1 Cantor Distribution
Let ; for instance, we could take as in the classical case. Let . Consider a mapping [5]
from the set of finite bitstrings () to the nonnegative reals. The bitstrings in of length are assumed to be equiprobable. Consider the generating function [1]
where denotes the length of the bitstring. Clearly
The quantity
is the Cantor moment for strings of length ; let denote the limit of this as . Denote the empty string by . From values
and employing the recurrence [6]
we have
for ; thus
Dividing both sides by , we have [5, 7, 8, 9]
because
and the singularity of is a simple pole. In particular, when ,
and, up to small periodic fluctuations [9, 10, 11],
as .
We merely mention a problem involving order statistics. Let denote the expected value of the minimum of
independent Cantordistributed random variables. It is known that
[12]in general. In the special case , it follows that
and, up to small periodic fluctuations [13],
as . If denotes the expected value of the maximum of variables, then
by symmetry.
2 Cantorsolus Distribution
We examine here the set of finite solus bitstrings (). Let
denote the Fibonacci numbers. The bitstrings in of length are assumed to be equiprobable. Clearly
From additional values
and employing the recurrence [6]
we have
for ; thus
The purpose of using multinomial coefficients here, rather than binomial coefficients as in Section 1, is simply to establish precedent for Section 3. Let be the Golden mean. Dividing both sides by , we have [1]
because
and the singularity of is a simple pole. In particular, when ,
and, up to small periodic fluctuations,
as . An integral formula in [1] for the preceding numerical coefficient involves a generating function of exponential type:
namely
(we believe that the fifth decimal given in [1] is incorrect, perhaps a typo). Unlike the formula for earlier, this expression depends on the sequence , , , … explicitly.
With regard to order statistics, it is known that [16]
in general. In the special case , we have, up to small periodic fluctuations,
as .
3 Cantormultus Distribution
We examine here the set of finite multus bitstrings (). Let
denote the second upper Fibonacci numbers [17]. The bitstrings in of length are assumed to be equiprobable. Clearly
From additional values
and employing the recurrence
we have
for ; thus
Let
be the second upper Golden mean [17, 18]. Dividing both sides by , we have
because
and the singularity of is a simple pole. In particular, when ,
but no asymptotics for are known. Order statistics likewise remain open.
4 Bitsums
We turn to a more fundamental topic: given a set of finite bitstrings, what can be said about the bitsum of a random of length ? If is unconstrained, i.e., if all strings are included in the sample, then
because a sum of independent Bernoulli() variables is Binomial(,). Expressed differently, the average density of s in a random unconstrained string is
, with a corresponding variance
.Let us impose constraints. If consists of solus bitstrings, then the total bitsum of all of length has generating function [19, 20]
and the total bitsum squared has generating function
hence has generating function
where is as in Section 2. Standard techniques [6] give asymptotics
for the average density of s in a random solus string and corresponding variance.
If instead consists of multus bitstrings, then the total bitsum of all of length has generating function [21]
and the total bitsum squared has generating function
hence has generating function
where is as in Section 3. We obtain asymptotics
for the average density of s in a random multus string and corresponding variance. Unsurprisingly and ; a clumping of s forces a higher density than a separating of s.
A famous example of an infinite aperiodic solus bitstring is the Fibonacci word [2, 3], which is the limit obtained recursively starting with and satisfying substitution rules , . The density of s in this word is [22], which exceeds the average but falls well within the onesigma upper limit . We wonder if an analogously simple construction might give an infinite aperiodic multus bitstring with known density.
5 Longest Bitruns
We turn to a different topic: given a set of finite bitstrings, what can be said about the duration of the longest run of s in a random of length ? If is unconstrained, then [6]
the Taylor expansion of the numerator series is [23]
and, up to small periodic fluctuations [24, 25],
as . Of course, identical results hold for , the duration of the longest run of s in .
If consists of solus bitstrings, then it makes little sense to talk about runs. For runs, over all , we have
and the Taylor expansion of the numerator series is [23]
where is as in Section 2.
If instead consists of multus bitstrings, then we can talk both about runs [23]:
and runs:
where is as in Section 3. Proof: the number of multus bitstrings with no runs of s has generating function [26]
we conclude by use of the summation identity
Study of runs of s proceeds analogously [27]. The solus and multus results here are new, as far as is known. Asymptotics would be good to see someday.
6 Acknowledgements
I am thankful to Alois Heinz for helpful discussions and for providing the generating function associated with via the Maple gfun package; R and Mathematica have been useful throughout. I am also indebted to a friend, who wishes to remain anonymous, for giving encouragement and support (in these dark days of the novel coronavirus outbreak).
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Steven Finch MIT Sloan School of Management Cambridge, MA, USA steven_finch@harvard.edu