A MARTINGALE INEQUALITY FOR THE SQUARE AND MAXIMAL FUNCTIONS BY LOUIS H.Y. CHEN TECHNICAL REPORT NO. 31 MARCH 15, 1979 PREPARED UNDER GRANT DAAG29-77-G-0031 FOR THE U.S. ARMY RESEARCH OFFICE Reproduction in Whole or in Part is Permitted for any purpose of the United States Government Approved for public release; distribution unlimited. DEPARTMENT OF STATISTICS STANFORD UNIVERSITY STANFORD, CALIFORNIA
A MARTINGALE INEQUALITY FOR THE SQUARE AND MAXIMAL FUNCTIONS By LOUIS H. Y. CHEN TECHNICAL REPORT NO. 31 MARCH 15, 1979 Prepared under Grant DAAG29-77-G-0031 For the U.S. Army Research Office Herbert Solomon, Project Director Approved for public release; distribution unlimited. DEPARTMENT OF STATISTICS STANFORD UNIVERSITY STANFORD, CALIFORNIA Partially supported under Office of Naval Research Contract N0001U-76-C-0475 (NR-OH2-267) and issued as Technical Report No. 270.
The findings in this report are not to be construed as an official Department of the Army position, unless so designated by other authorized documents.
A MARTINGALE INEQUALITY FOR THE SQUARE AND MAXIMAL FUNCTIONS By Louis H. Y. Chen University of Singapore and Stanford University 1. Introduction and Notation. An inequality involving a class of functions for weak martingales and nonnegative weak submartingales is proved. Three special cases are deduced, one of which generalizes and refines a result of Austin (1966). As an application of the inequality, the special cases are used to give easy proofs of Burkholder's (1966) L log L and L (for 1 < p < 2) inequalities for the square function of a martingale or a nonnegative submartingale. Although the inequality and the special cases are proved for weak martingales and nonnegative weak submartingales, they are also new for martingales and nonnegative submartingales. Weak martingales and weak submartingales were first defined in Nelson (1970). Examples of these can be found in Nelson (1970) and Berman (1976). For ease of reference, we give the definition here. A sequence f = (f.,f p,...) defined on a probability space is a weak martingale (or weak submartingale) if f is integrable and E(f _ f ) = (or > ) f a.s. for n > 1. Of course, a martingale (or submartingale) is a weak martingale (or weak submartingale). Throughout this paper, unless otherwise stated, f = (f,,fp,...) will denote a weak martingale or a nonnegative weak submartingale. As usual f Q = 0. The difference sequence of f will be denoted by
n p 1/2 d = (<L,d 1,...) Also S (f) = ( E <), S(f) = sup S (f), 2 n i=l 1 l<n<co n f* = sup f ± l, f* = sup fj and f = sup f for y l<i<n l<n<oo -^ l<.n<oo 1 < p < oo. All functions are real-valued, Borel measurable and defined on the real line. 2. The Main Results. We first derive an identity. Lemma 2.1. Let cp be a differentiable function whose derivative cp' is an indefinite integral of cp" such that qp(0) = cp' (0) = 0, cp" is a nonnegative and even function, and such that for n > 1, f cp' (f ) is ' * n n -i- _ integrable, Define K. (x) = (d.-x) if x >0 and = (d.-x) if x < 0, i > 1. Then for n > 1, c P(f n ) ^s integrable, and n /-m (2.1) Ecp (f ) > 2 E / i=l J -oo <P" ( f i _i +x ) K i ( x ) fe where equality holds in the weak martingale case. Furthermore for i > 1. f 1 P (2.2) / K i (x)dx = 5-df 2 i. J -co Proof. Since the proof of (2.2) is easy, we omit it here. Since rx /-x cp(0) = cp' (0) = 0, we have cp 1 (x) = / cp"(t)dt and cp(x) =/ cp'(t)dt. Jo. J 0 It follows that cp» is an odd function and cp an even function. Therefore
