双边规范级联：Wiener极限和变分数布朗极限的乘法精化途径
Bilateral Canonical Cascades: Multiplicative Refinement Paths to Wiener's and Variant Fractional Brownian Limits
摘要：
对于$t\in (0,1)$，原始密度为1，$b$为整数基数（$b\geq 2$ %），而$p\in (0,1)$为参数。第一构造阶段将单位间隔划分为$b$子间隔，并将每个子间隔中的密度与$\frac{1% }{2}+\frac{p}{2}$和${1/2}-\frac{p}{2}$的相应频率相乘1或-1。结果表明，由此产生的密度可以重整化，使$n\to \infty $（$n$是迭代次数）的签署措施在某种意义上收敛到一个非退化的限制。如果$H=1+\log_{b}$ $p>{1}/{2}$，因此$p>b^{{-1}/{% 2}}$，重整化产生一个鞅，收敛是强的，并且极限共享指数为$H$的分数布朗运动的Hölder和Hausdorff性质。如果$H\leq {1}/{2}$，因此$p\leq b^{{-1}/{2}%}$，则该鞅不收敛。然而，可以应用不同的归一化，对于$H\leq {1/2}$，可以应用于鞅本身，对于$H>% {1/2}$，可以应用于极限和有限近似之间的差异。在所有情况下，所得到的过程被发现弱收敛到维纳布朗运动，独立的$H$和$b$。因此，到通常的加性路径维纳测量，这个过程增加了一个无限的乘法路径。
英文摘要：
The original density is 1 for $t\in (0,1)$, $b$ is an integer base ($b\geq 2$%), and $p\in (0,1)$ is a parameter. The first construction stage divides the unit interval into $b$ subintervals and multiplies the density in each subinterval by either 1 or -1 with the respective frequencies of $\frac{1% }{2}+\frac{p}{2}$ and ${1/2}-\frac{p}{2}$. It is shown that the resulting density can be renormalized so that, as $n\to \infty $ ($n$ being the number of iterations) the signed measure converges in some sense to a non-degenerate limit. If $H=1+\log_{b}$ $p>{1}/{2}$, hence $p>b^{{-1}/{% 2}}$, renormalization creates a martingale, the convergence is strong, and the limit shares the Hölder and Hausdorff properties of the fractional Brownian motion of exponent $H$. If $H\leq {1}/{2}$, hence $p\leq b^{{-1}/{2}%}$, this martingale does not converge. However, a different normalization can be applied, for $H\leq {1/2}$ to the martingale itself and for $H>% {1/2}$ to the discrepancy between the limit and a finite approximation. In all cases the resulting process is found to converge weakly to the Wiener Brownian motion, independently of $H$ and of $b$. Thus, to the usual additive paths toward Wiener measure, this procedure adds an infinity of multiplicative paths.
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