Chung's Law and The Csáki Function

Journal of Theoretical Probability, Vol. 12, No. 2, 1999 Chung's Law and The Csaki Function1 Natalia Corn2,4 and Mikhail Lifshits3 Received August 14, 1997; revised January 10, 1998 We have found the limit for a Wiener process W and a class of" twice weakly differentiable functions h E C[0, 1], thus solving the problem of the convergence rate in Chung's func­ tional law for the so­called "slowest points". Our description is closely related to an interesting functional emerging from a large deviation problem for the Wiener process in a strip. KEY WORDS: Large deviation; Wiener process. 1. INTRODUCTION Let ( W ( t ) , t > 0 ) be a standard Wiener process and (C[0, 1], || • ||) be the space of continuous functions endowed with the supremum norm. We con­ sider the random functions Research supported by the Russian Foundation of Basic Research and the German Research Society. 2 Friedrich­Schiller­Universitat, Fak. fur Mathematik und Informatik, Ernst­Abbe­Platz 1­4, 07743, Jena, Germany. 3 197372, St.­Petersburg, Komendantskii, 22­2­49, Russia. 4 To whom correspondence should be addressed. 1 399 0894-9840/99/0400-0399$ 16.00/0 © 1999 Plenum Publishing Corporation Gorn and Lifshits 400 and the set which is called Strassen's set. The famous Strassen's law of the iterated logarithm (LIL) [cf. Strassen(15)] states that almost surely and It is naturally to ask for the convergence rate in both cases. Regarding (1.1) we refer to the papers Goodman and Kuelbs, (8,9) Grill,(11) Talagrand,(16) Mijnheer,(14) and Bulinskii and Lifshits.(2) We are going to consider the convergence rate in (1.2), i.e., for each h E S we are looking for E T ( h ) | 0 such that The following results about this convergence rate are known: (a) h = 0. The convergence rate can be derived from the Chung's law [cf. Chung (3)] i.e., we have here E T ( h ) = p/4(log 2 T) -1 . (b) |h|2 = f 1 0 ( h ' ( s ) ) 2 d s < 1 . In this case [cf. Csaki,(4) and De Acosta(6)] of internal elements of Strassen's set we have Therefore, the convergence is the same for all internal elements h but the corresponding constant increases to infinity when h approaches the boundary of S. Chung's Law and the Csaki Function 401 (c) |h| 2 = 1. This condition means that the limit element h is located on the border of Strassen's set. It turns out that the convergence rate essentially depends on h. It is known [cf. Goodman and Kuelbs (9) ] that for each h with |h| = 1, but It means that (Iog2 T) rate is attained, i.e., ­2/3 is the slowest possible convergence rate. This if and only if h' is of bounded variation on [0,1]. The correspondent elements are naturally called slowest with respect to Chung's law. Since the derivative of an absolutely continuous function h is not necessarily defined at each point of [0, 1], the expression "h' is of bounded variation" should be specified. Namely, we write Var(h') < i and say that h' is of bounded variation, if there exists a signed measure v of bounded variation on (0, 1 ] such that for all s E[0, 1] or, equivalently, It follows from the latter representation that h is absolutely continuous and the function of bounded variation h'(u) := — v[u, 1] is a version of the derivative of h. We call v the second derivative of h. Indeed, if h E C(2), then dv/du = h " ( u ) . I n the following we make use of the integration by parts formula 402 Gorn and Lifshits On the right­hand side and in other similar integrals with respect to v throughout this paper, the integration is carried over the semi­open interval [0,1). One can see from (1.3) that slowest elements belong to the image of the covariance operator of the Wiener measure. This observation gives a hint of the notion of the slowest elements in the general case of Gaussian measures [see details in Kuelbs et al.(13)]. Of course, on the border of the Strassen set there exist also the elements which are not the slowest. Moreover, in Grill(10) one can find the examples corresponding to all rates Er= (log2 T)­A with 2/3 <A< 1. One can ask for the exact convergence rate for the slowest elements, i.e., for constants Eh with In Kuelbs et al.,(13) the exact convergence rate was obtained for all slowest elements with respect to the Hilbert norm. For the uniform norm the solu­ tion was found by Csaki (4,5) for the following special cases: (1) if h is piecewise linear, then (2) if h is a quadratic function, then Eh can be expressed via the minimal eigenvalue of a certain differential equation (see Eq. (3.3) later). Our purpose is to obtain the value for Eh for all h with |h| = 1 and Var(h')< i. Reading Csaki,(5) one may get the wrong impression that the method applied there makes essential use of a specific feature of the quadratic func­ tion (the second derivative is constant). We show in this note that the result of Csaki(5) is a typical case of some more general phenomenon. 