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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