f\x\ f\x\ (2.3) 0 < tp(x) = / cp' (t)dt = x q>' ( x ) - / t <p (t)dt J 0 J 0 < x cp' (x) The Integrability of cp(f ) then follows from that of f cp' (f ). We also need the integrability of d.cp' (f.,) for i > 1. Since cp" > 0, cp' is nondecreasing. Therefore and Hence f. cp' ( f. )l( f. n > f. ) < f., cp' ( f. _ ). i IT Vl l-l' ' Vl l-l 1 ' i y ' i-l' Y Vl l-l' ' Id.cp^f.^)! = Idjcp-df^)! < if.icp'df.^d + if.^icp.df.^d < \f ± \V ( f ± l) + 21^1 cp- ( f ± _ ± \). This implies the integrability of d.cp' (f... ). We now derive (2.1). In the weak martingale case, the left hand side of (2.1) is equal to n (2.10 I E[cp(f.)-cp(f i _ 1 )3 n = E[cp(f J-cpCf^-^cp' (f^)] i=l = t E \J ( 9"(f i _ 1 +x)dxdy l(d i >0) +.X E {/ J cp"(f._ 1 +x)dxdyll(d i <0) l 3
In the nonnegative weak submartingale case, cp' (f, _ ) is nonnegative and (2.4) holds with the first equality replaced by " > ". Now cp" > 0. So we may reverse the order of the doubly integration in (2.k). By this, the extreme right hand side of (2.k) yields n /»oo Y. E / cp"(f i _ 1 +x)k i (x)dx i=l J-oo and the lemma is proved. In the case where f is a martingale or a nonnegative submartingale, let T be a stopping time. By replacing f in (2.1) by the stopped martingale or nonnegative submartingale f, we obtain n />oo (2.5) Ecp(f n ) > E I(T > i) I cp"(f i _ 1 -bc)k i (x)dx i=l -'-oo where again equality holds in the martingale case. If the differences p of f are mutually independent with zero means, cp(x) = x and T = inf{n: f > a] where a > 0, then (2.5) immediately yields Kblmogorov's two inequalities in the proof of the three series theorem. Theorem 2.1. Let \ r' be a nonnegative and even function which is nonincreasing on [0,oo), and let t(x) = J t' (t)dt. Then (2.6) ES d (f)r (f*) < 2 sup E f t( f ). n Proof. There is nothing to prove if the right hand side of (2.6) is infinite. So we assume it to be finite. Let K.(x) be as in Lemma 2.1. It is not difficult to see that, for i > 1, f.,+x lies
between f., and f. on {x: K. (x) > 0}. Now let cp" = f 1, X-X 1 1 XX C X <p"(t)dt = i r(x) and cp(x) = cp'(t)dt. Then the inte- 0 J 0 grability condition in Lemma 2.1 is satisfied and the lemma immediately yields BT(f)lr' (f*) < 2Bp(f_) < 2E fj*( fj ) < 2 sup EJfJ + ( f_i ) n n n n n ^ n n where the second inequality follows from (2.3). By letting n * oo and applying Fatou's lemma, the theorem is proved. We now deduce from (2.6) three special cases. Corollary 2.1. We have (2.7) B ä-öj2 < «Hfl! j 1+f* 2 (2.8) E ^-M<2 supejf log(l+ f ) ; 1+f* n a n 2-1 Proof. For (2.7), let f (x) = (l-h ) ; and for (2.8), let T r! (x) = (1+lxl)" 1. For (2.9), we first let i r' (x) = (a+ x ) P " 2 where a > 0 and then let a 1 0. The inequality (2.7) generalizes and refines a result of Austin (1966) who proved that the square function of an L -bounded martingale is square integrable on any set where the maximal function is bounded above. 5
