Doob's Maximal Inequality Research Articles

Martingale Hardy Orlicz–Lorentz–Karamata spaces and applications in Fourier analysis

Abstract We summarize some results as well as we prove some new results about the Orlicz–Lorentz–Karamata spaces and martingale Hardy Orlicz–Lorentz–Karamata spaces. More precisely, Doob's maximal inequality for submartingales and Burkholder–Davis–Gundy inequality are presented. We also show some fundamental martingale inequalities and modular inequalities. Additionally, based on atomic decompositions, duality theorems and fractional integral operators are discussed. As applications in Fourier analysis, we consider the Walsh–Fourier series on Orlicz–Lorentz–Karamata spaces. The dyadic maximal operators on martingale Hardy Orlicz–Lorentz–Karamata spaces are presented. The boundedness of maximal Fejér operator is proved, which further implies some convergence results of the Fejér means.

Detection of Arbitrage Opportunities and Forecasting in the Foreign Exchange Markets

In this paper, the existence of no arbitrage assumption over time and triangular arbitrage in a foreign exchange market is tested. Also, forecasting the exchange rates are studied. An optimal property of exchange rates returns to guarantee the no arbitrage assumption as well as for forecasting exchange rates is the martingale property. Some theoretical results under the risk neutral measure and their equivalents form in physical probability measure are given. Also, based on a real data set, it is seen that this assumption works well for forecasting purposes. Using the Doob maximal inequality, the accuracy of forecasts is surveyed. Then, a theoretical relation between beta market risks of exchange rates is surveyed. Finally, a conclusion section is also proposed.

Atomic decompositions and asymmetric Doob inequalities in noncommutative symmetric spaces

We provide asymmetric forms of Doob maximal inequality in the contexts of moments associated with convex functions and symmetric spaces of measurable operators. More precisely, let (M,τ) be a noncommutative measurable space equipped with a filtration (Mn)n≥1. Let (En)n≥1 be the conditional expectations corresponding to (Mn)n≥1. A sample result for the case of convex functions states that if Φ is an Orlicz function that is p-convex and q-concave for 1≤p≤q<2 and 1−p/2<θ<1, then for every x in the column Hardy space HΦc(M), there exist positive operators a,b, and a sequence of contractions (un)n≥1 in M such that for every n≥1,En(x)=aunb andτ(Φ(a1/(1−θ)))+τ(Φ(b1/θ))≲τ(Φ(Sc(x))). For the case of symmetric space, as an illustration, we obtain that if E is a separable symmetric Banach function space which is an interpolation of (Lp,Lq) for 1<p≤q<2, then for every z∈E(M), there exist positive operators α,β, two sequences of contractions (υn)n≥1 and (ηn)n≥1 in M such that for every n≥1,En(z)=αυn+ηnβ and‖α‖E(M)+‖β‖E(M)≲‖z‖E(M).Our approach is based on new algebraic atomic decompositions and improved Davis type decompositions which we develop for both the case of Hardy spaces associated with separable noncommutative symmetric spaces that are interpolation of the couple (Lp,Lq) for 1<p≤q<2 as well as the case of Hardy spaces associated with convex functions that are p-convex and q-concave for 1≤p≤q<2. Some other inequalities are also provided which are of independent interest.

New martingale inequalities and applications to Fourier analysis

Let (Ω,F,P) be a probability space and φ:Ω×[0,∞)→[0,∞) be a Musielak–Orlicz function. In this article, the authors prove that the Doob maximal operator is bounded on the Musielak–Orlicz space Lφ(Ω). Using this and extrapolation method, the authors then establish a Fefferman–Stein vector-valued Doob maximal inequality on Lφ(Ω). As applications, the authors obtain the dual version of the Doob maximal inequality and the Stein inequality for Lφ(Ω), which are new even in weighted Orlicz spaces. The authors then establish the atomic characterizations of martingale Musielak–Orlicz Hardy spaces Hφs(Ω), Pφ(Ω), Qφ(Ω), HφS(Ω) and HφM(Ω). From these atomic characterizations, the authors further deduce some martingale inequalities between different martingale Musielak–Orlicz Hardy spaces, which essentially improve the corresponding results in Orlicz space case and are also new even in weighted Orlicz spaces. By establishing the Davis decomposition on HφS(Ω) and HφM(Ω), the authors obtain the Burkholder–Davis–Gundy inequality associated with Musielak–Orlicz functions. Finally, using the previous martingale inequalities, the authors prove that the maximal Fejér operator is bounded from Hφ[0,1) to Lφ[0,1), which further implies some convergence results of the Fejér means; these results are new even for the weighted Hardy spaces.

