# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] 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 [2] Aldous, D.J.; Pitman, J., The standard additive coalescent, Ann. probab., 26, 1703-1726, (1998) · Zbl 0936.60064 [3] Berestycki, J., 2002. Ranked fragmentations. ESAIM, Probabilités et Statitistique 6, 157-176. Available via [4] Bertoin, J., A fragmentation process connected to Brownian motion, Probab. theory related fields, 117, 289-301, (2000) · Zbl 0965.60072 [5] Bertoin, J., Homogeneous fragmentation processes, Probab. theory related fields, 121, 301-318, (2001) · Zbl 0992.60076 [6] Bertoin, J., Self-similar fragmentations, Ann. inst. Henri Poincaré, 38, 319-340, (2002) · Zbl 1002.60072 [7] Beysens, D., Campi, X., Pefferkorn, E. (Eds.), 1995. Fragmentation Phenomena. World Scientific, Singapore. [8] Bingham, N.H.; Goldie, C.M.; Teugels, J.L., Regular variation, (1987), Cambridge University Press Cambridge [9] Chassaing, Ph.; Louchard, G., Phase transition for parking blocks, Brownian excursions and coalescence, Random struct. algorithms, 21, 76-119, (2002) · Zbl 1032.60003 [10] Haas, B., Loss of mass in deterministic and random fragmentations, Stochastic process. appl., 106, 245-277, (2003) · Zbl 1075.60553 [11] Haas, B. Regularity of formation of dust in self-similar fragmentations. Preprint. Available via · Zbl 1041.60058 [12] Jacod, J.; Shiryaev, A.N., Limit theorems for stochastic processes, (1987), Springer Berlin · Zbl 0635.60021 [13] Miermont, G. Self-similar fragmentations derived from the stable tree I: splitting at heights. Probab. Theory Related Fields (to appear). · Zbl 1042.60043 [14] Miermont, G. Self-similar fragmentations derived from the stable tree II: splitting at hubs. Preprint. Available via · Zbl 1071.60065 [15] Miermont, G., Schweinsberg, J., 2003. Self-similar fragmentations and stable subordinators. Séminaire de Probabilités XXXVII, to appear. Available via · Zbl 1038.60073 [16] Pitman, J., 2002. Combinatorial stochastic processes. Lecture notes for the St Flour summer school, to appear. Available via [17] 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.