AN INEQUALITY FOR TAIL PROBABILITIES OF MARTINGALES WITH BOUNDED DIFFERENCES

Transcription

1 Lithuanian Mathematical Journal, Vol. 4, No. 3, 00 AN INEQUALITY FOR TAIL PROBABILITIES OF MARTINGALES WITH BOUNDED DIFFERENCES V. Bentkus Vilnius Institute of Mathematics and Informatics, Akademijos 4, LT-600 Vilnius, Lithuania ( Abstract. Let M n = X 1 + +X n be a martingale with bounded differences X m = M m M m 1 such that P{ X m σ m }=1 with some nonnegative σ m. Write σ = σ σ n. We prove the inequalities P{M n x} c(1 (x/σ, P{M n >x} 1 c(1 ( x/σ with a constant c 1/(1 ( 3 5. The result yields sharp inequalities in some models related to the measure concentration phenomena. Keywords: probabilities of large deviations, martingale, bounds for tail probabilities, inequalities, bounded differences and random variables, measure concentration phenomena, product spaces, Lipschitz functions, Hoeffding s inequalities, Azuma s inequality. 1. INTRODUCTION AND RESULTS Let M n be a martingale (we define M 0 = 0 with bounded differences X j = M j M j 1 of sizes σ j such that P{ X j σ j }=1forallj. Hence, M n = X 1 + +X n. Write σ = σ1 + +σ n. Let (x = x be the standard normal distribution function. Write ϕ(t dt, ϕ(t = 1 { exp t }, π D(x = min { exp { x /}, c(1 (x }, where c is an absolute constant such that c 1/(1 ( 3 5. Our result is the following upper and lower bounds for the tail probabilities of M n. THEOREM 1.1. For x R, we have P{M n x} D(x/σ and P{M n >x} 1 D( x/σ. (1.1 Research supported by Max Planck Institute for Mathematics, Bonn. Translated from Lietuvos Matematikos Rinkinys, Vol. 4, No. 3, pp , July September, 00. submitted April 0, 001. Original article /0/ $ Plenum Publishing Corporation 55

2 56 V. Bentkus The upper bound 5 for c in (1.1 is not optimal and can be improved. Extending the methods of the author [], one can show that sup sup P{M n x} (1 (x/σ, n M n where sup is taken over all Bernoulli type martingales M n such that σ 1 = =σ n and all n (we call a martingale of Bernoulli type if the differences conditionally have two point distributions. Hence, the constant c in (1.1 must satisfy c. In the case of i.i.d. Rademacher random variables X 1,...,X n of size 1/ n such that P{X 1 = 1/ n}= P{X 1 = 1/ n}=1/ andσ = 1, we obviously have sup P{M n x} 1 (x, n which, for larger x, differs by the factor c 5 compared to (1.1. Theorem 1.1 improves the inequality of Hoeffding [4]. In our setting, his inequality (Theorem of Hoeffding [4] reads as P{M n x} exp { x /(σ }, x 0 (1. For larger x c 1 σ (with some absolute constant c 1, the bound (1.1 is better, since exp { x /(σ } c(1 (x/σ as x. For independent X 1,...,X n, the inequality of Pinelis [9] improves (1. (see (1.3 below. The inequalities of type (1. (and, hence, of (1.1 as well have extensive applications in combinatorics, operational research, computer science, random graphs (see McDiarmid [6]. The result applies to some simple models in the measure concentration phenomena for Lipschitz functions on product spaces (see Section 3, as well as for some nonlinear statistics. In these models, the bounds implied are the sharpest among the known ones. The space for improvements is restricted (cf. the discussion above. For statistical applications, optimal bounds for finite (that is, fixed n are of interest. In this sense, our result is not optimal and can be improved extending the methods of the author []. Theorem 1.1 shows that the martingale-type dependence does not influence much the bounds for tail probabilities compared to the independent and i.i.d. cases. Theorem 1.1 is insufficiently general. Extensions to more general settings are desirable for applications in measure concentration phenomena (see Section 3. This is even more desirable for applications in statistics (see Bentkus and Van Zuijlen [3], where bounds must be explicit and must depend explicitly on parameters of interest. The proof of Theorem 1.1 is based on induction in n and multiply applications of the Chebyshev inequality. We understand Chebyshev s inequality as f g if f g, and we always apply it with a linear function g. Our methods are well designed for applications where we have martingale-type dependence. Seemingly, for the first time inductional proofs without Fourier transform in the context of probabilities of large deviations were used by the author [1]. We hope that the method is robust against generalizations. However, one has not to underestimate the complexity of technicalities that can occur by extensions of Theorem 1.1. Usually, bounds for tail probabilities are proved using exponential functions (which can be interpreted as a kind of the Fourier Laplace transform together with Chebyshev s inequality, which most probably leads to losses in the accuracy. We do not use exponential functions. For independent X 1,...,X n such that σ 1 = = σ n, a bound P{M n x} B n (x with some function B n (x essentially smaller than c(1 (x, was obtained earlier by the author []. One can show that the bound B n (x is sharp on martingales for integer x and all n. Heuristically, the basic ideas and methods are already contained in this paper. In the context of estimates of the convergence rate in the Central Limit Theorem in general spaces, inductional proofs is an extensively used tool (see, for example, a book of Paulauskas and Račkauskas [7].

