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On the expected diameter of planar Brownian motion
Accepted Manuscript On the expected diameter of planar Brownian motion James McRedmond, Chang Xu
PII: DOI: Reference:
S0167-7152(17)30234-1 http://dx.doi.org/10.1016/j.spl.2017.07.001 STAPRO 7986
To appear in:
Statistics and Probability Letters
Received date : 4 July 2017 Accepted date : 6 July 2017 Please cite this article as: McRedmond, J., Xu, C., On the expected diameter of planar Brownian motion. Statistics and Probability Letters (2017), http://dx.doi.org/10.1016/j.spl.2017.07.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Manuscript Click here to view linked References
On the expected diameter of planar Brownian motion James McRedmonda,∗, Chang Xub a Department b Department
of Mathematical Sciences, Durham University, South Road, Durham DH1 3LE, UK. of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, UK.
Abstract Known results show that the diameter d1 of the trace of planar Brownian motion run for unit time satisfies 1.595 ≤ Ed1 ≤ 2.507. This note improves these bounds to 1.601 ≤ Ed1 ≤ 2.355. Simulations suggest that Ed1 ≈ 1.99. Keywords: Brownian motion, convex hull, diameter 2010 MSC: 60J65, 60D05
1. Introduction Let (bt , t ∈ [0, 1]) be standard planar Brownian motion, and consider the set b[0, 1] = {bt : t ∈ [0, 1]}. The Brownian convex hull H1 := hull b[0, 1] has been well-studied from L´evy (1948, §52.6, pp. 254–256) onwards; √ the expectations of the perimeter length `1 and area a1 of H1 are given by the exact formulae E`1 = 8π (due to Letac, 1978; Tak´acs, 1980) and Ea1 = π/2 (due to El Bachir, 1983). Another characteristic is the diameter d1 := diam H1 = diam b[0, 1] =
sup x,y∈b[0,1]
kx − yk,
for which, in contrast, no explicit formula is known. The exact formulae for E`1 and Ea1 rest on geometric integral formulae of Cauchy; since no such formula is available for d1 , it may not be possible to obtain an explicit formula for Ed1 . However, one may get bounds. By convexity, we have the almost-sure inequalities 2 ≤ `1 /d1 ≤ π, the extrema being the line segment and shapes of constant width (such as the disc). In other words, `1 `1 ≤ d1 ≤ . π 2
√ The formula of Letac (1978) and Tak´acs (1980) says that E`1 = 8π, so we get: p √ Proposition 1. 8/π ≤ Ed1 ≤ 2π. p √ Note that 8/π ≈ 1.5958 and 2π ≈ 2.5066. In this note we improve both of these bounds. 2. Lower bound For the lower bound, we note that b[0, 1] is compact and thus, as a corollary of Lemma 6 below, we have the formula d1 = sup r(θ), (1) 0≤θ≤π
where r is the parametrized range function given by r(θ) = sup (bs · eθ ) − inf (bs · eθ ) , 0≤s≤1
0≤s≤1
∗ Corresponding
author Email addresses: [email protected] (James McRedmond), [email protected] (Chang Xu)
Preprint submitted to Elsevier
4th July 2017
with eθ being the unit vector (cos θ, sin θ). Feller (1951) established that p Er(θ) = 8/π and E(r(θ)2 ) = 4 log 2,
(2)
and the density of r(θ) is given explicitly as
∞ 8 X (−1)k−1 k 2 exp{−k 2 r2 /2}, (r ≥ 0). f (r) = √ 2π k=1
(3)
p Combining (1) with (2) gives immediately Ed1 ≥ Er(0) = 8/π, which is just the lower bound in Proposition 1. For a better result, a consequence of (1) is that d1 ≥ max{r(0), r(π/2)}. Observing that r(0) and r(π/2) are independent, we get: Lemma 2. Ed1 ≥ E max{X1 , X2 }, where X1 and X2 are independent copies of X := r(0). It seems hard to explicitly compute E max{X1 , X2 } in Lemma 2, because although the density given at (3) is known explicitly, it is not very tractable. Instead we obtain a lower bound. Since max{x, y} =
