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On the growth rate of random Fibonacci sequences

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On the growth rate of random Fibonacci sequences. Beno t Rittaud (Universit Paris-13) with lise Janvresse & Thierry de la Rue (CNRS - Universit de Rouen) ... – PowerPoint PPT presentation

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Title: On the growth rate of random Fibonacci sequences


1
On the growth rate of random Fibonacci sequences
  • Benoît Rittaud (Université Paris-13)
  • with Élise Janvresse Thierry de la Rue (CNRS -
    Université de Rouen)

2
Random Fibonacci sequences
3
Random Fibonacci sequences
  • Classical Fibonacci sequence
  • Fn Fn1Fn2
  • (F0 a, F1 b)

4
Random Fibonacci sequences
  • Classical Fibonacci sequence
  • Fn Fn1Fn2
  • (F0 a, F1 b)
  • Main property Fn1/Fn goes to ? (1v5)/2, so

5
Random Fibonacci sequences
  • Classical Fibonacci sequence
  • Fn Fn1Fn2
  • (F0 a, F1 b)
  • Main property Fn1/Fn goes to ? (1v5)/2, so
  • (Fn)1/n ? ?

6
Random Fibonacci sequences
  • Random Fibonacci sequence
  • Fn Fn1?Fn2
  • (F0 a, F1 b)

7
Random Fibonacci sequences
  • Random Fibonacci sequence
  • Fn Fn1?Fn2
  • (F0 a, F1 b)
  • where the ? sign is obtained by tossing a
  • (balanced) coin for each n.

8
Random Fibonacci sequences
  • Random Fibonacci sequence
  • Fn Fn1?Fn2
  • (F0 a, F1 b)
  • where the ? sign is obtained by tossing a
  • (balanced) coin for each n.
  • Question what about the growth rate?

9
Random Fibonacci sequences
  • Two points of view

10
Random Fibonacci sequences
  • Two points of view
  • average growth rate E(Fn)1/n ? ?

11
Random Fibonacci sequences
  • Two points of view
  • average growth rate E(Fn)1/n ? ?
  • almost-sure growth rate E(Fn1/n)? ?

12
Random Fibonacci tree
13
Random Fibonacci tree
14
Random Fibonacci tree
15
Random Fibonacci tree
16
Random Fibonacci tree
17
Random Fibonacci tree
18
Random Fibonacci tree
  • Theorem 1
  • This tree (denote it by R) has the structure of
    the Fibonacci tree.

19
Random Fibonacci tree
  • Theorem 2
  • All pairs (a, b) of mutually primes integers
    appears exactly once in R with a as parent of b.

20
Random Fibonacci tree
  • Theorem 3
  • The walk from (1, 1) to
  • (a, b) in R can be obtained by considering the
    continued fraction expansion of a/b.

21
The average point of view
22
The average point of view
23
The average point of view
Sn 2Sn1Sn3
24
The average point of view
  • Since Sn 2Sn1Sn3, the growth rate in R is
    known.

25
The average point of view
  • Since Sn 2Sn1Sn3, the growth rate in R is
    known.
  • Consider the full tree as a union of (infinite)
    copies of R to evaluate the average growth rate
    of a random Fibonacci sequence.

26
The average point of view
  • Since Sn 2Sn1Sn3, the growth rate in R is
    known.
  • Consider the full tree as a union of (infinite)
    copies of R to evaluate the average growth rate
    of a random Fibonacci sequence.
  • The result is ?1 1.20556943, where
  • ?3 2?21 (? gt 1).

27
The almost-sure point of view
28
The almost-sure point of view
  • Divakar Viswanath (1999)
  • E(Fn1/n) ? 1.13198824

29
The almost-sure point of view
  • Divakar Viswanath (2000)
  • E(Fn1/n) ? ? 1.13198824
  • obtained by the computation of a (quite
  • tiresome) integral given by a formula of
  • Furstenberg.

30
The almost-sure point of view
  • A simplification

31
The almost-sure point of view
  • A simplification
  • where ?? is an explicit measure defined
  • inductively on Stern-Brocot intervals.

32
Generalizations
33
Generalizations
  • Unbalanced coin ( with probability p)

34
Generalizations
  • Unbalanced coin ( with probability p)
  • OK for both points of view.

35
Generalizations
  • Unbalanced coin ( with probability p)
  • OK for both points of view.
  • Crititical p
  • 1/3 for almost-sure, 1/4 for average

36
Generalizations
  • Unbalanced coin ( with probability p)
  • Linear case Fn Fn1 ?Fn2

37
Generalizations
  • Unbalanced coin ( with probability p)
  • Linear case Fn Fn1 ?Fn2
  • OK again (but irrelevant for average).

38
Generalizations
  • Unbalanced coin ( with probability p)
  • Linear case Fn Fn1 ?Fn2
  • ?-sequences Fn ?Fn1 ?Fn2

39
Generalizations
  • Unbalanced coin ( with probability p)
  • Linear case Fn Fn1 ?Fn2
  • ?-sequences Fn ?Fn1 ?Fn2
  • maybe hard in general, but

40
Generalizations
  • Unbalanced coin ( with probability p)
  • Linear case Fn Fn1 ?Fn2
  • ?-sequences Fn ?Fn1 ?Fn2
  • maybe hard in general, but
  • lets try with ? v2

41
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42
Generalizations
  • Unbalanced coin ( with probability p)
  • Linear case Fn Fn1 ?Fn2
  • ?-sequences Fn ?Fn1 ?Fn2
  • more generally,
  • the method works for any ? of the form
  • ? 2cos(p/k) (k integer)

43
Generalizations
  • Unbalanced coin ( with probability p)
  • Linear case Fn Fn1 ?Fn2
  • ?-sequences Fn ?Fn1 ?Fn2
  • more generally,
  • the method works for any ? of the form
  • 2cos(p/k) (k integer)
  • These are Rosen continued fraction.
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