 3.2, a sufficient condition for (5) is that R |f (x)| |x| dx < ∞.
The proof of Theorem 2.1 is then a perusing of the arguments of Theorem 1 in
[4] and is omitted.
7
Remark 4.1. Now, as in [4] based on Theorem 2.1 (i) and (ii) and the data
{B(u), 0 ≤ u ≤ t}, a point estimate for λ is provided by λ∗ (t), and an asymptotic
(1 − α) level (0 < α < 1) confidence interval for λ is
p
p
It ≡ (λ∗ (t) − σ̂t zα / N (t), λ∗ (t) + σ̂t zα / N (t))
where
N (t)
P (|Z| > zα ) = α,
Z ∼ N (0, 1),
σ̂t2 =
X
ξj2 /N (t) − λ∗ (t)2
j=1
and ξj =
R Tj
Tj−1
f (B(u))du, j ≥ 1.
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