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The paper deals with a test procedure able to state the compatibility of observed data with a reference model, by using an estimate of the volumetric part in the small-ball probability factorization which plays the role of a real complexity index. As a preliminary by-product we state some asymptotics for a new estimator of the complexity index. A suitable test statistic is derived and, referring to the U-statistics theory, its asymptotic null distribution is obtained. A study of level and power of the test for finite sample sizes and a comparison with a competitor are carried out by Monte Carlo simulations. The test procedure is performed over a financial time series.

Asymptotic factorizations for the small–ball probability (SmBP) of a Hilbert–valued random element X are established and discussed. In particular, given the first d principal components (PCs) and as the radius ε of the ball tends to zero, the SmBP is asymptotically proportional to (a) the joint density of the first d PCs, (b) the volume of the d–dimensional ball with radius ε, and (c) a correction factor weighting the use of a truncated version of the process expansion. Under suitable assumptions on the spectrum of the covariance operator of X and as d diverges to infinity when ε vanishes, some simplifications occur. In particular, the SmBP factorizes asymptotically as the product of the joint density of the first d PCs and a pure volume parameter. The factorizations allow one to define a surrogate intensity of the SmBP that, in some cases, leads to a genuine intensity. To operationalize the stated results, a non–parametric estimator for the surrogate intensity is introduced and it is proved that the use of estimated PCs, instead of the true ones, does not affect the rate of convergence. Finally, as an illustration, simulations in controlled frameworks are provided.

An unsupervised and a supervised classification approaches for Hilbert random curves are studied. Both rest on the use of a surrogate of the probability density which is defined, in a distribution-free mixture context, from an asymptotic factorization of the small-ball probability. That surrogate density is estimated by a kernel approach from the principal components of the data. The focus is on the illustration of the classification algorithms and the computational implications, with particular attention to the tuning of the parameters involved. Some asymptotic results are sketched. Applications on simulated and real datasets show how the proposed methods work.

Let \\(X\\) be a fuzzy set--valued random variable (\\frv{}), and \\(\\huku{X}\\) the family of all fuzzy sets \\(B\\) for which the Hukuhara difference \\(X\\HukuDiff B\\) exists \\(\\mathbb{P}\\)--almost surely. In this paper, we prove that \\(X\\) can be decomposed as \\(X(\\omega)=C\\Mink Y(\\omega)\\) for \\(\\mathbb{P}\\)--almost every \\(\\omega\\in\\Omega\\), \\(C\\) is the unique deterministic fuzzy set that minimizes \\(\\mathbb{E}[d_2(X,B)^2]\\) as \\(B\\) is varying in \\(\\huku{X}\\), and \\(Y\\) is a centered \\frv{} (i.e. its generalized Steiner point is the origin). This decomposition allows us to characterize all \\frv{} translation (i.e. \\(X(\\omega) = M \\Mink \\indicator{\\xi(\\omega)}\\) for some deterministic fuzzy convex set \\(M\\) and some random element in \\(\\Banach\\)). In particular, \\(X\\) is an \\frv{} translation if and only if the Aumann expectation \\(\\mathbb{E}X\\) is equal to \\(C\\) up to a translation. Examples, such as the Gaussian case, are provided.

The paper considers a particular family of set--valued stochastic processes modeling birth--and--growth processes. The proposed setting allows us to investigate the nucleation and the growth processes. A decomposition theorem is established to characterize the nucleation and the growth. As a consequence, different consistent set--valued estimators are studied for growth process. Moreover, the nucleation process is studied via the hitting function, and a consistent estimator of the nucleation hitting function is derived.

We propose a set-valued framework for the well-posedness of birth-and-growth process. Our birth-and-growth model is rigorously defined as a suitable combination, involving Minkowski sum and Aumann integral, of two very general set-valued processes representing nucleation and growth respectively. The simplicity of the used geometrical approach leads us to avoid problems arising by an analytical definition of the front growth such as boundary regularities. In this framework, growth is generally anisotropic and, according to a mesoscale point of view, it is not local, i.e. for a fixed time instant, growth is the same at each space point.