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1
Content available remote Doob's estimate for coherent random variables and maximal operators on trees
Cichomski Stanisław, Osękowski Adam
Probability and Mathematical Statistics
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2023
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Vol. 43, Fasc. 1
109--119
EN
Let ξ be an integrable random variable defined on (Ω, F, P). Fix k ∈ Z+ and let {Gji }1≤i≤n,1≤j≤k be a reference family of sub-σ-fields of F such that {Gji }1≤i≤n is a filtration for each j ∈ {1, . . . , k}. In this article we explain the underlying connection between the analysis of the maximal functions of the corresponding coherent vector and basic combinatorial properties of the uncentered Hardy-Littlewood maximal operator. Following a classical approach of Grafakos, Kinnunen and Montgomery-Smith, we establish an appropriate version of Doob’s celebrated maximal estimate.
2
Content available remote Burkholder inequality by Bregman divergence
Bogdan Krzysztof, Więcek Mateusz
Bulletin of the Polish Academy of Sciences. Mathematics
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2022
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Vol. 70, no. 1
83--92
EN
We prove the Burkholder inequality using Bregman divergence.
3
Content available remote Weighted Maximal Inequalities for Martingale Transforms
Brzozowski Michał, Osękowski Adam
Probability and Mathematical Statistics
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2021
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Vol. 41, Fasc. 1
89--114
EN
We study the weighted maximal L1-inequality for martingale transforms, under the assumption that the underlying weight satisfies Muckenhoupt’s condition A∞ and that the filtration is regular. The resulting linear dependence of the constant on the A∞ characteristic of the weight is optimal. The proof exploits certain special functions enjoying appropriate size conditions and concavity.
4
Content available remote Inequalities for second-order Riesz transforms associated with Bessel expansions
Osękowski Adam
Bulletin of the Polish Academy of Sciences. Mathematics
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2020
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Vol. 68, no. 1
75--88
EN
The paper contains the proofs of Lp, logarithmic and weak-type estimates for the second-order Riesz transforms arising in the context of multidimensional Bessel expansions. Using a novel probabilistic approach, which rests on martingale methods and the representation of Riesz transforms via associated Bessel-heat processes, we show that these estimates hold with constants independent of the dimension.
5
Content available remote Bellman functions and L^p estimates for paraproducts
Kovač V., Škreb K. A.
Probability and Mathematical Statistics
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2018
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Vol. 38, Fasc. 2
459--479
EN
We give an explicit formula for one possible Bellman function associated with the Lp boundedness of dyadic paraproducts regarded as bilinear operators or trilinear forms. Then we apply the same Bellman function in various other settings, to give self-contained alternative proofs of the estimates for several classical operators. These include the martingale paraproducts of Bañuelos and Bennett and the paraproducts with respect to the heat flows.
6
Content available remote An equivalent characterization of weak BMO martingale spaces
Zhou D., Li W., Jiao Y.
Probability and Mathematical Statistics
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2018
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Vol. 38, Fasc. 2
287--298
EN
In this paper, we give an equivalent characterization of weak BMO martingale spaces due to Ferenc Weisz (1998).
7
Content available remote Martingale approach for tail asymptotic problems in the generalized Jackson network
Miyazawa M.
Probability and Mathematical Statistics
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2017
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Vol. 37, Fasc. 2
395--430
EN
We study the tail asymptotic of the stationary joint queue length distribution for a generalized Jackson network (GJN for short), assuming its stability. For the two-station case, this problem has recently been solved in the logarithmic sense for the marginal stationary distributions under the setting that arrival processes and service times are of phase-type. In this paper, we study similar tail asymptotic problems on the stationary distribution, but problems and assumptions are different. First, the asymptotics are studied not only for the marginal distribution but also the stationary probabilities of state sets of small volumes. Second, the interarrival and service times are generally distributed and light tailed, but of phase-type in some cases. Third, we also study the case that there are more than two stations, although the asymptotic results are less complete. For them, we develop a martingale method, which has been recently applied to a single queue with many servers by the author.
8
Content available remote Weighted weak-type inequality for martingales
Osękowski A.
