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Multidimensional continued fractions and stable laws. (Fractions continues multidimensionnelles et lois stables.) (French) Zbl 0857.11035
The Jacobi-Perron algorithm is a multidimensional version of the classical continued fraction expansion. Starting from an $$x\in [0,1]^d$$ one computes $$\overline x = (x_2/x_1, x_3/x_1, \dots, 1/x_1)$$, and $$a(x) = [\overline x]$$, $$Tx = \{\overline x\}$$, where $$[ ]$$ and $$\{ \}$$ denote the integer and fractional part, respectively. The first approximant of $$x$$ is then defined as $$(1/a_d,a_1/a_d, \dots, a_{d-1}/a_d)$$, and the rest is $$Tx$$. One has $$x=(1/(a_d + (Tx)_d)$$, $$(a_1+(Tx)_1)/(a_d+(Tx)_d), \dots, (a_{d-1}+(Tx)_{d-1})/(a_d+(Tx)_d))$$. Continuing this procedure for all $$T^kx$$ as in the one dimensional case, one gets simultaneous rational approximations of the numbers $$x_1,\dots,x_d$$ of the form $$r_n=(p^1_n/q_n, \dots, p^d_n/q_n)$$. This algorithm relies on the iteration of the transformation $$T$$. It is well known (and goes back essentially to Gauss) that in the one dimensional case the measure $$T^n \lambda$$, where $$\lambda$$ is the Lebesgue measure on $$[0,1]$$, converges towards the measure $${1 \over \log 2} {dx \over 1+x}$$ which is the unique probability measure invariant under the transformation $$T$$, and absolutely continuous with respect to Lebesgue measure. This fact is extended in the paper to the multidimensional Jacobi-Perron algorithm, the density being now a piecewise analytic function on $$[0,1]$$.
Next the author considers the problem of the size of the integer vectors $$a(T^nx)$$, which are the coefficients in the continued fraction expansion of $$x$$, as $$n\to\infty$$. It is shown that for all multi-indices $$\alpha=(\alpha_1, \dots, \alpha_d)$$ where the $$\alpha_j$$ are positive real numbers, and $$\overline \alpha=\sum_j \alpha_j>1/2$$, the random variables $$1/n^{\overline \alpha} \sum^{n-1}_{k=0} (-1)^k a(T^k x)^\alpha$$ (where $$x$$ is chosen with Lebesgue measure) converge in distribution towards a multidimensional symmetric stable distribution of index $$1/ \overline \alpha$$. This extends a result due to Paul Lévy in the one-dimensional case.
Reviewer: Ph.Biane (Paris)

##### MSC:
 11J70 Continued fractions and generalizations 37A99 Ergodic theory 60F05 Central limit and other weak theorems
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##### References:
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