We investigate the convergence rate of optimal entropic cost $$v_\varepsilon $$ to transport as noise parameter $$\varepsilon \downarrow 0$$ . show that for a large class functions c on $${\mathbb {R}}^d\times {\mathbb {R}}^d$$ (for which plans are not necessarily unique or induced by map) and compactly supported $$L^{\infty }$$ marginals, one has -v_0= \frac{d}{2} \varepsilon \log (1/\varepsil...

Park and Pham's recent proof of the Kahn-Kalai conjecture was a major breakthrough in field graph hypergraph thresholds. Their result gives an upper bound on threshold at which probabilistic construction has $1-\epsilon$ chance achieving given monotone property. While their other parameters is optimal up to constant factors for any fixed $\epsilon$, it does not have dependence $\epsilon$ as $\e...

Let I ⊂ P(N) stand for an ideal containing finite sets. We discuss various kinds of statistical convergence and I-convergence for sequences of functions with values in R or in a metric space. For real valued measurable functions defined on a measure space (X,M, μ), we obtain a statistical version of the Egorov theorem (when μ(X) < ∞). We show that, in its assertion, equi-statistical convergence...

In this paper we investigate the relationship between the statistical (or generally I-convergence) of a series and the usual convergence of its subseries. We also give a counterexample which shows that Theorem 1 of the paper by B.C. Tripathy “On statistically convergent series”, Punjab.Univ. J.Math. 32 (1999), 1–8, is not correct.

The complete convergence is investigated for moving-average processes of doubly infinite sequence of negative dependence sub-gaussian random variables with zero means, finite variances and absolutely summable coefficients. As a corollary, the rate of complete convergence is obtained under some suitable conditions on the coefficients.