Bulletin of the Polish Academy of Sciences. Mathematics
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2017
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Vol. 65, no. 2
165--175
EN
Let X = (Xt) t ≥ 0 be a bounded martingale and let Y = (Yt) t ≥ 0 be differentially subordinate to X. We prove that if 1 ≤ p < ∞ and W = (Wt) t ≥ 0 is an Ap weight of characteristic [W] Ap, then ∥Y∥Lp, ∞ (W) ≤ Cp [W]Ap∥X∥L∞(W). The linear dependence on [W]Ap is shown to be the best possible. The proof exploits a weighted exponential bound which is of independent interest. As an application, a related estimate for the Haar system is established.
9
Content available remote Sharp inequalities for the Haar system and martingale transforms
Osękowski A.
Probability and Mathematical Statistics
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2016
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Vol. 36, Fasc. 1
147--164
EN
A classical result of Paley and Marcinkiewicz asserts that the Haar system on [0, 1] forms an unconditional basis in Lp provided 1 < p < ∞. The purpose of the paper is to study related weak-type inequalities, which can be regarded as a version of this property for p = 1. Probabilistic counterparts, leading to some sharp estimates for martingale transforms, are presented.
10
Content available remote A maximal inequality for stochastic integrals
Rapicki M.
Probability and Mathematical Statistics
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2016
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Vol. 36, Fasc. 2
311--333
EN
Assume that X is a càdlàg, real-valued martingale starting from zero, H is a predictable process with values in [−1, 1] and Y = ∫ HdX. This article contains the proofs of the following inequalities: (i) If X has continuous paths, then P(supt ≥ 0 Yt ≥ 1) ≤ 2E supt ≥ 0 Xt, where the constant 2 is the best possible. (ii) If X is arbitrary, then P(supt ≥ 0 Yt ≥ 1) ≤ cE supt ≥ 0 Xt; where c = 3.0446… is the unique positive number satisfying the equation 3c4 − 8c3 − 32 = 0. This constant is the best possible.
11
Content available remote Sharp Logarithmic Inequalities for Two Hardy-type Operators
Osękowski A.
Bulletin of the Polish Academy of Sciences. Mathematics
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2015
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Vol. 63, no. 3
237--247
EN
For any locally integrable f on Rn, we consider the operators S and T which average f over balls of radius |x| and center 0 and x, respectively: [WZÓR] for x ∈ Rn. The purpose of the paper is to establish sharp localized LlogL estimates for S and T. The proof rests on a corresponding one-weight estimate for a martingale maximal function, a result which is of independent interest.
12
Content available remote Sharp Weak-Type Inequality for the Haar System, Harmonic Functions and Martingales
Osękowski A.
Bulletin of the Polish Academy of Sciences. Mathematics
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2014
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Vol. 62, no 2
187--196
EN
Let (hk)k≥0 be the Haar system on [0,1]. We show that for any vectors ak from a separable Hilbert space H and any εk∈[−1,1], k=0,1,2,…, we have the sharp inequality ...[formula], where W([0,1]) is the weak-L∞ space introduced by Bennett, DeVore and Sharpley. The above estimate is generalized to the sharp weak-type bound ∥Y∥W(Ω)≤2∥X∥L∞(Ω), where X and Y stand for H-valued martingales such that Y is differentially subordinate to X. An application to harmonic functions on Euclidean domains is presented.
13
Content available remote Law of large numbers for monotone convolution
Wang J.-C., Wendler E.
Probability and Mathematical Statistics
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2013
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Vol. 33, Fasc. 2
225--231
EN
Using the martingale convergence theorem, we prove a law of large numbers for monotone convolutions μ1 ◃ μ2 ◃ . . . ◃ μn, where μj ’s are probability laws on R with finite variances but not required to be identical.
14
Content available remote Moment Inequality for the Martingale Square Function
Osękowski A.
Bulletin of the Polish Academy of Sciences. Mathematics
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2013
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Vol. 61, no 2
169--180
EN
Consider the sequence (Cn)n≥1 of positive numbers defined by C1=1 and Cn+1=1+C2n/4, n=1,2,…. Let M be a real-valued martingale and let S(M) denote its square function. We establish the bound E|Mn|≤CnESn(M), n=1,2,…, and show that for each n, the constant Cn is the best possible.
15
Content available remote Maximal Weak-Type Inequality for Orthogonal Harmonic Functions and Martingales
Osękowski A.