2. DEVIATION PROBABILITIES AND WIENER PROCESS IN A STRIP Our first aim is to explain how the probabilities appearing in Chung's law are related to some functional of Wiener process in a strip. Chung's Law and the Csaki Function 403 For all E we have where Making use of the Cameron­Martin formula, we obtain where the measure v is the second derivative of the function h. Further, the notation ||f||ab stands for sup uE[a,b] | f ( u ) | . Substituting W(u) = r W ( r - 2 u ) and t = r-2 = (log2 T ) 1 / 3 / 2 E 2 we can continue as follows 404 Gorn and Lifshits where C = 4E 3 h'(1). Recall that W is also a Wiener process. Our problem is thus reduced to the study of the behavior of the Wiener process in the strip [ — 1, 1] during the long interval [0, t]. Now we are going to obtain a large deviation result for the Wiener process in the strip related to (2.2). For this purpose, let us define Csaki function u:R 1 |->R 1 by introducing u ( a ) as the smallest eigenvalue of the Sturm­Liouville problem It turns out that u(.) is an even, concave, Lipschitz function with u(0) = p2/8 and asymptotic behavior u ( a ) ~ —|a| for a ­> i. For each signed measure v of bounded variation we denote its absolutely continuous part by vac and its singular part by vs. We write Var(v, A) for the variation of the measure v on the set A and denote Var(v) = Var(v, (0, 1)). Theorem 1. Let v be a finite signed measure on [0, 1) and W be a Wiener process. Then for each real C and for t ­> i it holds where and u(.) is the Csaki function. In the next section we will show how Theorem 1 provides the exact rate of convergence in Chung's law for the slowest elements. The proof of Theorem 1 is given in Section 4. In Sections 5 and 6, we obtain theoretical and numerical bounds for the Csaki function. 3. CHUNG'S LAW FOR THE SLOWEST POINTS Theorem 1 yields the following exact estimate in the Chung's law for all slowest points on the surface of the Strassen's ball S. Chung's Law and the Csaki Function 405 Theorem 2. Let h ES, |h| = 1, Var(h') < i and let the measure v be the second derivative of h. Then where Eh is the unique solution of the equation Proof. By combining (2.4) and (2.2), we get for each E > 0 where Since d(E) is a strictly increasing function, we have for large T for each E>Eh and for each E<Eh Now standard Borel­Cantelli arguments along pseudo­geometrical sequen­ ces Tn = exp{n(log n)A} (see, for example, Csaki(5)) finish the proof of Theorem 2. S Example 1. Let h be piecewise linear. Then v is discrete and 406 Gorn and Lifshits In this case Eh can be obtained explicitly; indeed and thus we have (1.5). Example 2. Let h(s) = (a/2) s2 + bs. Then vs = 0, dv ac /du = a, |h'( 1)| = |a + b|. In this case, Eq. (3.1) is equivalent to By the definition of the Csaki function, value of the problem u(4E3h|a|) is the smallest eigen­ Relation (3.2) yields that 4Eh3 is the smallest eigenvalue of the problem This is a representation of Eh as it was done by Csaki.(5) Example 3. Let h EC (2) Then vs = 0, dvac(u)/du = h"(u) and hence we have so that Eh can be obtained from the equation 4. PROOF OF THEOREM 1 We split our proof into several steps. 1. v(du) = adu. (This case corresponds to Csaki's(5) analysis of qua­ dratic functions). We introduce the notation E t , k ( . ) = E(.| W ( t ) = K] and consider the functional Chung's Law and the Csaki Function 407 where the function satisfies the Feynman­Kac equation [see Borodin and Salminen (1) ] The solution of this equation can be written as the series where ui are the eigenvalues of the problem enumerated so that u1<u2< ... and yi are the corresponding eigenfunc­ tions. Since the first term dominates in the representation and by f ­1 1 y 1 (x) e-Ctx dx = exp{ |C|t + o ( t ) } , we have Thus we obtain (2.4) for this special case with M ( v ) = f 0 1 u ( a ) d s = u ( a ) =u1. By the symmetry of the system (4.2) we have that u ( d ) = u ( — a ) . Holder inequality yields for all real a1 ,a2 and all A e(0, 1) that 408 Gorn and Lifshits Taking the logarithm and using the limit relation (2.4) with C = 0, we get the inequality which means that the function u is concave. Moreover, since u is an even function, we observe that it is increasing on ( —i, 0] and decreasing on [0, i). It is also clear that for arbitrary measures v1 and v2 one has In particular, applying this inequality to the measures v 1 (du) = a 1 du, v2(du) = a2 du and taking the limit, we get i.e., u(.) is a Lipschitz function. 