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References [1] Barbosa , E. G. , Dorea , C. C. Y. 2009 A note on the Lindeberg condition for convergence to stable laws in Mallows distance Bernoulli 15 922 – 924 10

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Abstract  

For the derivativesp (k)(x; α, γ) of the stable density of index α asymptotic formulae (of Plancherel Rotach type) are computed ask→∞ thereby exhibiting the detailed analytic structure for large orders of derivatives. Generalizing known results for the special case of the one-sided stable laws (O<α<1, γ=-α) the whole range for the index of stability and the asymmetry parameter γ is covered.

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Summary An integral analogue of the general almost sure limit theorem is presented. In the theorem, instead of a sequence of random elements, a continuous time random process is involved, moreover, instead of the logarithmical average, the integral of delta-measures is considered. Then the general theorem is applied to obtain almost sure versions of limit theorems for semistable and max-semistable processes, moreover for processes being in the domain of attraction of a stable law or being in the domain of geometric partial attraction of a semistable or a max-semistable law.

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We find a universal norming sequence in strong limit theorems for increments of sums of i.i.d. random variables with finite first moments and finite second moments of positive parts. Under various one-sided moment conditions our universal theorems imply the following results for sums and their increments: the strong law of large numbers, the law of the iterated logarithm, the Erdős-Rényi law of large numbers, the Shepp law, one-sided versions of the Csörgő-Révész strong approximation laws. We derive new results for random variables from domains of attraction of a normal law and asymmetric stable laws with index αЄ(1,2).

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Let X 1, X 2,… be independent, but not necessarily identically distributed random variables in the domain of attraction of a stable law with index 0<a<2. This paper uses M n=max 1 ? i ? n|X i| to establish a self-normalized law of the iterated logarithm (LIL) for partial sums. Similarly self-normalized increments of partial sums are studied as well. In particular, the results of self-normalized sums of Horváth and Shao[9]under independent and identically distributed random variables are extended and complemented. As applications, some corresponding results for self-normalized weighted sums of iid random variables are also concluded.

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Belkin, B. , A limit theorem for conditioned recurrent random walk attracted to a stable law, The Annals of Mathematical Statistics , 41 , No. 1, 146–163, 1970

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. I 1925 Simon, T. , Multiplicative strong unimodality for positive stable laws, Proc. Amer. Math. Soc

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Abstract  

A decisive paradigmatic change in the poetry of Lrinc Szab in 1927–1928 seems to be synchronous with a comparable alteration in world literature: a shift from classical modernity to hermetism, abstract objectivism, consciousness lyric and a new subjectivism, trends which anticipated present day styles and have survived in neoavant-gardist and poststructuralist poetics. By this move his poetic practice transcended the phase represented by Stefan George, Rainer Maria Rilke, or Mihly Babits and paralleled poetic events then unseen or unforeseen by Lrinc Szab. These included transmutations in the work of contemporary poets, Gottfried Benn, T. S. Eliot, Ezra Pound or Attila Jzsef. The novelty was not thematic but modal: a new discourse, a change from the homogeneous to the dialogical conception of the poem. In Lrinc Szab's vision of personality the voice of the actorincludes the voice of the rebellious individualof traditional poetry (with a sociological view, a pragmatical-pedagogical motivation and a psychological quest for truth) counterpoised by the voice of the spectatorexhibiting the stable laws of the universe (passing a logical judgment). It was a radical novelty in his poetry that the distinction and distance of the ego and the world ceased to be a precondition of aesthetic formation and were replaced by the polyphonic utterance of a You and the World, which living in, and at the same time distanced from, the ego enter into a dialogue in their poetic quiddity. The idea of “one justice” informing world history gave way to a demand for the “one's justice.” Lrinc Szab's sense of contradiction between the ego and history was reinforced by his study of Der Einzige und sein Eigentum by Max Stirner. In the light of this work he reconsidered the ideas of religion, fatherland, freedom, love, justice, and liberalism and came to confront all of the bonds of the individual personality and collective action. The poetic adaptation of Stirner's ideas presents a peculiar case of intertextuality: it encourages the poem's train of thought, tunes its mode of discourse, integrates textually in the course and rhetoric of creation, and occasionally informs the poems in their entirety (as a rule those of the highest poetic rank).

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Rimbert N., Sero-Guillaume O. Log-stable laws as asymptotic solutions to a fragmentation equation: application to the distribution of droplets in a high Weber-number spray, Physical Review E , Vol. 69, 2004, paper 056316

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