3 An inequality for tail probabilities 57 The history of inequalities for tail probabilities is very rich (see, for example, books Petrov [8] and Shorack and Wellner [10]. The names of Chernoff, Bennett, Prokhorov, Hoeffding and other come to mind. It seems that the most precise inequalities for sums of bounded random variables were obtained by Hoeffding [4] almost 40 years ago. We are not aware of improvements of (1. for martingales. For independent X 1,...,X n, Pinelis [9] has proved that with the constant c 0 = e 3 / The bound (1.1 yields P{ M n x} c 0 (1 (x/σ (1.3 P{ M n x} c(1 (x/σ. (1.4 The constant in (1.3 is better than that in (1.4. However, the value of c in (1.1 and (1.4 can be improved. Our methods are completely different from those Hoeffding [4] and Pinelis [9].. THE PROOF We write I(x= ϕ(t dt, B(x = ci(x for x R. x By the definition of c, wehaveci( 3 = 1. Hence, B( 3 = 1andB(x > 1forx 3, and the function B is strictly decreasing. Proof of Theorem 1. The lower bound in (1.1 follows from the upper one, just replacing M n by M n and considering complementary events. Hence, it suffices to prove only the upper bound. We apply induction in n. Without loss of generality we assume that σ = 1. Throughout we write 1 = τ +, where τ = σ 1 and = σ + +σ n. Due to the assumption σ = 1, it suffices to prove that since, by the Hoeffding inequality, P{M n x} exp{ x /} for all x 0. P{M n x} B(x, (.1 The case n = 1. In this case M 1 = X 1, and we have to prove that the upper bound in (.1 holds for all possible martingales M 1,thatis,thatP{X 1 x} B(x for arbitrary random variables X 1 such that EX 1 = 0 and P{ X 1 1} =1. If x 1, then B(x > 1, and we trivially have that P{X 1 x} 1 <B(x. If x>1, then P{X 1 x} =0 <B(x,since X 1 1, which concludes the proof for n = 1. The case n>1. By the induction assumption we have P{Z n 1 x} B(x/β for all x R (. and for any martingale sequence Z 0 = 0,Z 1,...,Z n 1 such that P{ Z k Z k 1 β k }=1andβ = β1 + + βn 1. We can assume throughout that 0 <τ<1, 0 <<1, x> 3. Indeed, in the cases where one of τ or is zero, the martingale M n contains at most n 1 nonzero differences, and the result holds by the inductional assumption. If x 3, then B(x 1, and we trivially have P{M n x} 1 B(x. Using (. and conditioning on X 1,wehave X1 P{M n x} =EP{X + +X n x X 1 X 1 } EB, (.3