3. Applications. In this section, we use Corollary 2.1 to give easy proofs of Burkholder's 1 log L and L (for 1 < p < 2) inequalities for the square function of a martingale or a nonnegative submartingale. These inequalities were first proved by Burkholder (1966). Since then different proofs have been given. (See, for example, Gordon (1972), Burkholder (1973), Chao (1973) and Garsia (1973).) Theorem 3»1«Let f = (f.,f 2,...) be a martingale or a nonnegative submartingale. Then «a 1 / 2 + (3.1) ES(f) < 2(^2-.) [1 + sup Elfjlog f n l ]. Proof. We shall use the following inequality which dates back to Young (1913). It can also be found in Doob (1953). + + 1 (3.2) a log b < a log a + be for a > 0 and b > 0. Replacing Lng f. 1. by Dy \ A ~ f. - in " (2.8) of Corollary 2.1 where X = Ef*, we obtain 2 (3.3 ) E 0 i < 2 sup EI f I log (1+Ä." 1 1 f I ) \+f " n n n which by (3.2) <2 sup[e f n log + f n l + (\e)" 1 (\+E f n l)] <2[1 + sup E f log + f ]. n n n
Now applying the Cauchy-Schwarz inequality to q2, x 1/2 1/2 ES(f) = E(S_L i) (v**) \+f and using (3.3) and the following inequality of Doob (1953) for submartingales, Hf\ < (^Cl + sup E f n log + fj], we obtain (3.1). This proves the theorem. Theorem 3*2. Let f = (f,,f p,...) be a martingale or a nonnegative submartingale. Then for 1 < p < 2, (3.10 S(f) p <2 l/2 p l/2 q f p where p" + q~ = 1. Proof. Applying Holder's inequality to i<><«ii;-.»$!^ptf, *> 1 "* P. we obtain 2, x2 P S(f) P < (E^M) ( f* P ) 1_ I P which by (2.9) of Corollary 2.1 1. 1
This together with the following inequality of Doob (1953) for sub- martingales, f*ll p < q fh p for 1< p < oo, p^ + q" 1 = 1, imply S._ W»p5 P- «sr^v"^ IWLsWW V P This proves the theorem. The absolute constants in (3.1) and (3.^) seem to be the lowest ever obtained. In (3.^)^ the order of magnitude of the constant as p -> 1 is the same as that obtained by Burkholder (1973). 8
References 1. Austin, D. G. (1966). A sample function property of martingales. Ann. Math. Statist., 37, 1396-1397. 2. Berman, Jeremy (1976). An example of a weak martingale. Ann. Probability k, 107-108. 3. Burkholder, D. L. (1966). Martingale transforms. Ann. Math. Statist., 37, 1 1 +9 1^-1504. h. Burkholder, D. L. (1973)«Distribution function inequalities for martingales. Ann. Probability 1, 19-^2. 5. Qiao, Jia-Arng (1973). A note on martingale square functions. Ann. Probability 1, IO59-IO60. 6. Doob, J. L. (1953). Stochastic Processes. Wiley, New York. 7- Garsia, Adriano M. (1973). Martingale Inequalities: Seminar Notes on Recent Progresses. Benjamin, Reading, Massachusetts. 8. Gordon, Louis (1972). An equivalent to the martingale square function inequality. Ann. Math. Statist., U3, 1927-193^. 9. Nelson, Paul I. (1970). A class of orthogonal series related to martingales. Ann. Math. Statist., Ul, 168I+-I69I+. 10. Young, W. H. (1913). On a certain series of Fouries. Proc. London Math. Soc, 11, 357-366. Department of Mathematics University of Singapore Bukit Timah Road Singapore 10
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UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (Wion Data Enfted) A Martingale Inequality for the Square and Maximal Functions An inequality for weak martingales and nonnegative weak submartingales is proved. Three special cases are deduced, one of which generalizes and refines a result of Austin (1966). As an application of the inequality, the special cases are used to give easy proofs of Burkholder's (1966) L log L and L (for 1 < p < 2) inequalities for the square function of a martingale or a nonnegative submartingale.. 270/31 - Author: Louis H.Y. Chen S/N 0102- LF-0)4-6601 UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGEfW*i*n Dmlä EMnmd)