On the Law of Free Subordinators
On the Law of Free Subordinators
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论自由下属法
DOI：
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发表时间：
2011
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通讯作者：
Noriyoshi Sakuma
中科院分区：
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作者：
Octavio Arizmendi;Takahiro Hasebe;Noriyoshi Sakuma
我们研究了表现为自由从属律的自由无限可分分布。这是在[0，inty)上支持的经典无限可分分布的自由模拟，称为自由正则测度。证明了自由正则测度类在自由乘法卷积、$0leq leq 1$的t次布尔幂、$tgeq 1$的t次自由乘法幂和弱收敛下是封闭的。此外，我们证明了一个对称分布是自由无限可分的当且仅当它的平方可以表示为自由泊松和自由正则测度的自由乘法卷积。这给出了两个新的关于经典卷积和自由卷积的可无限整除的分布的显式例子：chi^2(1)和F（1,1）。另一个结果是自由对易子运算保留了自由无限可除性。
We study the freely infinitely divisible distributions that appear as the laws of free subordinators. This is the free analog of classically infinitely divisible distributions supported on [0,infty), called the free regular measures. We prove that the class of free regular measures is closed under the free multiplicative convolution, t-th boolean power for $0leq tleq 1$, t-th free multiplicative power for $tgeq 1$ and weak convergence. In addition, we show that a symmetric distribution is freely infinitely divisible if and only if its square can be represented as the free multiplicative convolution of a free Poisson and a free regular measure. This gives two new explicit examples of distributions which are infinitely divisible with respect to both classical and free convolutions: chi^2(1) and F(1,1). Another consequence is that the free commutator operation preserves free infinite divisibility.