Download On moments of the Cantor and related distributions

Mathematica Aeterna, Vol. 5, 2015, no. 3, 493 - 502
On moments of the Cantor and related distributions
Pawe÷J. Szab÷
owski
Department of Mathematics and Information Sciences,
Warsaw University of Technology
ul. Koszykowa 75,
00-662 Warsaw, Poland
Abstract
We provide several simple recursive formulae for the moment sequence of
in…nite Bernoulli convolution. We relate moments of one in…nite Bernoulli convolution with others having di¤erent but related parameters. We give examples
relating Euler numbers to the moments of in…nite Bernoulli convolutions. One
of the examples provides moment interpretation of Pell numbers as well as new
identities satis…ed by Pell and Lucas numbers.
Mathematics Subject Classi…cation: Primary 11K38, 11B39, ; Secondary 60E10
Keywords: Moments of in…nite Bernoulli convolutions, moments of Cantor distribution, Silver ratio, Pell numbers, Lucas numbers.
1
Introduction
The aim of this note is to add a few simple observations to the analysis of
the distribution of the so called fatigue symmetric walk (term appearing in
[12]). These observations are based on the reformulation of known results
scattered through the literature. We however pay more attention to the moment sequences and less to the properties of distributions that produce these
moment sequences. It seems that the main novelty of the paper lies in the
probabilistic interpretation of Pell and Lucas numbers and easy proofs of some
identities satis…ed by these numbers. However in order to place these results
in the proper context we recall the de…nition and basic properties of in…nite
Bernoulli convolutions. In deriving properties of these convolutions we recall
some known, important results.
The paper is organized as follows. After recalling the de…nition and basic
facts about the fatigue random walks we concentrate on the moment sequences
of in…nite Bernoulli convolutions. We formulate a corollary of the results of
the paper expressed in terms of a moment sequence. This corollary formulated
494
Pawe÷J. Szab÷owski
in terms of number sequences provides identities of Pell and Lucas numbers of
even order (Remark 8).
2
In…nite Bernoulli Convolutions
Let fXn gn 1 be the sequence of i.i.d. random variables such that P (X1 = 1)
=
P P (X21 = 1) = 1=2: Further let fcn gn 1 be a sequence of reals such that
n 1 cn < 1: We de…ne random variable:
S=
X
cn X n :
n 1
By Kolmogorov Three-SeriesP
theorem S exists and moreover it is square in2
2
tegrable. We have ES =
n 1 cn . Obviously ES = 0: Let '(t) denote
characteristic
we have '(t) =
P function of S:
Q By the standard argument
Q
E exp(it
s 1 cn X n ) = E
n 1 exp(itcn Xn ) =
n 1 (exp(itcn )=2+exp( itcn )=2)
Q
= n 1 cos(tcn ):
We will concentrate on the special form of the sequence cn ; namely we will
assume that cn = n for some > 1: The reason for this is simple. There are
practically no results in the literature for other than cn = n sequences. The
problem of describing distributions appearing in ’fatigue random walk’is very
di¢ cult although simply formulated.
It is known (see [13]) that for all ; the distribution of S = S ( ) is continuous that is PS (fxg) = 0 for all x 2 R: Moreover it is also known (see
[16], [17]) that if for almostp 2 (1; 2] this distribution is absolutely continuous
and for almost all 2 (1; 2] it has square integrable density. Garsia in [7,
Theorem 1.8] showed examples of such leading to an absolutely continuous
distribution. Namely such 2 (1; 2) are the roots ofQmonic polynomials P
with integer coe¢ cients such that jP (0)j = 2 and
j i j>1 j i j = 2; where
f i g are the remaining roots of P .
There are known (see [4], [5]) countable many instances of 2 (1; 2] where
this distribution is singular. We will denote by ' the characteristic function
of S ( ) : Following [4] we know that the values such that ' (t) does not tend
to zero as t ! 1 are the so called Pisot or PV- numbers. Recall that those
are sole roots of such monic irreducible polynomials P with integer coe¢ cients
having the property that all other roots have absolute values less than 1.
Obviously we must then have P (0) = 1. For such 0 s the related distribution
is singular (by the Riemann–Lebesgue
are
p Lemma). Examples of such numbersp
the so called ’golden ratio’ 1 + 5 =2 or the so called ’silver ratio’1 + 2:
Moreover following [14] one knows that PV numbers are the only numbers
2 (1; 2] for which ' does not tend to zero. Of course singularity of the
distribution of S ( ) can occur for not being a PV number.
495
Cantor distribution
For > 2 it is known that the distribution of S is singular [10].
