Dynamical circle covering with homogeneous Poisson updating

Statistics and Probability Letters 78 (2008) 2400–2403 www.elsevier.com/locate/stapro

Dynamical circle covering with homogeneous Poisson updating Johan Jonasson ∗ Chalmers University of Technology, Sweden G¨oteborg University, Sweden Received 19 November 2007; received in revised form 4 March 2008; accepted 4 March 2008 Available online 8 March 2008

Abstract We consider a dynamical variant of Dvoretzky’s classical covering problem of the unit circumference circle, where the centers of the arcs are updated according to independent Poisson processes of unit intensity. This dynamical model was introduced (in greater generality) in Jonasson and Steif [Jonasson, J., Steif, J., 2008. Dynamical models for circle covering: Brownian motion and Poisson Q updating. Ann. Probab. 36, 739–764], where is was shown that when the length of the n’th arc is `n and we write u n := nk=1 (1 − `k ), then lim infn n(log n)u n < ∞ implies that the whole circle is a.s. covered at all times, whereas if P ` +···+` n /(n 2 log n) < ∞, then a.s. there are times at which the circle is not fully covered. 1 ne In this paper we modify the former condition to lim supn nu n < ∞. In particular this takes care of the natural border case `n = 1/n; no exceptional times exist. More generally, with the parametrization `n = c/n, there is no c for which there are exceptional times for which the model behaves differently than for the static case. c 2008 Elsevier B.V. All rights reserved.

MSC: 60K99

1. Introduction We first briefly introduce the classical static circle covering model. Let C denote the circle with circumference 1 and consider a decreasing sequence {`n }n≥1 of positive numbers approaching 0. Let In be the open arc of C of length `n with a center chosen uniformly at random independently of the other arcs. E := lim supn In and F := E c . By PLet ∞ the Borel–Cantelli Lemma, for each x ∈ C, P(x ∈ E) = 1 if and only if n=1 `n = ∞. Fubini’s Theorem yields immediately that in this case F has Lebesgue measure 0 a.s. Dvoretzky (1956) raised the question of whether, in the P ` = ∞ case, it was possible that F was nonempty and gave examples where this was indeed the case. There n n were a number of contributions to this question with the final result proved by Shepp (1972). (Kolmogorov’s 0-1 law tells us that P(F = ∅) ∈ {0, 1}.) Theorem 1.1 (L. Shepp). P(F = ∅) = 1 if and only if ∞ `1 +···+`n X e = ∞. n2 n=1 In particular if `n = c/n for all n, then P(F = ∅) = 1 if and only if c ≥ 1. ∗ Corresponding address: Chalmers University of Technology, Goteborg, Sweden.

E-mail address: [email protected]. URL: http://www.math.chalmers.se/∼jonasson. c 2008 Elsevier B.V. All rights reserved. 0167-7152/$ - see front matter doi:10.1016/j.spl.2008.03.001

J. Jonasson / Statistics and Probability Letters 78 (2008) 2400–2403

2401

The special cases `n = c/n for a constant c were known earlier, see Kahane (1959), Billard (1965) and Mandelbrot (1972). For a fuller introduction, see Jonasson and Steif (2008). Jonasson and Steif (2008) extended the classical model by including time dynamics. Two different models were considered. In the first of these, the centers of the arcs perform independent Brownian motions on C, each with variance 1. In the second model, one associates independent Poisson processes, or jump processes, with the different arcs, where the Poisson process associated with the nth arc has intensity `−α n for some parameter α ≥ 0. At the jumptimes of the nth arc, In is given a new center, chosen uniformly on C, independent of everything else. It was then asked, for each of these two models, if there are exceptional times at which we see different covering behavior from that which is seen in the earlier static model. Various results were shown. In the present paper, however, we will only focus on the latter model, i.e. the one with Poisson updating. Indeed, we will only be concerned with the perhaps most natural variant, namely with α = 0, i.e. each arc has updating intensity 1. In Jonasson and Steif (2008), the following result was shown. Here, of course, E t is the set of points of C that are covered by infinitely many arcs at time t, and Ft = E tc . Theorem 1.2. Consider the Poisson updating modelQ with α = 0. Assume that there are constants 0 < M0 < M1 < ∞ such that for all n, M0 /n < `n < M1 /n. Let u n := nk=1 (1 − `k ). Then (i) If lim infn n(log n)u n < ∞, then P(∃t ∈ [0, 1] : Ft 6= ∅) = 0. (ii) If ∞ `1 +`2 +···+`n X e < ∞, n 2 log n n=1 then P(∃t ∈ [0, 1] : Ft 6= ∅) = 1. This result is of course satisfying in the sense that it proves that there are `n ’s for which the circle is fully covered at any fixed time but where there are exceptional times at which this fails. This e.g. happens for `n = 1/n − 1/(n log n). On the other hand, it is unsatisfying since it does not, for the original and most natural parametrization `n = c/n, tell us what happens for the critical case c = 1. The point of this paper is to modify part (i) to the following result, which e.g. tells us that for `n = 1/n, there are no exceptional times. Theorem 1.3. If lim supn n u n < ∞, then P(∃t ∈ [0, 1] : Ft 6= ∅) = 0. There is still a very narrow gap between what is covered by Theorem 1.2(ii) and Theorem 1.3, e.g. when `n = 1/n − 1/(n log n log log n). It would of course be interesting to settle what happens for any {`n }. Remark on measurability. It is not immediately obvious that the set {∃t : Ft 6= ∅} is measurable. If the `n ’s are such that P(F 6= ∅) = 1 for the static case, then the complement of the interesting set is contained in a null set and hence measurable (after a completion of the probability space if necessary). We may thus assume that P(F 6= ∅) = 0 for the static case. Now note first that ) ) ( ( [\ [ \ c c {∃t : Ft 6= ∅} = ∃t : In (t) 6= ∅ = ∃t : In (t) 6= ∅ . n m≥n