1 (x + y + |x − y|) 2
we get
1 E max{X1 , X2 } = EX + E|X1 − X2 |. (4) 2 Thus with Lemma 2, the lower bound in Proposition 1 is improved given any non-trivial lower bound for E|X1 − X2 |. Using the fact that for any c ∈ R, if m is a median of X, E|X − c| ≥ E|X − m|, we see that E|X1 − X2 | ≥ E|X − m|. Again, the intractability of the density at (3) makes it hard to exploit this. Instead, we provide the following as a crude lower bound on E|X1 − X2 |. Lemma 3. For any a, h > 0, E|X1 − X2 | ≥ 2h P(X ≤ a) P(X ≥ a + h). Proof. We have E|X1 − X2 | ≥ E [|X1 − X2 |1{X1 ≤ a, X2 ≥ a + h}] + E [|X1 − X2 |1{X2 ≤ a, X1 ≥ a + h}] ≥ h P(X1 ≤ a) P(X2 ≥ a + h) + h P(X2 ≤ a) P(X1 ≥ a + h) = 2h P(X ≤ a) P(X ≥ a + h)
which proves the statement. This lower bound yields the following result. Proposition 4. For a, h > 0 define 4 π2 9π 2 π2 4 4 exp − 2 − exp − 2 1 − exp − . g(a, h) := h π 2a 3π 2a π 8(a + h)2 p Then Ed1 ≥ 8/π + g(1.492, 0.337) ≈ 1.6014.
Proof. Consider
Z := sup |bs · e0 |. 0≤s≤1
Then it is known (see Jain and Pruitt, 1975) that for x > 0, 4 π2 4 9π 2 4 π2 exp − 2 − exp − 2 ≤ P(Z < x) ≤ exp − 2 . π 8x 3π 8x π 8x 2
(5)
Moreover, we have Z ≤ X ≤ 2Z. Since X ≤ 2Z, we have
4 π2 9π 2 4 exp − 2 , P(X ≤ a) ≥ P(Z ≤ a/2) ≥ exp − 2 − π 2a 3π 2a
by the lower bound in (5). On the other hand, 4 π2 P(X ≥ a + h) ≥ P(Z ≥ a + h) ≥ 1 − exp − , π 8(a + h)2 by the upper bound in (5). Combining pthese two bounds and applying p Lemma 3 we get E|X1 −X2 | ≥ 2g(a, h). So from (4) and the fact that EX = 8/π by (2) we get Ed1 ≥ 8/π + g(a, h). Numerical evaluation using MAPLE suggests that (a, h) = (1.492, 0.337) is close to optimal, and this choice gives the statement in the proposition. 3. Upper bound We also improve the upper bound in Proposition 1. √ Proposition 5. Ed1 ≤ 8 log 2 ≈ 2.3548. Proof. First, we claim that d21 ≤ r(0)2 + r(π/2)2 .
(6)
It follows from (6) and (2) that E(d21 ) ≤ E(X12 + X22 ) = 2E(X 2 ) = 8 log 2. The result now follows by Jensen’s inequality. It remains to prove the claim (6). Note that the diameter is an increasing function, that is, if A ⊆ B then diam A ≤ diam B. Note also, that by the definition of r(θ), b[0, 1] ⊆ z + [0, r(0)] × [0, r(π/2)] =: Rz for some z ∈ R2 . Since the diameter of the set Rz is attained at the diagonal, p diam Rz = r(0)2 + r(π/2)2 , for all z ∈ R2 , and we have diam b[0, 1] ≤ diam Rz , the result follows. 4. Final remark and auxiliary lemma We make one further remark about second moments. In the proof of Proposition 5, we saw that E(d21 ) ≤ 8 log 2 ≈ 5.5452. A bound in the other direction can be obtained from the fact that d21 ≥ `21 /π 2 , and we have (see Wade and Xu, 2015, §4.1) that E(`21 ) = 4π
Z
π/2
−π/2
dθ
Z
0
∞
du cos θ
cosh(uθ) tanh sinh(uπ/2)
(2θ + π)u 4
≈ 26.1677,
which gives E(d21 ) ≥ 2.651. Finally, for completeness, we state and prove the lemma which was used to obtain equation (1). Lemma 6. Let A ⊂ Rd be a nonempty compact set, and let rA (θ) = supx∈A (x · eθ ) − inf x∈A (x · eθ ). Then diam A = sup rA (θ). 0≤θ≤π
3
Proof. Since A is compact, for each θ there exist x, y ∈ A such that rA (θ) = x · eθ − y · eθ
= (x − y) · eθ ≤ kx − yk.