Algebraic Davis Decomposition and Asymmetric Doob Inequalities

In this paper we investigate asymmetric forms of Doob maximal inequality. The asymmetry is imposed by noncommutativity. Let $(\M,\tau)$ be a noncommutative probability space equipped with a weak-$*$ dense filtration of von Neumann subalgebras $(\M_n)_{n \ge 1}$. Let $\E_n$ denote the corresponding family of conditional expectations. As an illustration for an asymmetric result, we prove that for $1 < p < 2$ and $x \in L_p(\M,\tau)$ one can find $a, b \in L_p(\M,\tau)$ and contractions $u_n, v_n \in \M$ such that $$\E_n(x) = a u_n + v_n b \quad \mbox{and} \quad \max \big\{ \|a\|_p, \|b\|_p \big\} \le c_p \|x\|_p.$$ Moreover, it turns out that $a u_n$ and $v_n b$ converge in the row/column Hardy spaces $\H_p^r(\M)$ and $\H_p^c(\M)$ respectively. In particular, this solves a problem posed by Defant and Junge in 2004. In the case $p=1$, our results establish a noncommutative form of Davis celebrated theorem on the relation between martingale maximal and square functions in $L_1$, whose noncommutative form has remained open for quite some time. Given $1 \le p \le 2$, we also provide new weak type maximal estimates, which imply in turn left/right almost uniform convergence of $\E_n(x)$ in row/column Hardy spaces. This improves the bilateral convergence known so far. Our approach is based on new forms of Davis martingale decomposition which are of independent interest, and an algebraic atomic description for the involved Hardy spaces. The latter results are new even for commutative von Neumann algebras.

Rosenthal-type inequalities for the maximum of partial sums of stationary processes and examples

The aim of this paper is to propose new Rosenthal-type inequalities for moments of order higher than 2 of the maximum of partial sums of stationary sequences including martingales and their generalizations. As in the recent results by Peligrad et al. [Proc. Amer. Math. Soc. 135 (2007) 541-550] and Rio [J. Theoret. Probab. 22 (2009) 146-163], the estimates of the moments are expressed in terms of the norms of projections of partial sums. The proofs of the results are essentially based on a new maximal inequality generalizing the Doob maximal inequality for martingales and dyadic induction. Various applications are also provided.

Random martingales and localization of maximal inequalities

Let ( X , d , μ ) be a metric measure space. For ∅ ≠ R ⊆ ( 0 , ∞ ) consider the Hardy–Littlewood maximal operator M R f ( x ) = def sup r ∈ R 1 μ ( B ( x , r ) ) ∫ B ( x , r ) | f | d μ . We show that if there is an n > 1 such that one has the “microdoubling condition” μ ( B ( x , ( 1 + 1 n ) r ) ) ≲ μ ( B ( x , r ) ) for all x ∈ X and r > 0 , then the weak ( 1 , 1 ) norm of M R has the following localization property: ‖ M R ‖ L 1 ( X ) → L 1 , ∞ ( X ) ≍ sup r > 0 ‖ M R ∩ [ r , n r ] ‖ L 1 ( X ) → L 1 , ∞ ( X ) . An immediate consequence is that if ( X , d , μ ) is Ahlfors–David n-regular then the weak ( 1 , 1 ) norm of M R is ≲ n log n , generalizing a result of Stein and Strömberg (1983) [47]. We show that this bound is sharp, by constructing a metric measure space ( X , d , μ ) that is Ahlfors–David n-regular, for which the weak ( 1 , 1 ) norm of M ( 0 , ∞ ) is ≳ n log n . The localization property of M R is proved by assigning to each f ∈ L 1 ( X ) a distribution over random martingales for which the associated (random) Doob maximal inequality controls the weak ( 1 , 1 ) inequality for M R .

