UND INFORMATIK. Maximal φ-inequalities for Nonnegative Submartingales. G. Alsmeyer und U. Rösler

ANGEWANDTE MATHEMATIK UND INFORMATIK Maximal φ-inequalities for Nonnegative Submartingales G. Alsmeyer und U. Rösler Universität Münster und Universität Kiel e-mail: gerolda@math.uni-muenster.de, roesler@math.uni-kiel.de Bericht Nr.??/2-S UNIVERSITÄT MÜNSTER

Maximal φ-inequalities for Nonnegative Submartingales Gerold Alsmeyer Institut für Mathematische Statistik Fachbereich Mathematik Westfälische Wilhelms-Universität Münster Einsteinstraße 62 D-4849 Münster, Germany Uwe Rösler Mathematisches Seminar Christian-Albrechts-Universität Kiel Ludewig-Meyn-Straße 4 D-2498 Kiel, Germany Let M n n be a nonnegative submartingale and def = max kn M k, n the associated maximal sequence. For nondecreasing convex functions φ :[, [, with φ = Orlicz functions we provide various inequalities for Eφ in terms of EΦ am n where, for a, Φ a x def = x s a a φ r r dr ds, x >. Of particular interest is the case φx = xfor which a variational argument leads us to EM n + E log xdx /2 2. A further discussion shows that the given bound is better than Doob s classical bound e e +EM n log + M n whenever EM n + e 2.78. AMS 99 subject classifications. 6G42, 6E5. Keywords and phrases. Nonnegative submartingale, maximal sequence, Orlicz and Young function, Choquet representation, convex function inequality.

2. Introduction Let M n n be a nonnegative submartingale and def = max kn M k, n the associated maxima. Moment inequalities for the sequence n in terms of M n n are usually based on Doob s maximal inequality which states that P >t M n dp. t { >t} for n and t>. If p>, a combination of. with Hölder s inequality shows EM np p p E p.2 p for n [4, p. 255f], the constant being sharp see [3, p. 4]. In case p = one finds with. that E e +EM n log + M n.3 e for n. Clearly, these results apply to M n n if M n n is a martingale. Orlicz and Young functions..2 and.3 are only special cases within a whole class of convex function inequalities based on Doob s inequality which is the main topic of this paper. Let C denote the class of Orlicz functions, that is unbounded, nondecreasing convex functions φ :[, [, with φ =. If the right derivative φ is also unbounded then φ is called a Young function and we denote by C the subclass of such functions. Given any probability space Ω, A,P, each φ C induces the semi-banach space L φ P, φ of φ-integrable random variables X on Ω, A, P, where X φ def = inf{ >: EΦ X / } defines the underlying semi-norm. L φ P, φ is called an Orlicz space and equals the space of α-times integrable functions L α P, α in case φx =x α for some α [,. Since φx = x φ sds xφ x by convexity, the numbers p = p φ def xφ x = inf x> φx and p = p φ def xφ x = sup x> φx.4 are both in [, ]. φ is called moderate [5, p. 62] if p < or, equivalently [7, Thm. 3..], if for some and then all > there exists a finite constant c such that φx c φx, x..5 This property is shared by all φ C which are also regularly varying at infinity at order α [2], thus including φx =x α for α [,. Examples of non-moderate Orlicz functions are φx = expx α for any α.

