Aim of this paper is to give an extensive treatment of bipartite mean field spin systems, ordered and disordered: at first, bipartite ferromagnets are investigated, achieving an explicit expression for the free energy trough a new minimax variational principle. Furthermore via the Hamilton-Jacobi technique the same free energy structure is obtained together with the existence of its thermodynam...

is atomic in nature . Hence, whenever μ is continuous, we have Tn =1 almost surely for all n. This is in stark contrast with the Glivenko-Cantelli norm, which is known to converge to zero almost surely as n --~ oo (by the Glivenko--Cantelli theorem) . If μ is atomic, it is quite obvious that Tn --~ 0 almost surely as n --~ 0. In order for Tn to be small when μ is nonatomic, we should not use th...

Mahmoody et al. (TCC 2016-A) showed that basing indistinguishability obfuscation (IO) on a wide range of primitives in a black-box way is as hard as basing public-key cryptography on one-way functions. The list included any primitive P that could be realized relative to random trapdoor permutation or degree-O(1) graded encoding oracle models in a secure way against computationally unbounded pol...

Contents Preface 5 Chapter 1. Probability, measure and integration 7 1.1. Probability spaces, measures and σ-algebras 7 1.2. Random variables and their distribution 18 1.3. Integration and the (mathematical) expectation 30 1.4. Independence and product measures 54 Chapter 2. Asymptotics: the law of large numbers 71 2.1. Weak laws of large numbers 71 2.2. The Borel-Cantelli lemmas 77 2.3. Strong...

In this paper, we will discuss the concept of almost sure convergence for specic groups of fuzzyrandom variables. For this purpose, we use the type of generalized Chebyshev inequalities.Moreover, we show the concept of almost sure convergence of weighted average pairwise NQDof fuzzy random variables.

In statistics as in life, many things become clearer when we consider context. Statisticians’ use of context itself becomes clearer, in fact, when we consider the past century. It was anathema to them prior to 1906, when Markov [22] proved the weak law of large numbers applies to chains of dependent events over finite domains (i.e., finite-state Markov processes). He published several papers on...

In a recent paper Salem and Zygmund [1] proved the following result : Put 2TCv a " = avn~ _ 2n (v = 0, 1,. . ., 2n)-{-1 and denote the 99,'(t) the v-th Rademacher function. Denote by L.(t, 0) the unique trigonometric polynomial (in 0) of degree not exceeding n for Denote Mjt) = max I L,, (t, 0)1. Then for almost all t 050<2a M,, t) lim BLOCKIN~ <_ 2. (log n)a n=~ P. ERDÖS I am going to prove th...

Some of the most important Italian contributions to probability and statistics came to light in the WW1 years 1915-1918: the Cantelli’s theory of stochastic convergence together with the first correct and (rather) general version of the strong law of large numbers, and the Gini theory of statistical association. These studies were influential in determining the characteristics of the subsequent...

The bootstrap Glivenko-Cantelli and bootstrap Donsker theorems of Giné and Zinn (1990) contain both necessary and sufficient conditions for the asymptotic validity of Efron’s nonparametric bootstrap. In the more general case of exchangeably weighted bootstraps, Praestgaard and Wellner (1993) and Van der Vaart and Wellner (1996) give analogues of the sufficiency half of the Theorems of Giné and ...