On the expected diameter of planar Brownian motion

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1 On the expected diameter of planar Brownian motion James McRedmond a Chang Xu b 30th March 018 arxiv: v1 [math.pr] 1 Jul 017 Abstract Knownresultsshow that thediameter d 1 ofthetrace of planarbrownian motion run for unit time satisfies Ed This note improves these bounds to Ed Simulations suggest that Ed Keywords: Brownian motion; convex hull; diameter. 010 Mathematics Subject Classifications: 60J65 (Primary) 60D05 (Secondary). Let (b t,t [0,1]) be standard planar Brownian motion, and consider the set b[0,1] = {b t : t [0,1]}. The Brownian convex hull H 1 := hullb[0,1] has been well-studied from Lévy [5, 5.6, pp. 5 56] onwards; the expectations of the perimeter length l 1 and area a 1 of H 1 are given by the exact formulae El 1 = 8π (due to Letac and Tákacs [,6]) and Ea 1 = π/ (due to El Bachir [1]). Another characteristic is the diameter d 1 := diamh 1 = diamb[0,1] = sup x y, x,y b[0,1] for which, in contrast, no explicit formula is known. The exact formulae for El 1 and Ea 1 rest on geometric integral formulae of Cauchy; since no such formula is available for d 1, it may not be possible to obtain an explicit formula for Ed 1. However, one may get bounds. By convexity, we have the almost-sure inequalities l 1 /d 1 π, the extrema being the line segment and shapes of constant width (such as the disc). In other words, l 1 π d 1 l 1. The formula of Letac and Takács [,6] says that El 1 = 8π, so we get: Proposition 1. 8/π Ed 1 π. a Department of Mathematical Sciences, Durham University, South Road, Durham DH1 3LE, UK. b Department of Mathematics and Statistics, University of Strathclyde, 6 Richmond Street, Glasgow G1 1XH, UK. 1

2 Note that 8/π and π In this note we improve both of these bounds. 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 d 1 = sup r(θ), (1) where r is the parametrized range function given by r(θ) = sup (b s e θ ) inf (b s e θ ), 0 s 1 0 s 1 with e θ being the unit vector (cosθ,sinθ). Feller [] established that and the density of r(θ) is given explicitly as Er(θ) = 8/π and E(r(θ) ) = log, () f(r) = 8 ( 1) k 1 k exp{ k r /}, (r 0). (3) π k=1 Combining (1) with () gives immediately Ed 1 Er(0) = 8/π, which is just the lower bound in Proposition 1. For a better result, a consequence of (1) is that d 1 max{r(0), r(π/)}. Observing that r(0) and r(π/) are independent, we get: Lemma. Ed 1 Emax{X 1,X }, where X 1 and X are independentcopies of X := r(0). It seems hard to explicitly compute Emax{X 1,X } in Lemma, 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 ) we get Emax{X 1,X } = EX + 1 E X 1 X. () Thus with Lemma, the lower bound in Proposition 1 is improved given any non-trivial lower bound for E X 1 X. 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 X 1 X 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 X 1 X. Lemma 3. For any a,h > 0, Proof. We have E X 1 X hp(x a)p(x a+h). E X 1 X E[ X 1 X 1{X 1 a,x a+h}] which proves the statement. +E[ X 1 X 1{X a,x 1 a+h}] hp(x 1 a)p(x a+h)+hp(x a)p(x 1 a+h) = hp(x a)p(x a+h),

3 This lower bound yields the following result. Proposition. For a,h > 0 define ( } g(a,h) := h { π exp π { })(1 a 3π exp 9π π { }) a exp π. 8(a+h) Then Ed 1 8/π +g(1.9,0.337) Proof. Consider Z := sup b s e 0. 0 s 1 Then it is known (see [3]) that for x > 0, } { π exp π } { 8x 3π exp 9π P(Z < x) π } { 8x exp π. (5) 8x Moreover, we have Z X Z. Since X Z, we have P(X a) P(Z a/) { } π exp π a 3π exp { 9π by the lower bound in (5). On the other hand, P(X a+h) P(Z a+h) 1 } { π exp π, 8(a+h) by the upper bound in (5). Combining these two bounds and applying Lemma 3 we get E X 1 X g(a,h). So from () and the fact that EX = 8/π by () we get Ed 1 8/π+g(a,h). Numerical evaluation using MAPLE suggests that (a,h) = (1.9,0.337) is close to optimal, and this choice gives the statement in the proposition. We also improve the upper bound in Proposition 1. Proposition 5. Ed 1 8log.358. a }, Proof. First, we claim that It follows from (6) and () that d 1 r(0) +r(π/). (6) E(d 1 ) E(X 1 +X ) = E(X ) = 8log. 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 diama diamb. Note also, that by the definition of r(θ), b[0,1] z +[0,r(0)] [0,r(π/)] =: R z for some z R. Since the diameter of the set R z is attained at the diagonal, diamr z = r(0) +r(π/), for all z R, and we have diamb[0,1] diamr z, the result follows. 3

4 We make one further remark about second moments. In the proof of Proposition 5, we saw that E(d 1 ) 8log A bound in the other direction can be obtained from the fact that d 1 l 1 /π, and we have (see [7,.1]) that π/ E(l 1 ) = π dθ π/ 0 ducosθ cosh(uθ) sinh(uπ/) tanh ( ) (θ+π)u , which gives E(d 1).651. Finally, for completeness, we state and prove the lemma which was used to obtain equation (1). Lemma 6. Let A R d be a nonempty compact set, and let r A (θ) = sup x A (x e θ ) inf x A (x e θ ). Then diama = sup r A (θ). Proof. Since A is compact, for each θ there exist x,y A such that r A (θ) = x e θ y e θ = (x y) e θ x y. So sup r A (θ) sup x,y A x y = diama. It remains to show that sup r A (θ) diama. 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 andlet z = y x be such that ẑ := z/ z = e θ0 for θ 0 [0,π]. Then as required. sup r A (θ) r A (θ 0 ) y e θ0 x e θ0 = z ẑ = z = diama, Acknowledgements The authors are grateful to Andrew Wade for his suggestions on this note. The first author is supported by an EPSRC studentship. References [1] M. El Bachir, L enveloppe convexe du mouvement brownien, Ph.D. thesis, Université Toulouse III Paul Sabatier, [] W. Feller, The asymptotic distribution of the range of sums of independent random variables, Ann. Math. Statist. (1951) 7 3. [3] N.C. Jain and W.E. Pruitt, The other law of the iterated logarithm, Ann. Probab. 3 (1975) [] G. Letac, Advanced problem 630, Amer. Math. Monthly 85 (1978) 686.

5 [5] P. Lévy, Processus Stochastiques et Mouvement Brownien, Gauthier-Villars, Paris, 198. [6] L. Takács, Expected perimeter length, Amer. Math. Monthly 87 (1980) 1. [7] A.R. Wade and C. Xu, Convex hulls of random walks and their scaling limits, Stochastic Process. Appl. 15 (015)