2. Now we consider the measures v(du) = l(u) du, where l(u) = Emj=1aj1 [bj­1,bj) (u) with 0 = b0<b1<... <b m =1 being a partitioning of the interval [0, 1], We have to investigate the quantity Chung's Law and the Csaki Function 409 Introducing the notations ,Ft = c{W t },F < t = c{ Ws , s<t} and denoting lj = bj — b j - 1 , we have by the Markov property of the Wiener process Let us consider the function Applying the Feynman­Kac formula as in (4.1) and writing the solution of the corresponding partial differential equation as the series, we obtain that where u i (m) are the eigenvalues of the boundary value problem 410 Gorn and Lifshits enumerated so that u1(m) <u2(m) < ...,yi(m) are the corresponding eigenfunc­ tions normalized in L2[ — 1, 1]­norm and We repeat the same procedure and obtain where again satisfies the Feynman­Kac equation. Solving the corresponding boundary value problem, we obtain where and ui(m­1), yi(m-1) are the eigenvalues and eigenfunctions of the boundary value problem (4.5) with am replaced by a m - 1 . After m — 2 similar itera­ tions we finally land at p(t, C, v) = U1(0, 0), with the coefficients where Chung's Law and the Csaki Function 411 and ui(j), yi(j) are the eigenvalues and normalized eigenfunctions of the boundary value problem (4.5) with am replaced by aj. We show now that the term related to the minimal eigenvalues deter­ mines the behavior of the sum when t tends to infinity. Normalization assumption and the Cauchy inequality yield and Moreover, the eigenfunctions y1(j) corresponding to the minimal eigen­ values are strictly positive on (—1,1) (Sturm­Liouville oscillation theorem, cf. Kamke, (12) p.260). Hence, and for each fixed E > 0 where CE = f-1-1+E y 1 (m) (x) dx>0. These estimates lead to the following bounds for p ( t , C, v): where Since the eigenfunctions y i (1) (x) are bounded altogether and uk(j)=k2 (cf. Kamke, (12) p. 262­263) we have by the dominated convergence theorem limt­>i Fu= C with C = [IIm-1j=1 C1,1(j)]y1(1)(0) and therefore 412 Gorn and Lifshits On the other hand, where and If E <inf 1 < j < m [(u2(j)-u1(j)lj]/|C|, then by the dominated convergence theorem limt­>i FL(t) = CCE and we obtain Since E can be chosen arbitrary small we finally have that Thus we have proved Theorem 1 for v(du) = l(u)du Emj=1aj1[bj-1,bj)(u). with l(u) — 3. We consider now an arbitrary measure v = l(u)du, l EL 1 [0, 1]. It can be approximated in the variation distance by a sequence of measures for which the formula (2.4) is valid. Using the Lipschitz property (4.3), we easily obtain (2.4) for the measure v with M(v) = lim m _ >i M(v m ). 4. Now let us consider v with a singular component: v = vac+vs. Applying (2.4) to vac we get the upper bound. Since Chung's Law and the Csaki Function 413 we immediately have the estimate The proof of lower bound is more technical, so we omit certain details. Let D >0 be a small fixed number. For each measure v and d E (0, D) we define the dilation measure vd as vd(A) : = v ( A ( 1 + d ) ) . It is easy to see that for all d E (0, D) one has Now let us consider the mixtures of dilations, By the Holder inequality we have Recall that for each v the measure VD is absolutely continuous with the density moreover, for each absolutely continuous measure p(du) = p(u) du it holds ppD­>p in L1 when D­>0. Applying (2.4) to the measure VD, we get and thus 414 Gorn and Lifshits The following statement will be useful for the analysis of the singular com­ ponents. Lemma 1. Let v be a singular measure on [0, 1] and L be the Lebesgue measure. Then there exist measurable sets { A D , D >0} such that We omit the elementary proof of this lemma. Now we bring together all the necessary parameters. (1) Let E > 0 be an arbitrary small number. (2) We choose M = M(E) so large that for |p|> M one has (cf. (5.5) in Section 5) (3) We