So sup0≤θ≤π rA (θ) ≤ supx,y∈A kx − yk = diam A. It remains to show that sup0≤θ≤π rA (θ) ≥ diam A. This is clearly true if A consists of a single point, so suppose that A contains at least two points. Suppose that the diameter of A is achieved by x, y ∈ A and let z = y − x be such that zˆ := z/kzk = eθ0 for θ0 ∈ [0, π]. Then sup rA (θ) ≥ rA (θ0 ) ≥ y · eθ0 − x · eθ0
0≤θ≤π
= z · zˆ = kzk = diam A, as required. Acknowledgements The authors are grateful to Andrew Wade for his suggestions on this note. The first author is supported by an EPSRC studentship. References M. El Bachir, “L’enveloppe convexe du mouvement brownien,” Ph.D. thesis, Universit´e Toulous III—Paul Sabatier, 1983. W. Feller, “The asymptotic distribution of the range of sums of independent random variables,” Ann. Math. Statist., vol. 22, pp. 427–432, 1951. N. C. Jain and W. E. Pruitt, “The other law of the iterated logarithm,” Ann. Probab., vol. 3, pp. 1046–1049, 1975. G. Letac, “Advanced problem 6230,” Amer. Math. Monthly, vol. 85, p. 686, 1978. P. L´evy, Processus Stochastiques et Mouvement Brownien.
Gauthier-Villars, Paris, 1948.
L. Tak´acs, “Expected perimeter length,” Amer. Math. Monthly, vol. 87, p. 142, 1980. A. R. Wade and C. Xu, “Convex hulls of random walks and their scaling limits,” Stochastic Process. Appl., vol. 125, pp. 4300–4320, 2015.
4
PII: DOI: Reference:
S0167-7152(17)30234-1 http://dx.doi.org/10.1016/j.spl.2017.07.001 STAPRO 7986
To appear in:
Statistics and Probability Letters
Received date : 4 July 2017 Accepted date : 6 July 2017 Please cite this article as: McRedmond, J., Xu, C., On the expected diameter of planar Brownian motion. Statistics and Probability Letters (2017), http://dx.doi.org/10.1016/j.spl.2017.07.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Manuscript Click here to view linked References
On the expected diameter of planar Brownian motion James McRedmonda,∗, Chang Xub a Department b Department
of Mathematical Sciences, Durham University, South Road, Durham DH1 3LE, UK. of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, UK.
Abstract Known results show that the diameter d1 of the trace of planar Brownian motion run for unit time satisfies 1.595 ≤ Ed1 ≤ 2.507. This note improves these bounds to 1.601 ≤ Ed1 ≤ 2.355. Simulations suggest that Ed1 ≈ 1.99. Keywords: Brownian motion, convex hull, diameter 2010 MSC: 60J65, 60D05
1. Introduction Let (bt , t ∈ [0, 1]) be standard planar Brownian motion, and consider the set b[0, 1] = {bt : t ∈ [0, 1]}. The Brownian convex hull H1 := hull b[0, 1] has been well-studied from L´evy (1948, §52.6, pp. 254–256) onwards; √ the expectations of the perimeter length `1 and area a1 of H1 are given by the exact formulae E`1 = 8π (due to Letac, 1978; Tak´acs, 1980) and Ea1 = π/2 (due to El Bachir, 1983). Another characteristic is the diameter d1 := diam H1 = diam b[0, 1] =