To simplify notation we will write supp X; where X is a random variable
meaning supp PX ; where PX denotes a distribution of X: Similarly X Y
denotes a random variable whose distribution is a convolution of distributions
of X and Y:
We have a simple Lemma.
1
Lemma 1 i) supp(S ( ))
; 11 :
1
In particular:
ia) if = 2 then S U ([ 1; 1]) and
ib) if = 3 then supp(S + 1=2) is equal to the Cantor set.
In general if is aPpositive integer then supp(S + 1=(
1)) consist of all
j
where rj 2 f0; 2g. Moreover the distribution
numbers of the form j 1 rj
of (S( ) + 1=(
1)) is ’uniform’on this set.
ii) 8k 1 :
k
Y1
k
' ( t) = ' (t)
cos j t :
(1)
j=0
Pk
i 1
Si ( k ); where Si ( ) ( i = 1; : : : ; k) are
iii) 8k
1 : S( )
i=1
i.i.d. random variables each having distribution S ( ) : Consequently ' (t) =
Q
k
j 1
t .
j=1 ' k
iv) Let us denote mn ( ) = ES( )n : Then 8n 1 : m2n 1 ( ) = 0 and
n 1
X
2n
(k)
m2j ( )W2(n
1 j=0 2j
1
m2n ( ) =
2kn
n Q
(1)
(k)
with m0 = 1, where Wn = 1; Wn ( ) = dtd n ( kj=1 cosh(
P
P
= 2k1 1 1i1 =0;:::;ik 1 =0 (1 + kj=1 (2ij 1) j )2n :
In particular we have:
m2n ( ) =
m2n ( ) =
1
2n
m2j ( )
j=0
);
j 1
t))
t=0
2n
;
2j
(2)
2(n j)
n 1
X 2(n j)
X
2n
m2j ( )
2l
1 j=0 2j
l=0
1
4n
v) 8k
1 : m2k ( )=
k th Euler number.
1
2k
1
Pk
1
j=0
2k
2j
2j
E2(k
j) m2j (
2l
:
(3)
); where Ek denotes
P
n
i) First of all notice that 1 1 =
; hence S + 1 1 =
n 1 1=
n (Xn + 1): Now since P (Xn + 1 = 0) = P (Xn + 1 = 2) = 1=2 we
Proof.
P
1
n 1
1
n 1
X
j) (
496
Pawe÷J. Szab÷owski
see that supp(S + 1 1 )
[0; 2 1 ]: Notice also that if = 2 then S + 1 =
P
2 n 1 21n Yn ; where P (Yn = 0) = P (Yn = 1) = 1=2: In other words (S + 1)=2
is a number chosen from [0; 1] with ’equal chances’that is (S +1)=2 has uniform
distribution on [0; 1]:
When = 3 we see that S + 1=2 is a number that can be written with
the help of ’0’ and ’2’ in ternary expansion. In other words S + 1=2 is number
drawn from the Cantor set with equal chances. For an integer we argue in
the similar way.
Qk 1
Q
cos j t .
ii) We have 'S ( k t) = n 1 cos( k t 1n ) = 'S (t) j=0
iii) Fix integer k. Notice that we have:
X
n
S( )=
Xn =
X
kj
Xkj +
j 1
n 1
X
kj+1
Xkj
1
+ ::: +
j 1
X
kj+k 1
Xkj
k+1
=
j 1
k
X
m 1
m=1
X
j
k
Xkj
m+1 :
j 1
Now since by assumption all Xi are i.i.d. we deduce that Si ( k ) are i.i.d.
random variables with distribution de…ned by ' k (t): Hence we have ' (t) =
Qk
j 1
t):
j=1 ' k (
iv) First of all we notice that 'S (t) is an even function hence all derivatives
of odd order at zero are equal to 0. Secondly let
(t) denote the moment
generating function of S( ): It is easy to notice that (s) = 'S( ) ( is): Let
(2n)
us denote m2n ( ) =
(0): Basing on the elementary formula
1
cosh( ) cosh( ) = (cosh( + ) + cosh (
2
));
we can easily obtain by induction the following identity:
k
Y1
j
cosh( t) =
j=0
Since (cosh t)(2n)
t=0
=
1
2k 1
2n
1
X
i1 =0;:::;ik
cosh(t(1 +
2k
1
i1 =0;:::;ik
!(2n)
cosh( j t)
cosh t 1 +
1 =0
j=1
1 =0
j=0
1
X
1) j )):
(2ij
, we have
k
Y1
1
k
X
k
X
j=1
(2ij
=
t=0
1)
j
!!!(2n)
(k)
= W2n ( ) :
t=0
497
Cantor distribution
Now using Leibnitz formula for di¤erentiation applied to (1) we get
f (2n) ( k t)
2kn
=
2n
k 1
X
2n Y
( cosh
j
j=0
i=0
i
t)(j) f (2n
j)
(t) :
Setting t = 0 and using the fact that all derivatives of both f and cosh t of
odd order at zero are zeros we get the desired formula.
