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On small masses in self-similar fragmentations. (English) Zbl 1075.60092
A self-similar fragmentation is a Markov process in the space \(\mathcal S_1\) of decreasing sequences \(x_1\geq x_2\geq \dots \geq 0\) of total mass \(\sum x_i=1\). It is assumed that the fragmentation starts from the initial configuration \(X(0)=(1,0,0,\dots )\), the process \(X\) is self-similar with some index \(\alpha >0\), and distinct fragmentations have independent evolutions.
The law of the process \(X\) is determined by the self-similarity index \(\alpha \) and a so-called dislocation measure \(\nu \) on \(\mathcal S_1\) which describes the rates of dislocation of a unit mass into some sequence \(x=(x_1,x_2,\dots )\in \mathcal S_1\).
The aim of this paper is to describe the asymptotic behavior of random quantities \(N(\varepsilon ,t)\) and \(M(\varepsilon ,t)\) where \(N(\varepsilon ,t)\) is the number of those fragments whose mass is greater than \(\varepsilon >0\) at time \(t>0\), i.e. \(N(\varepsilon ,t)=\operatorname {card}\,\{i:X_i(t)>\varepsilon \}\), and \(M(\varepsilon ,t)\) is the total mass of all those fragments whose mass is smaller than \(\varepsilon \) at time \(t\), i.e. \(M(\varepsilon ,t)=\sum _{i=1}^\infty X_i(t)I_{\{X_i(t)<\varepsilon \}}\). Let us define functions \(\varphi (\varepsilon )=\int _{\mathcal S_1}(\operatorname {card}\,\{i:x_i>\varepsilon \}-1)\, d\nu \) and \(f(\varepsilon )=\int _{\mathcal S_1}\sum _{i=1}^\infty x_iI_{\{x_i<\varepsilon \}}\, d\nu \). By the main theorem of the paper, if \(0<\beta <1\), then \((† )\) \(\varphi \) is regularly varying as \(\varepsilon \to 0\) with index \(-\beta \) if and only if \((‡ )\) \(f\) is regularly varying as \(\varepsilon \to 0\) with index \(1-\beta \). Moreover, if \((† )\) or \((‡ )\) is true and there exists \(k\in \mathbb N\) obeying \(\nu \{x_k>0\}=0\), then \[ \lim _{\varepsilon \to 0}\frac {N(\varepsilon ,t)}{\varphi (\varepsilon )}=\lim _{\varepsilon \to 0}\frac {M(\varepsilon ,t)}{f(\varepsilon )}=\int _0^t\sum _{i=1}^\infty X_i^{\alpha +\beta }(s)\,ds \] almost surely, and the limit is positive and finite (and random except for the case \(\alpha +\beta =1\)).

60J25 Continuous-time Markov processes on general state spaces
60F15 Strong limit theorems
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