sup x,y∈b[0,1]
kx − yk,
for which, in contrast, no explicit formula is known. The exact formulae for E`1 and Ea1 rest on geometric integral formulae of Cauchy; since no such formula is available for d1 , it may not be possible to obtain an explicit formula for Ed1 . However, one may get bounds. By convexity, we have the almost-sure inequalities 2 ≤ `1 /d1 ≤ π, the extrema being the line segment and shapes of constant width (such as the disc). In other words, `1 `1 ≤ d1 ≤ . π 2
√ The formula of Letac (1978) and Tak´acs (1980) says that E`1 = 8π, so we get: p √ Proposition 1. 8/π ≤ Ed1 ≤ 2π. p √ Note that 8/π ≈ 1.5958 and 2π ≈ 2.5066. In this note we improve both of these bounds. 2. Lower bound For the lower bound, we note that b[0, 1] is compact and thus, as a corollary of Lemma 6 below, we have the formula d1 = sup r(θ), (1) 0≤θ≤π
where r is the parametrized range function given by r(θ) = sup (bs · eθ ) − inf (bs · eθ ) , 0≤s≤1
0≤s≤1
∗ Corresponding
author Email addresses: [email protected] (James McRedmond), [email protected] (Chang Xu)
Preprint submitted to Elsevier
4th July 2017
with eθ being the unit vector (cos θ, sin θ). Feller (1951) established that p Er(θ) = 8/π and E(r(θ)2 ) = 4 log 2,
(2)
and the density of r(θ) is given explicitly as
∞ 8 X (−1)k−1 k 2 exp{−k 2 r2 /2}, (r ≥ 0). f (r) = √ 2π k=1
(3)
p Combining (1) with (2) gives immediately Ed1 ≥ Er(0) = 8/π, which is just the lower bound in Proposition 1. For a better result, a consequence of (1) is that d1 ≥ max{r(0), r(π/2)}. Observing that r(0) and r(π/2) are independent, we get: Lemma 2. Ed1 ≥ E max{X1 , X2 }, where X1 and X2 are independent copies of X := r(0). It seems hard to explicitly compute E max{X1 , X2 } in Lemma 2, because although the density given at (3) is known explicitly, it is not very tractable. Instead we obtain a lower bound. Since max{x, y} =
1 (x + y + |x − y|) 2
we get
1 E max{X1 , X2 } = EX + E|X1 − X2 |. (4) 2 Thus with Lemma 2, the lower bound in Proposition 1 is improved given any non-trivial lower bound for E|X1 − X2 |. Using the fact that for any c ∈ R, if m is a median of X, E|X − c| ≥ E|X − m|, we see that E|X1 − X2 | ≥ E|X − m|. Again, the intractability of the density at (3) makes it hard to exploit this. Instead, we provide the following as a crude lower bound on E|X1 − X2 |. Lemma 3. For any a, h > 0, E|X1 − X2 | ≥ 2h P(X ≤ a) P(X ≥ a + h). Proof. We have E|X1 − X2 | ≥ E [|X1 − X2 |1{X1 ≤ a, X2 ≥ a + h}] + E [|X1 − X2 |1{X2 ≤ a, X1 ≥ a + h}] ≥ h P(X1 ≤ a) P(X2 ≥ a + h) + h P(X2 ≤ a) P(X1 ≥ a + h) = 2h P(X ≤ a) P(X ≥ a + h)
which proves the statement. This lower bound yields the following result. Proposition 4. For a, h > 0 define 4 π2 9π 2 π2 4 4 exp − 2 − exp − 2 1 − exp − . g(a, h) := h π 2a 3π 2a π 8(a + h)2 p Then Ed1 ≥ 8/π + g(1.492, 0.337) ≈ 1.6014.
