Abstract
In this paper, we prove a theorem that confirms, under a supplementary condition, a conjecture concerning random permutations of sequences of partitions of the unit interval.
1 Introduction
The general study of uniformly distributed sequences of partitions was initiated in [1], inspired by a beautiful construction and result from the study by Kakutani [2]. The subject is closely related to the theory of uniformly distributed sequences, initiated in [3]. There are two classical references for the subject: [4] and [5].
Kakutani took the interval
Kakutani proved that this sequence of partitions (denote it by
for every continuous function
In other words, the discrete measure concentrated in the points
The construction has been generalized in [1]. Let
In the first step, the longest interval(s) of
If
The following theorem includes the results of the study by Kakutani ([1], Theorem 2.7]).
Theorem 1
The sequence
There are interesting connections between the theory of u.d. sequences of partitions and u.d. sequences of points. This connection is far reaching in the construction of a significant subclass of
The present paper is concerned with a result in the domain of uniformly distributed sequences of partitions, related to a proposition by von Neumann for uniformly sequences of points [7].
Theorem 2
If
One of the consequences of von Neumann’s result is that there are many u.d. sequences of points.
We will now introduce the definitions we need.
Definitions
Given a partition
The diameter of
Given a sequence of partitions
If
We will denote by
2 Main results
In a previous paper [8], we proved the following result.
Proposition 3
If
In the same paper, we made the following conjecture.
Conjecture
If
We need some preliminary calculations.
Let
Consider a sequence
Select at random from the
Obviously,
hence,
It is easy to see that the second moment of
This, together with the independence of the
But this is not what we were looking for.
If we want to identify a class for which the conjecture is true, we have to make some assumptions on the sequence
A simple sufficient assumption is expressed as follows:
Theorem 4
If the series of squares of diameters of
Proof
We have
Apply now the Čebišëv inequality. By our assumption, we have, for every
Recalling that
almost surely for every
The set
Observe now that
for every
In other words, the empirical distribution function
On the other hand, convergence in distribution is known to be equivalent to weak convergence, so the desired conclusion follows.□
Acknowledgements
The author wishes to thank the referees for their useful comments.

Conflict of interest: The author states no conflict of interest.
References
[1] A. Volčič, A generalization of Kakutani’s splitting procedure, Ann. Mat. Pura Appl. 190 (2011), no. 1, 45–54, 10.1007/s1023101001363Search in Google Scholar
[2] S. Kakutani, A problem on equidistribution on the unit interval [0,1], Measure Theory (Proc. Conf., Oberwolfach, 1975), Lecture Notes in Mathematics, vol. 541, Springer, Berlin, 1976, pp. 369–375. 10.1007/BFb0081068Search in Google Scholar
[3] H. Weyl, Über ein Problem aus dem Gebiete der diophantischen Approximationen, Nach. Ges. Wiss. Göttingen, Math.Phys. Kl. 21 (1914), 234–244. Search in Google Scholar
[4] L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Pure and Applied Mathematics, WileyInterscience, New YorkLondonSidney, 1974. Search in Google Scholar
[5] M. Drmota and R. F. Tichy, Sequences, Discrepancies and Applications, Lecture Notes in Mathematics, vol. 1651, Springer Verlag, Berlin, 1997. 10.1007/BFb0093404Search in Google Scholar
[6] I. Carbone, Discrepancy of LSsequences of partitions and points, Ann. Mat. Pura Appl. 191 (2012), no. 4, 819–844. 10.1007/s102310110208zSearch in Google Scholar
[7] J. von Neumann, Gleichmässig dichte Zahlenfolgen, Mat. Fiz. Lapok 32 (1925), 32–40. Search in Google Scholar
[8] I. Carbone and A. Volčič, A von Neumann theorem for uniformly distributed sequences of partitions, Rend. Circ. Mat. Palermo 60 (2011), no. 1–2, 83–88. 10.1007/s122150110030xSearch in Google Scholar
[9] K. L. Chung, A Coure in Probability Theory, third edn, Academic Press, San Diego, 2001. Search in Google Scholar
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