3 Given a Young function φ, the right continuous inverse ψ x def = inf{y : φ y >x}of φ is also unbounded and thus ψx def = x ψ s ds again an element of C, called the conjugate Young function to φ. Obviously, this conjugation is reflexive. A simple geometric argument shows [5, p. 63] that xφ x = φx + φ x ψ s ds = φx + ψφ x, x..6 and, by reflexivity, the same identity holds true with the roles of φ and ψ interchanged. With the help of.6, ψx p xψ x and ψ φ x x we infer as in [5, p. 69] that ψ xφ x = φx+ψφ x φx+ p φ xψ φ x φx+ ψ p xφ x ψ for x and thus p φ = inf x> xφ x φx p ψ Thm. 3..] so that we have the identity p ψ. The reverse inequality can also be shown [7, p φ = p ψ p ψ, or equivalently p φ + p ψ =..7 As further results stated in [7, Thm. 3..] we mention that for any φ C with p = p φ the assertions φx p φx for all > and x>;.8 φx x p.9 hold true, and that for moderate φ with p = p φ furthermore φx p φx for all > and x>;. φx. x p. The following two inequalities, the first of which may also be found in [5, p. 69], are easily deduced from another inequality stated as 2.2 in the next section. This latter inequality emerges as a special case of one of our results, see Corollary 2.2, but can also be derived by different arguments based on Doob s inequality and the Choquet representation of a function in C. For the interested reader this is briefly demonstrated in Appendix. Proposition.. Let φ be an Orlicz function with p = p φ > and M n n be a nonnegative submartingale. Then M n φ p p M n φ.2

for each n. Ifφis also moderate, i.e. p = p φ <, then furthermore 4 EφM n p p EφM n.3 p for each n. 2. Main Results Inequalities of type.3 are also the content of our main results to be presented in this section. They are based upon integration of a variational variant of Doob s inequality. to be proved in Proposition 3.2, namely P M n >t P M n />sds = t t t E + t 2. for all n, t>and,. Under additional contraints on φ we will see that by good choices of the constant in.3 can be improved considerably. We will also derive an inequality for E in terms of EM n log + M n which in many situations strictly beats.3. The following two subclasses of C will be of interest hereafter. We shall denote by C the set of all differentiable φ C whose derivative is concave or convex, and by C the set of φ C such that φ x x is integrable at and thus in particular φ =. Put C def = C C. Given φ C and a, define Φ a x def = x s a a φ r r dr ds, x > 2.2 and note that Φ a [a, C for a>. If φ C the same holds true for Φ def = Φ, whereas Φ otherwise. The function Φ will be of great importance in our subsequent analysis. If φ C then Φ is obviously again an element from this class. If in addition φ is concave or convex the same holds true for Φ x = x Φ x = φ x x φ r r dr, hence φ C implies Φ C. Use to see that φ and Φ are related through the differential equation xφ x Φx = φx, x 2.3 under the initial conditions φ = φ = Φ = Φ =. Ifφx=x p for some p>, then Φx = p xp, in particular Φ = φ in case φx =x 2. If φx=xthen Φ, but we have Φ x =xlog x x +. Further properties of Φ and its relation to φ are collected in Lemma 3. at the beginning of Section 3 where it will be seen particularly that Φ and φ grow at the same order of magnitude unless φ or its conjugate are non-moderate. Theorem 2.. Let M n n be a nonnegative submartingale and φ C. Then EφM n φa + EΦ am n / 2.4

for all a,, and n, in particular = 2 5 EφM n φa + EΦ a 2M n 2.5 for all a>and n. IfM n n is a positive martingale with M n+ cm n for some c> and all n, and if EΦ a M <, then Eφ c EΦ a M n EΦ a M 2.6 for all n. Of course, inequality 2.4 with a = is of interest only when Φ <, thus for φ C. The conditions on M n n implying 2.6 were given by Gundy [6] to demonstrate that the bound in.3 cannot be much improved, see 2.7 below and also [8, p. 7f]. If φx =x p for some p>, then Φx = p xp implies in 2.4 with a = EM np p p EMp n 2.7 for all n and,. Elementary calculus shows that p def p = argmin, = p p With this value of in 2.7 Doob s L p -inequality.2 comes out again. For an extension of it consider nonnegative increasing functions φ on [, such that φ /γ is also convex for some γ>. As before let M n n be a nonnegative submartingale. Then the same holds true for φ /γ M n n, the associated sequence of maxima being φ /γ n. Consequently,.2 implies γ γ Eφ EφM n 2.8 γ for each n. Interesting examples are φx =x p log r + x for p> and r choose γ = p, as well as φx =e rx for r>. For the latter example any γ>will do and since γ γ γ decreases to e as γ, we obtain Ee rm n eee rm n 2.9 for all n and r>. We will show in the next section that Φx p φx for each φ C with p = p φ > φ C. With this at hand the subsequent corollary follows from Theorem 2.. Corollary 2.2. Given the situation of Theorem 2., let φ C be such that p = p φ >. Then Eφ p EφM n/ 2.

