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01463nam a2200205Ia 4500 |
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10.1016-j.aim.2022.108398 |
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220425s2022 CNT 000 0 und d |
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|a 00018708 (ISSN)
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| 245 |
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|a A phase transition for tails of the free multiplicative convolution powers
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| 260 |
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|b Academic Press Inc.
|c 2022
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| 856 |
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|z View Fulltext in Publisher
|u https://doi.org/10.1016/j.aim.2022.108398
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|a We study the behavior of the tail of a measure μ⊠t, where ⊠t is the t-fold free multiplicative convolution power for t≥1. We focus on the case where μ is a probability measure on the positive half-line with a regularly varying right tail i.e. of the form x−αL(x), where L is slowly varying. We obtain a phase transition in the behavior of the right tail of μ⊞t between regimes α<1 and α>1. Our main tool is a description of regular variation of the tail of μ in terms of the behavior of the corresponding S-transform at 0−. We also describe the tails of ⊠ infinitely divisible measures in terms of the tails of corresponding Lévy measure, treat symmetric measures with regularly varying tails and prove the free analog of the Breiman lemma. © 2022 Elsevier Inc.
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|a Free multiplicative convolution
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|a Infinite divisibility
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|a Regular variation
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|a S-transform
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|a Tails
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|a Kołodziejek, B.
|e author
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|a Szpojankowski, K.
|e author
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| 773 |
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|t Advances in Mathematics
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