choose E1 =E1(E, M) so small that (a) (M + 2)E1<E; (b) L ( A ) <E 1 => Var(v ac , A) < E (using the absolute continuity of vac.) (4) Finally, introducing the notations we choose D =D(E,E 1 ) so small that (a) ||pacD-Pac||1<E (using L1— convergence pDac-> p ac ); (b) E A : L ( A ) < E 1 , Var(vsD, A ) > V a r ( v s ) ­ E , Var(vsD,A C )<E (see Lemma 1). Now we are able to estimate M(v D ). By our previous definitions and the Lipschitz continuity of u we have Chung's Law and the Csaki Function 415 Thus we have and comparing this inequality with (4.7) we obtain the lower bound Finally, remarking that p ( t , C, v) = p(t, 0, v C ) with v C ( A ) : = v ( A ) - C 1 { 1 we have E A} 416 Gorn and Lifshits 5. ASYMPTOTIC BOUNDS FOR THE CSAKI FUNCTION We used the Csaki function u(.) in the estimation of small ball probabilities for the Wiener measure. It is curious to notice that one may proceed in inverse direction getting estimates for u from probabilistic estimates [cf. Grill,(10) and Kuelbs et al.(13)]. Application of (2.4) with C = 0, v(du) = a du gives Since it is clear that the lower bound for u(a) is straightforward to obtain: for all a E R, In order to calculate the upper bound, we make use of the following nontrivial general result which can be easily obtained by the same methods as in Grill,10 Lemma 4; Kuelbs et al.(13) Theorem 1*. Let X be a centered Gaussian vector taking values in a separable Banach space C. Let K: C* |—> C denote its covariance operator and • | the RKHS­norm related to the distribution of X. Assume that the small ball estimates hold for some positive c0, p. Lemma 2. Let g E C*, h = Kg E C, x>0 and A E [0, 1). Then we have for large D Chung's Law and the Csaki Function 417 Elementary optimization over the parameter A yields with We will apply these estimates in the simple setting X=W, C= C[0, 1], c0 = p2/8, p = 2, x=1. Moreover, for given a E R1, we define an element h E C[0, 1 ] by h(u) = (a/2) u2 - au. For this h we have Consequently, we have the bound with On the other hand, we can apply the Cameron­Martin equalities (2.1), (2.2), and state that for each t>0, D = t3/2, r = D - 1 / 3 and h as before, one has The comparison of (5.3) and (5.4) gives 418 Gorn and Lifshits From this inequality the upper estimate for the Csaki function follows: Our numeric calculations show that this upper estimate is very close to the exact value. 6. COMPUTATION OF THE CSAKI FUNCTION To compute the Csaki function, we fix a > 0 and first look for the eigenvalues of the problem The substitution k = (2/a2)1/3 (ax + u) leads to the equivalent problem The solutions of this differential equation are given by linear combinations of the following functions [cf. Kamke (12) ] where J1/3( .), Yl/3( .) are the Bessel functions of the first and second kind and I1/3( .), K 1/3 ( .) are the modified Bessel functions of the first and second Chung's Law and the Csaki Function 419 Fig. 1. The Csaki function (solid line) and its upper and lower estimations (dashed lines). kind, respectively. Substituting the boundary conditions, we obtain that the necessary and sufficient condition for u to be an eigenvalue of (6.1) is where A = 23/2/3a. Solving these equations numerically (we used MatLab software), we obtain u ( a ) as presented in Fig.l. Now we are able to com­ pute the value of Eh for arbitrary external h. For example, for the external function h(s) = ( R 5 / 3 ) s 3 our calculations give the value Eh ~ 0.5077 and for h(s) = (R2/p) sin(ps) the value Eh ~ 0.5704. 7. CONCLUDING REMARKS Remark 1. The method used in this paper can be also applied to the analysis of convergence rate for slowest points for d­dimensional Wiener process. In this case a d­dimensional boundary value problem with zero boundary conditions on the sphere has to be solved instead of the boundary value problem (4.2). Remark 2. Let X 1 , X 2 , . . . be iid random variables with zero means and unit variances. If 420 Gorn and Lifshits holds, then the partial sum process clusters at every h E S with the Wiener process rate [cf. Einmahl and Goodman(7)]. So we have that our result is also valid for partial sum processes satisfying the moment condition (7.1). ACKNOWLEDGEMENT The authors are deeply grateful to Prof. W. Linde for his encouraging support of this work, REFERENCES 1. Borodin, A. N., and Salminen, P. (1996). 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