for all, and n. Ifφis also moderate p < then for each n. 6 Eφ p p p p p EφM n 2. A comparison of the constants going with EφM n in 2. and.3 shows that the one in 2. is strictly better unless p = p see Lemma A.2 in Appendix 2 for a rigorous proof. For large p, p and β def = p /p we also have that p p p p p βe, while p p p e β. Putting q = p p and choosing = q in 2., we obtain EφM n EφqM n 2.2 for each n and any φ C with p>. It is this inequality which will easily give the assertions of Proposition. see Section 3. By a similar argument as the one leading to 2.8, Doob s inequality.2 can be used to get refinements of.3 and 2. for functions φ C. However, the main tool for this is not 2. but rather a suitable Choquet representation of φ exploited in a similar manner as in [] for the special case of φ C with concave derivative. Theorem 2.3. Given the situation of Theorem 2., let φ C, k and φ k the k-th order derivative of φ. Then φ k C φ C implies EφM n k+ k + EφM n 2.3 k as well as for each n. M n φ k + k M n φ 2.4 In case φx =x p for integral p>wehaveφ p C and thus that 2.3 coincides with.2. We finally turn to the case φx =xfor which Φ but Φ x =xlog x x + as mentioned earlier. A more general version of inequality 2.4 proved in Section 3 Proposition 3.2 will be used to derive the following alternative to Doob s L -inequality.3. Theorem 2.4. If M n n is a nonnegative submartingale, then E b + b b E log xdx 2.5

7 for all b>and n. The value of b which minimizes the right hand side equals b def = + E M n log xdx /2 and gives EM n + /2 2 E log xdx 2.6 If M n n is a positive martingale with M n+ cm n for some c>and all n, and if EM log + M <, then E EM n log + M n EM log + M 2.7 c for all n. Since x log ydy=xlog+ x x for x, inequality 2.5 may be restated as E b + b EM n log + M n EM n + 2.8 b for all n and b>. For the special choices b = EM n + + and b = e, this yields for each n E +EM n + EM n + EM n log + M n 2.9 and E e + e EM n log + M n EM n +, 2.2 e respectively, where the right hand side of 2.9 is to be interpreted as if M n a.s. Note that even the choice b = e, though only suboptimal, leads to a better bound for E than in.3 whenever EM n + e 2.78. The previous upper bounds for EM n may be further improved if m def = EM >. To see this note that ˆM def n n =, M m, M m,... forms again a nonnegative submartingale whose maxima ˆM n satisfy = M n m for n. Consequently, 2.5 and 2.8 for ˆM n n give Corollary 2.5. EM n E ˆM n+ b + b ˆM n If M n n is a nonnegative submartingale with m = EM >, then b+ /m b E b b E m log+ log xdx m + 2.2 E m /2 2 for all b>and n, in particular /m E + E log xdx 2.22