v) We use the result of [18] that states that for each N; the inverse of the
2i
lower triangular matrix of degree N N with (i; j) entry 2j
is the lower
2i
triangular matrix with (i; j)th entry equal to 2j E2(i j) :
Remark 1 Formula (2) is known in a slightly di¤erent form. It appeared in
[8], [6] and [1].
n
o
(k)
Remark 2 Notice that polynomials Wn ( )
satisfy the following rek;n 1
cursive relationship for k > 1 :
Wn(k)
n
X
2n
( )=
2j
j=0
2j
(k 1)
Wj
( );
(1)
with Wn ( ) = 1: Hence its generating functions k (t; ) satisfy the following
relationship
k (t; ) =
k 1 ( t; ) cosh t;
P
(k)
t2n
Wn ( ):
where we have have denoted: k (t; ) = n 0 (2n)!
Remark 3 Notice that the above mentioned lemma provides an example of two
singular distributions whose convolution is a uniform distribution. Namely we
have S(4) 2S(4) = S(2): Similarly we have S(2) = S (8) 2S(8) 4S(8) or
S(2) = S(2k ) : : : 2k 1 S(2k ): The …rst example was already noticed by Kersher
and Wintner in [10, equation 22a].
Remark 4 We can deduce even more from these examples, namely, following
the result of Kersher [9, p. 451], that characteristic functions 'n (t) of S (n)
(where n is an integer > 2) do not tend to zero as t ! 1: Thus since we have
'4 (t)'4 (2t) = sin t=t and '8 (t)'8 (2t)'8 (4t) = sin t=t we deduce that if tk ! 1
is a sequence such that j'4 (tk )j > " > 0 for suitable " then '4 (2tk ) ! 0:
Similarly if tk ! 1 such that j'8 (tk )j > " > 0 then '8 (2tk )'8 (4tk ) ! 0.
Similar observations can be made can be made in more general situation. It
is known from the papers of Erd½os [4], [5] the situation that j' (tk )j > " > 0
for some sequence tk ! 1 occurs when
is a Pisot number (brie‡y PVnumber). On the other hand as it is known roots of Pisot numbers are not
498
Pawe÷J. Szab÷owski
Pisot, hence using the above mentioned result of Salem that j' 1=k (t)j ! 0 as
t ! 1; where is some PV number and k > 1 any integer. But we have
Q
(j 1)=k
' 1=k (tn ) = kj=1 '
tn ! 0; where tn ! 1 is such a sequence that
j' (tn )j > " > 0:
Remark 5 One knows that if
= q=p where p and q are relatively prime
integers and p > 1 then ' (t) = O((log jtj) ) where
= (p; q) > 0 as
t ! 1 (see [9, equation (3)]). Besides we know that the distribution of S ( )
is singular. Hence from our considerations it follows that if = (q=p)1=k for
some integer k; then ' (t) = O((log jtj) k ): Is it also singular?
3
Moment sequences
To give a connection of certain moment sequences with some known integer
sequences let us remark that some moment sequences satisfy the following
identities:
Remark 6 i)
n
X
2n
9 m2n (3) =
m2j (3);
2j
j=0
n
n
X
2n
81 m2n (3) =
m2j (3)(24(n
2j
j=0
n
j) 1
+ 22(n
j) 1
):
ii)
n
p
n
p
5 m2n
25 m2n
5
5
n
X
p
2n
=
m2j ( 5);
2j
j=0
n
X
p
2n
=
m2j ( 5)4n j L2(n
2j
j=0
j) =2;
where Ln denotes n th Lucas number, de…ned below.
Proof. i) The …rst assertion is a direct application
(2) while in proving
Pn 2n of
j
the second one we use (3) and the fact that j=0 2j 9 = 4n (4n + 1)=2 which
is elementary to prove by generating function method and which is known
(see [21] seq. no. A026244). ii) Again the
follows (2) while
Pn…rst2nstatement
j
n
the second follows (3) and the fact that j=0 2j 5 = 4 Tn (3=2); where Tn
denotes the Chebyshev polynomial of the …rst kind. (This identity is also
elementary to prove by generating function method and which is known (see
499
Cantor distribution
[21] seq. no. A099140). Further we use the fact that Tn (3=2) = L2n =2: ([21],
seq. no. A005248).