n

m≥n

(t)c

Hence it suffices to show that the set {∃t : ∩n In 6= ∅} is measurable. Letting Sn := {(x, t) ∈ C × [0, 1] : x ∈ In (t)c } and S = ∩n Sn , this set can be written as {S 6= ∅}. Observe that by compactness, \ N {∩n Sn 6= ∅} = {∩n=1 Sn 6= ∅}. N

Now, the whole dynamical model can be defined in a straightforward way (as in Jonasson and Steif (2008), page 2-3) in terms of a set {X i j }i,∞j=1 of independent C-valued random variables (the center-points of the arcs) and a set N S 6= ∅} can {Yi j }i,∞j=1 of independent exponential random variables (the times between updatings). The set {∩n=1 n then clearly be expressed as a countable union of events expressed in terms of these random variables. Therefore this set is measurable and hence so is {∩n Sn 6= ∅}. Finally the set S and ∩n Sn may differ only on sets of the type Inc × {t0 } where t0 is a jump time for some arc. The number of such sets is countable, and each of them is almost surely empty since P(F 6= ∅) = 0 for the static case. Hence, possibly after a completion of the probability space, {S 6= ∅} is measurable.

2402

J. Jonasson / Statistics and Probability Letters 78 (2008) 2400–2403

2. Proof of Theorem 1.3 c Consider the static model. Let µ be Lebesgue measure on C = R/Z, let U = ∩∞ n=1 In , i.e. the set of points n c of C that are not covered by any arc, and let Un = ∩k=1 Ik . It is easily seen that P(F 6= ∅) = 1 if and only if P(U 6= ∅) = limn→∞ P(Un 6= ∅) = limn→∞ P(µ(Un ) > 0) > 0. The key ingredient in the proof of Theorem 1.1 is the fact that E[µ(Un )|Un 6= ∅] ≥ 21 E[µ(Un )|0 ∈ Un ]. This follows from the lemma on page 146 of Kahane (1985) in the following way.

Eµ(Un ) 1 2 P(µ(Un ) >

2E[µ(Un )|Un 6= ∅] =

≥ 0)

Eµ(Un ) P(µ(Un ∩ [0, 21 ]) > 0)

≥ E[µ(Un )|0 ∈ Un ]

where the last inequality follows from the lemma applied with  = 1/2. Hence, with A x := {x ∈ Un }, P(Un 6= ∅) =

Eµ(Un ) 2Eµ(Un ) 2P(A0 ) 2P(A0 )2 ≤ =R =R , E[µ(Un )|Un 6= ∅] E[µ(Un )|0 ∈ Un ] C P(A x |A0 )dx C P(A x ∩ A0 )dx

where the last two equalities follow from Fubini’s Theorem. Now consider the dynamical model on the time interval [0, h]. Let, for each n, Z n = Z n (h) be the indicator that arc In does not update during the time interval [0, h]. Let In0 = Z n In (t) and let Un0 = ∩nk=1 (In0 )c and U 0 = ∩n Un0 . Then clearly, if Un0 is empty, then so is Un (t) for every t ∈ [0, h]. Now, given the Z n ’s, the Un0 ’s correspond to a static model with some of the original arcs removed. Hence, putting A0x := {x ∈ Un0 } and writing PZ for conditional probability given Z 1 , Z 2 , . . ., we have PZ (µ(Un0 ) > 0) ≤ 2 R C