4 58 V. Bentkus since, for given X 1, the sequence Z 0 = 0,Z 1 = X,...,Z n 1 = X + +X n is a martingale sequence with differences such that P{ Z k Z k 1 σ k+1 }=1. To simplify the notation, write t ϱ(t = ci, EB X1 = Eϱ(X 1. Note that ϱ depends on n and x, which is not reflected in the notation. We have to prove that Eϱ(X 1 B(x. The function ϱ is convex in the interval (, 3], since we assume that x 3 and the function z I(z is convex for z 0. The random variable X 1 satisfies the inequality P{ X 1 τ} =1. The function ϱ is convex in the interval [ τ,τ], since τ 1andϱ is convex in (, 3]. Therefore, the linear function u(t def = τ t τ ϱ( τ+ t + τ τ ϱ(τ satisfies u( τ = ϱ( τ and u(τ = ϱ(τ and is an upper bound for ϱ(t, ϱ(t u(t for all t τ. (.4 Using (.4 and EX 1 = 0, we obtain Eϱ(X 1 Eu(X 1 = c + τ I + c τ I. (.5 Inequality (.5 reduces the proof of (.1 to checking that Let us prove (.6. We have w(τ def = I + τ + I τ + τ xτ + 1 τ w (τ = ϕ 3 ϕ I(x 0. (.6 xτ 1 3. (.7 A bit later we shall prove that w (τ 0for0 τ 1. Therefore, w(τ is a decreasing function of τ 0, and the proof of (.6 reduces to checking that w(0 0. But w(0 = 0, which concludes the proof of (.6. Let us prove that w (τ 0for0 τ 1. Using (.7 and the fact that ϕ(t = (π 1/ exp{ t /}, the inequality w (τ 0 is equivalent to Hence, we have to check (.8. We have v(τ def = xτ (xτ 1 exp {xa} 0, A = A(τ def = τ. (.8 v (τ = x + x exp {xa}+x(xτ 1A exp {xa}, v (0 = 0, and with ( v (τ = x exp {xa} xa + (xτ 1A + x(xτ 1A (.9 A = 1 + τ 4, A = (3τ + τ 3 6. (.10 Below we shall show that v (τ 0. This means that τ v(τ is a convex function such that v(0 = 0. Therefore, for τ 0, this function can assume only nonnegative values, which proves (.8.

5 An inequality for tail probabilities 59 It remains to show that v (τ 0. Due to (.9 and (.10, this is equivalent to the inequality x(1 + τ 4 + (xτ 1(3τ + τ x(xτ 1(1 + τ 8 0. (.11 Multiplying by 8,using = 1 τ, and collecting similar terms, we see that (.11 is equivalent to with Dividing by τ, inequality (.1 is equivalent to ax + bx + d 0 (.1 a = τ(1 + τ, b = 4τ 4, d = τ(3 τ τ 4. y(x def = (1 + τ x 4τ 3 x 3 + τ + τ 4 0. (.13 The function x y(x is a quadratic increasing function of x x τ,where It is easy to verify that x τ = τ 3 (1 + τ. x τ 1 for all 0 τ 1. Hence, the function x y(x from (.13 is an increasing function of x 1. Therefore, for x 3, inequality (.13 is implied by the stronger inequality y( 3 0, that is, by the inequality y( 3 = 8τ + 4τ 4 4 3τ 3 0 for 0 τ 1. (.14 The verification of (.14 is elementary. This proves (.11 and the theorem. 3. SOME APPLICATIONS TO THE MEASURE CONCENTRATION To simplify the notation, instead of a product of arbitrary metric (or measurable spaces, we consider only the case of the n-dimensional unit cube [ 1, 1] n. Let Z 1,...,Z n be independent random variables such that P{ Z j 1} =1. Let a function f :[ 1, 1] n R satisfy the separate Lipschitz condition in each of the variables, that is, f(x1,...,x j 1,x,x j+1,...,x n f(x 1,...,x j 1,y,x j+1,...,x n σj x y. (3.1 Recall that σ = σ 1 + +σ n. COROLLARY 3.1. Let f satisfy the Lipschitz condition (3.1. Then, for x 0, we have P{f Ef + x} D, P{f Ef x} D, (3. σ σ where f = f(z 1,...,Z n. If EZ j = 0 for all j, then the bound (3. improves to P{f Ef + x} D, P{f Ef x} D. (3.3 σ σ