Proof. Consider
Z := sup |bs · e0 |. 0≤s≤1
Then it is known (see Jain and Pruitt, 1975) that for x > 0, 4 π2 4 9π 2 4 π2 exp − 2 − exp − 2 ≤ P(Z < x) ≤ exp − 2 . π 8x 3π 8x π 8x 2
(5)
Moreover, we have Z ≤ X ≤ 2Z. Since X ≤ 2Z, we have
4 π2 9π 2 4 exp − 2 , P(X ≤ a) ≥ P(Z ≤ a/2) ≥ exp − 2 − π 2a 3π 2a
by the lower bound in (5). On the other hand, 4 π2 P(X ≥ a + h) ≥ P(Z ≥ a + h) ≥ 1 − exp − , π 8(a + h)2 by the upper bound in (5). Combining pthese two bounds and applying p Lemma 3 we get E|X1 −X2 | ≥ 2g(a, h). So from (4) and the fact that EX = 8/π by (2) we get Ed1 ≥ 8/π + g(a, h). Numerical evaluation using MAPLE suggests that (a, h) = (1.492, 0.337) is close to optimal, and this choice gives the statement in the proposition. 3. Upper bound We also improve the upper bound in Proposition 1. √ Proposition 5. Ed1 ≤ 8 log 2 ≈ 2.3548. Proof. First, we claim that d21 ≤ r(0)2 + r(π/2)2 .
(6)
It follows from (6) and (2) that E(d21 ) ≤ E(X12 + X22 ) = 2E(X 2 ) = 8 log 2. The result now follows by Jensen’s inequality. It remains to prove the claim (6). Note that the diameter is an increasing function, that is, if A ⊆ B then diam A ≤ diam B. Note also, that by the definition of r(θ), b[0, 1] ⊆ z + [0, r(0)] × [0, r(π/2)] =: Rz for some z ∈ R2 . Since the diameter of the set Rz is attained at the diagonal, p diam Rz = r(0)2 + r(π/2)2 , for all z ∈ R2 , and we have diam b[0, 1] ≤ diam Rz , the result follows. 4. Final remark and auxiliary lemma We make one further remark about second moments. In the proof of Proposition 5, we saw that E(d21 ) ≤ 8 log 2 ≈ 5.5452. A bound in the other direction can be obtained from the fact that d21 ≥ `21 /π 2 , and we have (see Wade and Xu, 2015, §4.1) that E(`21 ) = 4π
Z
π/2
−π/2
dθ
Z
0
∞
du cos θ
cosh(uθ) tanh sinh(uπ/2)
(2θ + π)u 4
≈ 26.1677,
which gives E(d21 ) ≥ 2.651. Finally, for completeness, we state and prove the lemma which was used to obtain equation (1). Lemma 6. Let A ⊂ Rd be a nonempty compact set, and let rA (θ) = supx∈A (x · eθ ) − inf x∈A (x · eθ ). Then diam A = sup rA (θ). 0≤θ≤π
3
Proof. Since A is compact, for each θ there exist x, y ∈ A such that rA (θ) = x · eθ − y · eθ
= (x − y) · eθ ≤ kx − yk.
So sup0≤θ≤π rA (θ) ≤ supx,y∈A kx − yk = diam A. It remains to show that sup0≤θ≤π rA (θ) ≥ diam A. This is clearly true if A consists of a single point, so suppose that A contains at least two points. Suppose that the diameter of A is achieved by x, y ∈ A and let z = y − x be such that zˆ := z/kzk = eθ0 for θ0 ∈ [0, π]. Then sup rA (θ) ≥ rA (θ0 ) ≥ y · eθ0 − x · eθ0
0≤θ≤π
= z · zˆ = kzk = diam A, as required. Acknowledgements The authors are grateful to Andrew Wade for his suggestions on this note. The first author is supported by an EPSRC studentship. References M. El Bachir, “L’enveloppe convexe du mouvement brownien,” Ph.D. thesis, Universit´e Toulous III—Paul Sabatier, 1983. W. Feller, “The asymptotic distribution of the range of sums of independent random variables,” Ann. Math. Statist., vol. 22, pp. 427–432, 1951. N. C. Jain and W. E. Pruitt, “The other law of the iterated logarithm,” Ann. Probab., vol. 3, pp. 1046–1049, 1975. G. Letac, “Advanced problem 6230,” Amer. Math. Monthly, vol. 85, p. 686, 1978. P. L´evy, Processus Stochastiques et Mouvement Brownien.
Gauthier-Villars, Paris, 1948.
L. Tak´acs, “Expected perimeter length,” Amer. Math. Monthly, vol. 87, p. 142, 1980. A. R. Wade and C. Xu, “Convex hulls of random walks and their scaling limits,” Stochastic Process. Appl., vol. 125, pp. 4300–4320, 2015.
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