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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 $$\alpha +\beta =1$$).

##### MSC:
 60J25 Continuous-time Markov processes on general state spaces 60F15 Strong limit theorems
##### Keywords:
strong limit theorems
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##### References:
  Aldous, D.J., Deterministic and stochastic models for coalescence (aggregation, coagulation)a review of the Mean-field theory for probabilists, Bernoulli, 5, 3-48, (1999) · Zbl 0930.60096  Aldous, D.J.; Pitman, J., The standard additive coalescent, Ann. probab., 26, 1703-1726, (1998) · Zbl 0936.60064  Berestycki, J., 2002. Ranked fragmentations. ESAIM, Probabilités et Statitistique 6, 157-176. Available via  Bertoin, J., A fragmentation process connected to Brownian motion, Probab. theory related fields, 117, 289-301, (2000) · Zbl 0965.60072  Bertoin, J., Homogeneous fragmentation processes, Probab. theory related fields, 121, 301-318, (2001) · Zbl 0992.60076  Bertoin, J., Self-similar fragmentations, Ann. inst. Henri Poincaré, 38, 319-340, (2002) · Zbl 1002.60072  Beysens, D., Campi, X., Pefferkorn, E. (Eds.), 1995. Fragmentation Phenomena. World Scientific, Singapore.  Bingham, N.H.; Goldie, C.M.; Teugels, J.L., Regular variation, (1987), Cambridge University Press Cambridge  Chassaing, Ph.; Louchard, G., Phase transition for parking blocks, Brownian excursions and coalescence, Random struct. algorithms, 21, 76-119, (2002) · Zbl 1032.60003  Haas, B., Loss of mass in deterministic and random fragmentations, Stochastic process. appl., 106, 245-277, (2003) · Zbl 1075.60553  Haas, B. Regularity of formation of dust in self-similar fragmentations. Preprint. Available via · Zbl 1041.60058  Jacod, J.; Shiryaev, A.N., Limit theorems for stochastic processes, (1987), Springer Berlin · Zbl 0635.60021  Miermont, G. Self-similar fragmentations derived from the stable tree I: splitting at heights. Probab. Theory Related Fields (to appear). · Zbl 1042.60043  Miermont, G. Self-similar fragmentations derived from the stable tree II: splitting at hubs. Preprint. Available via · Zbl 1071.60065  Miermont, G., Schweinsberg, J., 2003. Self-similar fragmentations and stable subordinators. Séminaire de Probabilités XXXVII, to appear. Available via · Zbl 1038.60073  Pitman, J., 2002. Combinatorial stochastic processes. Lecture notes for the St Flour summer school, to appear. Available via  Schweinsberg, J., Applications of the continuous-time ballot theorem to Brownian motion and related processes, Stochastic process. appl., 95, 151-176, (2001) · Zbl 1060.60046
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