6 60 V. Bentkus Inequalities (3.3 are rather precise already in the case of the linear function f(x= σ 1 x 1 + +σ n x n, (see Introduction. The Lipschitz condition is just a device to control the sizes of the differences X j = E j+1,...,n f(z 1,...,Z n E j,...,n f(z 1,...,Z n, where E j,...,n f(z 1,...,Z n = E ( f(z 1,...,Z n Z 1,...,Z j 1 (3.4 stands for the conditional expectation given Z 1,...,Z j 1. The bound (3.3 holds in the case of a product of arbitrary measurable spaces and arbitrary functions f such that the differences X j from (3.4 have the sizes σ j, that is, P{ X j σ j }=1. Proof of Corollary 3.1. Note that f Ef = X 1 + +X n is a martingale sequence with the differences X j. In order to apply Theorem 1.1, we have to estimate the sizes of X j. Let Y j be an independent copy of Z j, independent from other random variables as well. Let E be the conditional expectation given Z 1,...,Z j. Then, applying the Lipschitz condition (3.1, we have X j E f(...,zj 1,Z j,z j+1,... f(...,z j 1,Y j,z j+1,... σ j E Z j Y j. (3.5 To prove (3. we use the trivial estimate E Z j Y j. Then (3.5 yields X j σ j,andtheorem1.1 implies (3.. To prove (3.3 we note that E x Y 1 for all 1 x 1 (3.6 if a random variable Y satisfies EY = 0andP{ Y 1} =1. Using (3.6, estimate (3.5 yields X j σ j,and an application of Theorem 1.1 provides (3.3. Let us prove (3.6. It is clear that u(t def = x t v(t def 1 + x (1 t 1 x (1 + t = + (3.7 for x 1and t 1. Indeed, the function t u(t is a convex function of t. The function t v(t is a linear function of t such that u( 1 = v( 1 and u(1 = v(1. Hence, (3.7 holds. Replacing t by Y, taking the expectation in (3.7 and using EY = 0, we obtain E x Y 1 + x + 1 x 1 for x 1, which proves (3.6. The inequalities of types (3. and (3.3 have extensive applications in combinatorics and random graphs (see McDiarmid [6]. The known inequalities have the form P{f Ef + x} exp{ x /(σ }, which, for larger x, is worse than (3.3. The models we considered are rather simple ones in the measure concentration. The bounds (3. and (3.3 seem to be the sharpest among the known ones in these models. See Talagrand [11] and Ledoux [5] for a nice introduction into the topic.

7 An inequality for tail probabilities 61 REFERENCES 1. V. Bentkus, Large deviations in Banach spaces, Probab. Theory Appl., 31(4, ( V. Bentkus, An inequality for large deviation probabilities of sums of bounded i.i.d.r.v., Lith. Math. J., 41(, ( V. Bentkus and M. van Zuijlen, Upper confidence bounds for mean, Ann. Statist. (to appear. 4. W. Hoeffding, Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc., 58, ( M. Ledoux, Concentration of measure and logarithmic Sobolev inequalities, in: Séminaire de Probabilités, XXXIII, Springer, Berlin (1999, pp C. McDiarmid, On the method of bounded differences, London Math. Soc. Lecture Note Ser., 141, ( V. Paulauskas and A. Račkauskas, Approximation Theory in the Central Limit Theorem. Exact Results in Banach Spaces, Kluwer, Dordrecht ( V. V. Petrov, Sums of Independent Random Variables, Springer, New York ( I. Pinelis, Extremal probabilistic problems and Hotelling s T test under a symmetry assumption, Ann. Statist., (4, ( G. R. Shorack and J. A. Wellner, Empirical Processes with Applications to Statistics, Wiley, New York ( M. Talagrand, Concentration of measure and isoperimetric inequalities in product spaces, Inst. Hautes Études Sci. Publ. Math., 81, (1995. (Translated by V. Bentkus