We obtain lower bounds for ℙ(ξ ≥ 0) and ℙ(ξ > 0) under assumptions on the moments of a centered random variable ξ. The estimates obtained are shown to be optimal and improve results from the literature. They are then applied to obtain probability lower bounds for second order Rademacher chaos.

Recently Balakrishnan and Iliopoulos [Ann. Inst. Statist. Math. 61 (2009)] gave sufficient conditions under which the maximum likelihood estimator (MLE) is stochastically increasing. In this paper we study test plans which are not considered there and we prove that the MLEs for those plans are also stochastically ordered. We also give some applications to the estimation of reliability.

We prove that for λ ∈ [0,1] and A, B two Borel sets in ${ℝ}^n$ with A convex, ${\Phi }^{-1}\left({\gamma }_n(\lambda A+(1-\lambda )B)\right)\ge \lambda {\Phi }^{-1}\left({\gamma }_n\left(A\right)\right)+(1-\lambda ){\Phi }^{-1}\left({\gamma }_n\left(B\right)\right)$, where ${\gamma }_n$ is the canonical gaussian measure in ${ℝ}^n$ and ${\Phi }^{-1}$ is the inverse of the gaussian distribution function.

By using large deviation techniques, we prove a Strassen type law of the iterated logarithm, in Hölder norm, for Lévy's area process.

We improve the constants in the Men’shov-Rademacher inequality by showing that for n ≥ 64, $E\left(su{p}_{1\le k\le n}\right|{\sum }_{i=1}^k{X}_i|²\le 0.11(6.20+log₂n)²$ for all orthogonal random variables X₁,..., Xₙ such that ${\sum }_{k=1}^{n}E\left|X_k\right|²=1$.

In this paper, we introduce a new linear programming second-order stochastic dominance (SSD) portfolio efficiency test for portfolios with scenario approach for distribution of outcomes and a new SSD portfolio inefficiency measure. The test utilizes the relationship between CVaR and dual second-order stochastic dominance, and contrary to tests in Post [Post] and Kuosmanen [Kuosmanen], our test detects a dominating portfolio which is SSD efficient. We derive also a necessary condition for SSD efficiency...

The Riesz transforms of a positive singular measure $\nu \in M\left({\mathbf{R}}^n\right)$ satisfy the weak type inequality$$m\left[\sum _{j=1}^n|R_j\nu |\>\lambda \right]\ge \frac{C\parallel \nu \parallel }{\lambda },\lambda \>0$$where $m$ denotes Lebesgue measure and $C$ is a positive constant only depending on $m$.

Let X be a submartingale starting from 0, and Y be a semimartingale which is orthogonal and strongly differentially subordinate to X. The paper contains the proof of the sharp estimate $\mathbb{P}\left(su{p}_{t\ge 0}\right|Y_t\left|\ge 1\right)\le 3.375...\parallel X\parallel ₁$. As an application, a related weak-type inequality for smooth functions on Euclidean domains is established.