Bulletin of the Polish Academy of Sciences. Mathematics
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2013
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Vol. 61, no 3-4
209--218
EN
Assume that u, v are conjugate harmonic functions on the unit disc of C, normalized so that u(0)=v(0)=0. Let u∗, |v|∗ stand for the one- and two-sided Brownian maxima of u and v, respectively. The paper contains the proof of the sharp weak-type estimate... [formula]. Actually, this estimate is shown to be true in the more general setting of differentially subordinate harmonic functions defined on Euclidean domains. The proof exploits a novel estimate for orthogonal martingales satisfying differential subordination.
16
Content available remote Two-Echelon Reservoir Inventory Management with Forecast Updates: Perspective from Operations of Multireservoir in Interbasin Water Diversion Projects
Xu Y., Wang L., Chen Z., Xia G., Yang Y.
Przegląd Elektrotechniczny
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2012
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R. 88, nr 9b
84-91
EN
We consider a finite-horizon, periodic-review inventory model with inflow forecasting updates following the martingale model of forecast evolution (MMFE) in multiresevoirs. This model introduces a new method of determining an operating policy in which the policy is based on the dynamic programming (DP) model with a physical equation and a recursive equation. It adequately considers the internal relationship among multireservoirs in inter-basin water diversion projects (IBWDP) and calculates the expected benefits from future operation. The stochastic nature of the inflow is taken into account by considering the correlation between the streamflows of each pair of consecutive time intervals based on MMFE. According to interdependence, the probability of transition from a given state or stage to its succeeding ones can be calculated. Finally, to assess the effectiveness of the policies, the model is compared with other model and is applied to the Chinese South-North Water Diversion project.
PL
Analizowano model okresowej inwentaryzacji wraz z przewidywaniem nawodnienia w systemie wielu rezerwuarów. Wprowadzono programowanie dynamiczne uwzględniające wewnętrzne relacje między rezerwuarami w dywersyjnych projektach wodnych. Model sprawdzono na przykładzie chińskiego projektu systemu wodnego północ-południe.
17
Content available remote A Note on the Burkholder–Rosenthal Inequality
Osękowski A.
Bulletin of the Polish Academy of Sciences. Mathematics
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2012
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Vol. 60, no 2
177--185
EN
Let df be a Hilbert-space-valued martingale difference sequence. The paper is devoted to a new, elementary proof of the estimate... [formula].
18
Content available remote Weak-type inequality for the martingale square function and a related characterization of Hilbert spaces
Osękowski A.
Probability and Mathematical Statistics
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2011
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Vol. 31, Fasc. 2
227--238
EN
Let f be a martingale taking values in a Banach space B and let S(f) be its square function. We show that if B is a Hilbert space, then P(S(f) ≥1)≤√e∥f∥1and the constant √e is the best possible. This extends the result of Cox, who established this bound in the real case. Next, we show that this inequality characterizes Hilbert spaces in the following sense: if B is not a Hilbert space, then there is a martingale f for which the above weak-type estimate does not hold.
19
Content available remote Sharp inequalities for the square function of a nonnegative martingale
Osękowski A.
Probability and Mathematical Statistics
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2010
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Vol. 30, Fasc. 1
61--72
EN
We determine the optimal constants Cp and C*p p such that the following holds: if f is a nonnegative martingale and S(f) and f* denote its square and maximal functions, respectively, then ǁS(f)ǁp ≤Cp ǁfǁp; p < 1; and ǁS(f)ǁp ≤C*p ǁf*ǁp; p ≤1.
20
Content available remote Sharp ratio inequalities for a conditionally symmetric martingale
Osękowski A.
Bulletin of the Polish Academy of Sciences. Mathematics
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2010
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Vol. 58, no 1
65-77
EN
Let ƒ be a conditionally symmetric martingale and let S(ƒ) denote its square function. (i) For p, q > 0, we determine the best constants Cp,q such that [wzór...]. Furthermore, the inequality extends to the case of Hilbert space valued ƒ. (ii) For N = 1,2,... and q > 0, we determine the best constants C'N,q such that [wzór...]. These bounds are extended to sums of conditionally symmetric variables which are not necessarily integrable. In addition, we show that neither of the inequalities above holds if the conditional symmetry is not assumed.
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