when choosing the minimizing b compare 2.6. 8 3. Proofs We begin with a collection of some useful properties of the function Φ = Φ in 2.2 associated with an element φ of C. Lemma 3.. For each φ C with p = p φ > the function Φ satisfies Φx φx, x. 3. p If φ is moderate, i.e. p = p φ <, then Φx p φx, x. 3.2 Finally, for each φ C the inequality holds true. lim inf x Φx x log x > 3.3 Φx The final inequality shows that lim x x = whenever φ C is a function close to the identity in the sense that φx = oxlog x asx. Another class of examples comprises φx =rx log r + x log r + x for r in which case Φx =+x log r + x use the differential equation 2.3. Proof. Using xφ x pφx, we obtain by partial integration s φ r r dr = φs s + s φr r 2 dr φs s + p s φ r r dr and thus s φ r r dr p φs p s p φ s for s. An integration of this inequality with respect to s obviously implies 3.. The second inequality 3.2 is obtained analogously when utilizing that p < implies xφ x p φx for all x. As to 3.3, choose a> such that φ a >. Then Φx Φ a x = x s a a φ r dr ds = φ a x logx/a x + a for all x a. The proofs of Theorem 2. and 2.4 are based on the following more general proposition.

9 Proposition 3.2. Let M n n be a nonnegative submartingale and φ C. Then inequality 2. holds true and furthermore EφM n φb+ Φ a Φ a b Φ {M n />b} ab b dp 3.4 for all n, a, b > and,. If φ x x valid for b =. is integrable at, i.e. φ C, then 3.4 remains Proof. Doob s maximal inequality gives P >t M n dp t { >t} = t = t t P M n >t,m n >sds P M n >tds + t P M n >t + t t t P M n >sds PM n />sds 3.5 and thus P >t P M n />sds t t for all n, t> and,, which is 2.. With this result we further infer EφM n φb+ φb+ = φb+ = φb+ = φb+ b φ tpm n >tdt b b b {M n />b} φ t PM n />sds dt t t s φ t dt P M n />sds b t Φ as Φ ab PM n />sds Φ a Φ a b Φ ab b dp for all n, b, t >,, and a>, where a = may also be chosen if φ x x at. is integrable Proof of Theorem 2.. Inequality 2.4 follows directly from Proposition 3.2 if we choose b = a and recall that Φ a a =Φ aa =. Hence we are left with the proof of 2.6. If M n n is a positive martingale satisfying M n+ cm n for all n and some c>, then P >t M n dp ct { >t} ct {M >t} M dp

for all n and t>, see [8, p. 72]. Consequently, P >t M n dp M dp ct {M n >t} ct {M >t} = P M n >s PM >s ds ct t + P M n >t PM >t c 3.6 for all n and t>. Assuming EΦ a M <, and also Eφ < there is nothing to prove otherwise, 2.6 now follows upon integration over, with respect to t of both sides of 3.6 multiplied with φ t. We must only note that φ t P M n >t PM >t dt = EφM n EφM because φ is convex and EφM k EφM n < for k n. We continue with a short proof of Proposition. stated in the Introduction. Proof of Proposition.. Let φ C with p = p φ >, put q def = p p and recall from 2.2 that Eφ def EφqM n for each n. Setting γ n = M n φ, this inequality implies as well as use also. EφM n/q n EφM n / n = EφM n EφqM n q p EφM n. The second inequality proves.3 while the first one yields φ q n and thus assertion.2. Proof of Theorem 2.3. The basic tool for the proof of Theorem 2.3 is to convert the assumption of smooth convexity φ k C into a suitable Choquet decomposition of φ. Each φ C can be written as φx = x t + Q φ dt [, where Q φ dt =φ δ + φ dt. Hence, if φ C, then φx = = = = x x φ y dy y t + Q φ dt dy [, [, [, x y t + dy Q φ dt x t + 2 Q φ dt. 2

