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A new notion of typicality for arbitrary probability measures on standard Borel spaces is proposed, which encompasses the classical notions of weak and strong typicality as special cases. Useful lemmas about strong typical sets, including…

The aim of this paper is to discuss various concentration inequalities for U-statistics and most recent results. A special focus will be on providing proofs for bounds on the U-statistics using classical concentration inequalities, which,…

In this monograph, we prove an asymptotic approximation for integrals of probability densities over sets in finite dimensional euclidean space, which are far away from the origin (asymptotic sets). We use this approximation to investigate…

Asymptotic equivalence theory developed in the literature so far are only for bounded loss functions. This limits the potential applications of the theory because many commonly used loss functions in statistical inference are unbounded. In…

We introduce p-equivalence by asymptotic probabilities, which is a weak almost-equivalence based on zero-one laws in finite model theory. In this paper, we consider the computational complexities of p-equivalence problems for regular…

The problem is addressed of defining the values of functions, whose variables tend to infinity, from the knowledge of these functions at asymptotically small variables close to zero. For this purpose, the extrapolation by means of different…

In this article, we provide a comprehensive analysis of the asymptotic behavior of Bell numbers, enhancing and unifying various results previously dispersed in the literature. We establish several explicit lower and upper bounds. The main…

The conditional distribution of the next outcome given the infinite past of a stationary process can be inferred from finite but growing segments of the past. Several schemes are known for constructing pointwise consistent estimates, but…

Our goal is to finally settle the persistent problem in Diophantine Approximation of finding best linear approximates. Classical results from the theory of continued fractions provide the solution for the special homogeneous case in the…

We consider the asymptotic properties of Approximate Bayesian Computation (ABC) for the realistic case of summary statistics with heterogeneous rates of convergence. We allow some statistics to converge faster than the ABC tolerance, other…

We prove the claim in the title under mild conditions which are usually satisfied when trying to establish asymptotic normality. We assume strictly stationary and absolutely regular data.

Weak-type quasi-norms are defined using the mean oscillation or the mean of a function on dyadic cubes, providing discrete analogues and variants of the corresponding quasi-norms on the upper half-space previously considered in the…

We consider the phenomenon of classicalization in nonlinear sigma models with both positive and negative target space curvature and with any number of derivatives. We find that the theories with only two derivatives exhibit a weak form of…

A large literature has grown up around the proposed use of 'weak measurements' (i.e., unsharp measurements followed by post-selection) to allegedly provide information about hidden ontological features of quantum systems. This paper…

The aim of this work is to expose some asymptotic series associated to some expressions involving the volume of the n-dimensional unit ball. All proofs and the methods used for improving the classical inequalities announced in the final…

Classical probability theory supports probability measures, assigning a fixed positive real value to each event, these measures are far from satisfactory in formulating real-life occurrences. The main innovation of this paper is the…

The purpose of this article is to present a general method to find limiting laws for some renormalized statistics on random permutations. The model considered here is Ewens sampling model, which generalizes uniform random permutations. We…

The aim of this article is to establish asymptotic distributions and consistency of subsampling for spectral density and for magnitude of coherence for non-stationary, almost periodically correlated time series. We show the asymptotic…

We establish common fixed point theorems for two pairs of weakly compatible self-mappings using an auxiliary function of two variables. Unlike classical results, our theorems do not assume continuity of the mappings and require completeness…

A family of random matrices is said to converge strongly to a limiting family of operators if the operator norm of every noncommutative polynomial of the matrices converges to that of the limiting operators. Recent developments surrounding…