Bernoulli随机性与有偏正态性
Bernoulli Randomness and Biased Normality
摘要：
我们可以考虑$μ$ -马丁-洛夫随机性，对于$2^ω$上的概率测度$μ$，例如给定$p \in (0, 1)$的伯努利测度$μ_p$。研究了$n^ω$中参数为$p_0, p_1, \dotsc, p_{n-1}$的序列的Bernoulli随机性，并引入了一种有偏正态性.我们证明了每个Bernoulli随机实在有偏意义下是正规的，这就有了在$n^ω$中有偏正规实的集合具有满Bernoulli测度的推论.给出了由正规序列计算有偏正规序列的一个算法，从而给出了有偏正规实数的显式例子。我们研究了随机性在迭代函数系统中的应用。最后，我们列出了一些有关伯努利随机性和有偏正态性的进一步问题。
英文摘要：
One can consider $μ$-Martin-Löf randomness for a probability measure $μ$ on $2^ω$, such as the Bernoulli measure $μ_p$ given $p \in (0, 1)$. We study Bernoulli randomness of sequences in $n^ω$ with parameters $p_0, p_1, \dotsc, p_{n-1}$, and we introduce a biased version of normality. We prove that every Bernoulli random real is normal in the biased sense, and this has the corollary that the set of biased normal reals has full Bernoulli measure in $n^ω$. We give an algorithm for computing biased normal sequences from normal sequences, so that we can give explicit examples of biased normal reals. We investigate an application of randomness to iterated function systems. Finally, we list a few further questions relating to Bernoulli randomness and biased normality.
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