PZ (A00 )2 . PZ (A0x ∩ A00 )dx

We also have PZ (A0x ∩ A00 ) =

n Y

(1 − 2`k + (`k − x)+ ) Z k .

k=1

Observe that for any a ∈ (0, 1/2], 1 − 2`k + a > (1 − `k )2 + a > (1 − `k )2 (1 + a) > (1 − `k )2 (ea − a 2 ) > (1 − `k )2 ea (1 − a 2 ). Since M0 /k < `k < M1 /k, it follows that PZ (A0x

∩

A00 )

>

n  Y

2 (`k −x)+

(1 − `k ) e

(1 − `2k )

Zk

= const

n P Z k (`k −x)+ 0 2 k=1 PZ (A0 ) e ,

k=1

where the constant is independent of n. By the condition M0 /n < `n < M1 /n, ! ∞ ∞ X X 2 Z n `n = `n VarZ 1 < ∞. Var n=1

n=1

Therefore Chebyshev’s inequality and the fact that EZ k = e−h > 1 − h imply that n X k=1

Z k (`k − x)+ >

bMX 0 /xc∧n

Z k `k − M1 > (1 − h)

k=1

bM 0 /xc X

`k − M 1

k=1

with probability tending to 1 as n → ∞. Hence  −1 0 /xc Z (1−h) bMP `k −M1   k=1 P(µ(Un0 ) > 0) < o(1) + const  e dx  . C

J. Jonasson / Statistics and Probability Letters 78 (2008) 2400–2403

2403

Since lim supn nu n < ∞, we have supn nu n > M for some constant M. Therefore e`1 +···+`n > M −1 n and hence Z e

(1−h)

bMP 0 /xc k=1

`k −M1

dx > const

C

Z

1

x −(1−h) dx = const · h −1 .

0

Thus P(µ(Un0 ) > 0) < o(1) + Ch for a constant C independent of n, and hence, by letting n → ∞, P(µ(U 0 ) > 0) < Ch. Thus, with B(h) = {∃t ∈ [0, h] : U (t) 6= ∅}, we have P(B(h)) < Ch. This entails that, for any positive integer k, the expected number of time intervals [( j − 1)/k, j/k), j = 1, 2, . . . , k that contain a time point t for which U (t) occurs, is bounded by C. Putting Nn for the number of such time intervals and T for the set of t’s for which U (t) occurs, it is easy to see that |T | ≤ lim infn Nn . Hence by Fatou’s Lemma, T is a.s. finite. However, since the process studied (i.e. the whole dynamical circle covering model) is a reversible Markov process, we conclude, by combining Theorem 6.7 in Getoor and Sharp (1984) and (2.9) in Fitzsimmons and Getoor (1988), that T is a.s. empty.  Acknowledgment Research partly supported by the Swedish Research Council. References ´ Billard, P., 1965. S´eries de Fourier al´eatoirement born´ees, continues, uniform´ement convergentes. Ann. Sci. Ecole Norm. Sup. 82 (3), 131–179. Dvoretzky, A., 1956. On covering a circle by randomly placed arcs. Proc. Natl. Acad. Sci. USA 42, 199–203. Fitzsimmons, P.J., Getoor, R.K., 1988. On the potential theory of symmetric Markov processes. Math. Ann. 281, 495–512. Getoor, R.K., Sharp, M.J., 1984. Naturality, standardness, and weak duality for Markov processes. Zeits. Wahr. verw. Gebiete 67, 1–62. Jonasson, J., Steif, J., 2008. Dynamical models for circle covering: Brownian motion and Poisson updating. Ann. Probab. 36, 739–764. Kahane, J.P., 1959. Sur le recouvrement d’un cercle par des arcs dispos´es au hasard. C. R. Acad. Sci. Paris 248, 184–186. Kahane, J.P., 1985. Some Random Series of Functions, second edn. Cambridge University Press, Cambridge. Mandelbrot, B., 1972. On Dvoretzky coverings for the circle. Z. Wahr. Verw. Gebiete 22, 158–160. Shepp, L.A., 1972. Covering the circle with random arcs. Israel J. Math. 11, 328–345.