Doobs inequality for non-commutative martingales

Let $1\le p<\8$ and $(x_n)_{\nen}$ be a sequence of positive elements in a non-commutative $L_p$ space and $(E_n)_{\nen}$ be an increasing sequence of conditional expectations, then the $L_p$ norm of \sum_n E_n(x_n) can be estimated by c_p times the $L_p$ norm of \sum_n x_n. This inequality is due to Burkholder, Davis and Gundy in the commutative case. By duality, we obtain a version of Doob's maximal inequality for $1<p\le \8$.

Maximal inequalities for demimartingales and a strong law of large numbers

Chow's maximal inequality for (sub)martingales is extended to the case of demi(sub)martingales introduced by Newman and Wright (Z. Wahrsch. Verw. Geb. 59 (1982) 361–371). This result serves as a “source” inequality for other inequalities such as the Hajek–Renyi inequality and Doob's maximal inequality and leads to a strong law of large numbers. The partial sum of mean zero associated random variables is a demimartingale. Therefore, maximal inequalities and a strong law of large numbers are obtained for associated random variables as special cases.

Best Bounds in Doob's Maximal Inequality for Bessel Processes

Let ((Zt), Pz) be a Bessel process of dimension α>0 started at z under Pz for z⩾0. Then the maximal inequalityEz(max0⩽t⩽τZpt)⩽pp−(2−αp/(2−α)Ez(Zpτ)−pp−(2−α)zpis shown to be satisfied for all stopping times τ for (Zt) with Ez(τp/2)<∞, and all p>(2−α)∨0. The constants (p/(p−(2−α)))p/(2−α) and p/(p−(2−α)) are the best possible. If λ is the greater root of the equation λ1−(2−α)/p−λ=(2−α)/(cp−c(2−α)), the equality is attained in the limit through the stopping timesτλ, p=inf{t>0:Zpt⩽λmax0⩽r⩽tZpr}when c tends to the best constant (p/(p−(2−α)))p/(2−α) from above. Moreover we show that Ez(τq/2λ, p)<∞ if and only if λ>((1−(2−α)/q)∨0)p/(2−α). The proof of the inequality is based upon solving the optimal stopping problemV∗(z)=supτEz(max0⩽t⩽τZpt−cZpτ)by applying the principle of smooth fit and the maximality principle. In addition, the exact formula for the expected waiting time of the optimal strategy is derived by applying the minimality principle. The main emphasis of the paper is on the explicit expressions obtained.

On Doob's maximal inequality for Brownian motion

If B = ( B t ) t ⩾ 0 is a standard Brownian motion started at x under P x for x ⩾ 0, and τ is any stopping time for B with E x ( τ) < ∞, then for each p > 1 the following inequality is shown to be sharp: E x max 0⩽t⩽τ |B t| p ⩽ p p−1 p E x|B t| p− p p−1 x p. The sharpness is realized through the stopping times of the form τ λ,ϵ= inf t >0| max 0⩽s⩽t B s−λB t⩾ϵ for which it is computed: E 0(τλ,ϵ= ϵ 2 λ(2−λ) whenever ε > 0 and 0 < λ < 2. Hence, for the stopping time σ λ,ϵ= inf t>0 max 0⩽s⩽t |B s|−λ|B t|⩾ϵ which is shown to be a convolution of τ λ, λε with the first hitting time of ε by | B| = (| B t |) t ⩾ 0 , we have E 0(σλ,ϵ)= 2ϵ 2 (2−λ) for all ε > 0 and all 0 < λ < 2. The method of proof relies upon the principle of smooth fit and the maximality principle for a Stephan problem with moving (free) boundary, and Itô-Tanaka's formula (being applied two dimensionally). The main emphasis is on the explicit formulas obtained throughout.

Associated random variables and martingale inequalities

Many of the classical submartingale inequalities, including Doob's maximal inequality and upcrossing inequality, are valid for sequences Sj such that the (Sj+1-Sj's are associated (positive mean) random variables, and for more general “demisubmartingales”. The demisubmartingale maximal inequality is used to prove weak convergence to the two-parameter Wiener process of the partial sum processes constructed from a stationary two-parameter sequence of associated random variables Xijwith \(\mathop \sum \limits_i \mathop \sum \limits_j {\text{Cov}}\left( {X_{{\text{oo}}} ,X_{ij} } \right) < \infty \).