An Embedded Model Estimator for Non-Stationary Random Functions Using Multiple Secondary Variables

An algorithm for non-stationary spatial modelling using multiple secondary variables is developed herein, which combines geostatistics with quantile random forests to provide a new interpolation and stochastic simulation. This paper introduces the method and shows that its results are consistent and similar in nature to those applying to geostatistical modelling and to quantile random forests. The method allows for embedding of simpler interpolation techniques, such as kriging, to further condition the model. The algorithm works by estimating a conditional distribution for the target variable at each target location. The family of such distributions is called the envelope of the target variable. From this, it is possible to obtain spatial estimates, quantiles and uncertainty. An algorithm is also developed to produce conditional simulations from the envelope. As they sample from the envelope, realizations are therefore locally influenced by relative changes of importance of secondary variables, trends and variability.

The space $D$ in several variables: random variables and higher moments

We study the Banach space $D([0,1]^m)$ of functions of several variables that are (in a certain sense) right-continuous with left limits, and extend several results previously known for the standard case $m=1$. We give, for example, a description of the dual space, and we show that a bounded multilinear form always is measurable with respect to the $\sigma$-field generated by the point evaluations. These results are used to study random functions in the space. (I.e., random elements of the space.) In particular, we give results on existence of moments (in different senses) of such random functions, and we give an application to the Zolotarev distance between two such random functions.

RECONSTRUCTION OF GAUSSIAN RANDOM FUNCTIONS FROM SPECTRUM DATA

It is known that a stationary random process is represented as a superposition of harmonic oscillations with real frequencies and uncorrelated amplitudes. In the study of nonstationary processes, it is natural to have increasing or declining oscillationсs. This raises the problem of constructing algorithms that would allow constructing broad classes of nonstationary processes from elementary nonstationary random processes. A natural generalization of the concept of the spectrum of a nonstationary random process is the transition from the real spectrum in the case of stationary to a complex or infinite multiple spectrum in the nonstationary case. There is also the problem of describing within the correlation theory of random processes in which the spectrum has no analogues in the case of stationary random processes, namely, the spectrum point is real, but it has infinite multiplicity for the operator image of the corresponding operator, and when the spectrum itself is complex. Reconstruction of the complex spectrum of a nonstationary random function is a very important problem in both theoretical and applied aspects. In the paper the procedure of reconstruction of random process, sequence, field from a spectrum for Gaussian random functions is developed. Compared to the stationary case, there are wider possibilities, for example, the construction of a nonstationary random process with a real spectrum, which has infinite multiplicity and which can be distributed over the entire finite segment of the real axis. The presence of such a spectrum leads, in contrast to the case of a stationary random process, to the appearance of new components in the spectral decomposition of random functions that correspond to the internal states of «strings», i.e. generated by solutions of systems of equations in partial derivatives of hyperbolic type. The paper deals with various cases of the spectrum of a non-self-adjoint operator , namely, the case of a discrete spectrum and the case of a continuous spectrum, which is located on a finite segment of the real axis, which is the range of values of the real non-decreasing function a(x). The cases a(x)=0, a(x)=a0, a(x)=x and a(x) is a piecewise constant function are studied. The authors consider the recovery of nonstationary sequences for different cases of the spectrum of a non-self-adjoint operator promising since spectral decompositions are a superposition of discrete or continuous internal states of oscillators with complex frequencies and uncorrelated amplitudes and therefore have deep physical meaning.

Expectations

The expectation is defined, applying integration concepts to probability measures. Leading examples are given and the different characterizations of the expectation of a function are compared. The Markov and Jensen inequalities are given and consideration of multivariate distributions then leads to the treatment of the Cauchy–Schwarz, Hölder, Liapunov, Minkowski, and Loève inequalities. The final section treats the calculus of random functions of a real variable.

How to Construct Quantum Random Functions

Pseudorandom functions ( PRFs ) are one of the foundational concepts in theoretical computer science, with numerous applications in complexity theory and cryptography. In this work, we study the security of PRFs when evaluated on quantum superpositions of inputs. The classical techniques for arguing the security of PRFs do not carry over to this setting, even if the underlying building blocks are quantum resistant. We therefore develop a new proof technique to show that many of the classical PRF constructions remain secure when evaluated on superpositions.

Long range dependence of heavy-tailed random functions

We introduce a definition of long range dependence of random processes and fields on an (unbounded) index space $T\subseteq \mathbb{R}^d$ in terms of integrability of the covariance of indicators that a random function exceeds any given level. This definition is specifically designed to cover the case of random functions with infinite variance. We show the value of this new definition and its connection to limit theorems via some examples including subordinated Gaussian as well as random volatility fields and time series.