An inductive argument now gives that φx = [, x t + k+ k +! Q φ kdt 3.7 for each φ C with φ k C. Put ϕ k,t x def = x t+ k+ k+! for k and t [,. Note that is convex for each t. Thus we infer with the help of 3.7 and the argument which ϕ /k+ k,t proved 2.8 that EφM n = = [, Eϕ k,t M n Q φ kdt k+ k + Eϕ k,t M n Q k φ kdt [, k+ k + EφM n k for each n. Replacing with /γ def n in the previous estimation, where γ n = k+ k M n φ, further gives 2.4 by a similar argument as in the proof of Proposition.. Proof of Theorem 2.4. If φx =x, then Φ x =xlog x x + and Φ x = log x for x>. Inequality 3.4 with these functions reads E b + {M n >b} log M n + b log b M n dp = b + M n log M n M n log + log b + +b dp {M n >b} for all b> and,. Choose b> and = b to obtain 2.5 in its equivalent form 2.8. 2.6 follows by elementary calculus. Finally, if M n n is a positive martingale with M n+ cm n for some c> and all n, and if EM log + M <, then a use of the first inequality in 3.6 gives after partial integration E M n dp M dp dt ct {M n >t} {M >t} M = c E M n t dt M = EM n log + M n EM log + M c t dt for all n and hence the asserted inequality 2.7. Appendix Inequality 2.2 from which Proposition. was deduced may be viewed as a specialization of inequality A.2 below to the pair X, Y =M n,. The purpose of this short

2 appendix is to provide a proof of A.2 based upon a Choquet representation of the involved convex function φ. Lemma A.. Let X, Y be nonnegative random variables satisfying Doob s inequality tpy t E {Y t} X for all t. Then EφY EφqX holds for each Orlicz function φ, where q def = q φ = p φ p φ. A. A.2 Proof. If p = p φ =,thusq=, there is nothing to prove. So let p> and put V = qx. Write φ in Choquet decomposition, that is φx = x t + φ dt, x. Then we obtain EφV EφY = E = E = E + E [, V t + Y t + φ dt [, = E {V Y } V t φ dt [,V ] V t φ dt [,V ] V t φ dt [,Y ] + E V Y φ dt [,Y ] = I + I 2. [,Y ] [,Y ] [,Y ] Y t φ dt V t φ dt Y t φ dt V t φ dt + {V<Y} Y,V ] V,Y ] t V φ dt It is easily seen that I. So it remains to prove that I 2 = EV φ V EY φ Y. To this end write EY φ Y peφy = pe Y t + φ dt [, = peyφ Y pe tφ dt {Y t} which, after rearranging terms and using A., leads to EY φ Y qe tφ dt qe {Y t} {Y t} Xφ dt = EV φ Y and thus the desired conclusion.

3 Appendix 2 We claimed in Section 2 that y y y x y y x A.3 x holds true for all y x> and that the inequality is strict unless x = y. A.3 may be rewritten as y x y Taking logarithms we arrive at y logx logy y x. y y log x log y. The desired conclusion is now an immediate consequence of the following lemma. Lemma A.2. For each y>, the function f y :,y] IR, f y x def = y logx logy y log x log y is strictly decreasing with f y y =. Proof. It suffices to note that for all x,y. f yx = y x y x < References [] ALSMEYER, G. and RÖSLER, U. 23. The best constant in the Topchii-Vatutin inequality for martingales. To appear in Statist. Probab. Letters. [2] BINGHAM, N.H., GOLDIE, C.M. and TEUGELS, J.L. 987. Regular Variation. Cambridge Univ. Press, Cambridge. [3] BURKHOLDER, D.L. 99. Explorations in martingale theory and applications. Ecole d Eté de Probabilité de Saint Flour XIX -989, Ed. P. Hennequin, Lect. Notes Math. 464, -66. Springer, Berlin. [4] CHOW, Y.S. and TEICHER, H. 997. Probability Theory: Independence, Interchangeability, Martingales 3rd Edition. Springer, New York. [5] DELLACHERIE, C. and MEYER, P.A. 982. Probabilities and Potential B. Theory of Martingales. North-Holland, Amsterdam. [6] GUNDY, R.F. 969. On the class L log L, martingales, and singular integrals. Studia Math. 33, 9-8. [7] LONG, R. 993. Martingale Spaces and Inequalities. Peking University Press and Vieweg, Wiesbaden. [8] NEVEU, J. 975. Discrete-Parameter Martingales. North-Holland, Amsterdam.