We also have the following Lemma.
Lemma 2 8n
1; k
m2n ( ) =
2:
n
X
i1 ;:::;ik
(2n)!
(2i1 )! : : : (2ik )!
=0
2(i2 +2i3 :::(k 1)ik )
k
Y
m2ij ( k ):
(4)
j=1
In particular:
n
X
2n
m2n ( ) =
2j
j=0
m2n ( ) =
X
i;j=0
i+j n
2j
m2j ( 2 )m2n
(2n)!
(2i)!(2j)!(2(n i
2i 4j
m2j ( 3 )m2i ( 3 )m2n
2j (
2
(5)
);
(6)
j))!
2i 2j (
3
):
Proof. (4) follows directly Lemma 1, iii).
As a corollary we get the following four observations:
Corollary 3 We have:
P
i) 8n 1 : 4n = nj=0
p
ii) S
2 has density
2n+1
2j+1
and 1 =
Pn
j=0
2n+1
2j+1
4j E2(n
j) :
p
p
8
if p jxj
2 p1;
< p p 2=4
g(x) =
2( 2 + 1 jxj)=8 if
2 1 jxj p 2 + 1;
:
0
if
jxj > 1 + 2:
p
iii) Let us denote n = ( 2 + 1)n , then
p
p
1
m2n ( 2) = 2n+2
2n+2 =(4 2(n + 1)(2n + 1)):
Proof. Since for = 2; the random variable S is distributed as U [ 1; 1] and
1
its moments are equal to ES 2n = 2n+1
: Now we use Lemma 1 iii) and iv).
p
p
ii) From the proof of Lemma 2 it follows that S
2
S (2)p+ 2S (2) :
R1
Now keeping in mind that S (2) U ( 1; 1) we deduce that g(x) = 82 1 h(x
p
p
2
if jxj
2;
4
t)dt; where h (x) =
: iii) By straightforward calcula0 if otherwise:
p
p Rp
p Rp
p
p
R 2+1 2n
2 1 2n
2+1
tions we get m2n ( 2) = 2 0
x g(x)dx = 22 0
x dx+ 42 p2 1 x2n ( 2+
1 x)dx:
500
Pawe÷J. Szab÷owski
Remark 7 Let
formulae:
(2), (3)pand observe by direct calcup (5),
Pnus apply
2n j
2n
lation that 2 j=0 2j 2 = (1 + 2) + (1
2)2n . We get the following
identities: 8n 1 :
n
X
p
2n
m2n ( 2) =
2j (2n
j=0
1
=
2n
1
=
where
n
= (1 + ( 1)n )(
n
4n
+
1
n
2j
2j + 1)(2j + 1)
n 1
X
p
2n
m2j ( 2)
1 j=0 2j
n 1
X
p
2n
m2j ( 2)
1 j=0 2j
2(n j) ;
)=4:
Now let us recall the de…nition of the so called Pell and Pell–Lucas numbers.
Using sequence n the Pell numbers fPn g and the Pell–Lucas numbers fQn g
are de…ned
p
(7)
Pn = ( n + ( 1)n+1 n 1 )=(2 2);
n
Qn = n + ( 1) n 1 ;
(8)
where n is de…ned in 3,iii).
Using these de…nitions we can rephrase the assertions of Corollary 3 and
Remark 7 adding to recently discovered ([15], [2]) new identities satis…ed by the
Pell and the Pell-Lucas numbers and of course a probabilistic interpretation of
Pell numbers.
p
P2n+2
Remark 8 i) m2n ( 2) = (2n+2)(2n+1)
; 2n = Q2n =2:
ii) 8n 1 :
P2n+2
Q2n
n 1
2
2n 1
2
P2n
P2n
n
X
2n + 2 j
2;
=
2j + 1
j=0
(9)
n
X
2n j
= 2
2;
2j
j=0
(10)
n
X
2n
=
P2j ;
2j
j=0
n
X
2n
=
P2j Q2(n
2j
j=0
n
X
p
2n
(1 + 2)2j = 2n
2j
j=0
1
(11)
(12)
j) ;
+ 2n 2 Q2n + 2n
1
p
2P2n :
(13)
Cantor distribution
501
Proof.
last statement requires justi…cation. First we …nd that
Pn 2n Only the
n
j=0 2j Q2j = 2 (1 + Q2n =2) using (10). Then we use (7), (8) and (11).
References
[1] Arnold, Barry C. The generalized Cantor distribution and its corresponding inverse distribution. Statist. Probab. Lett. 81 (2011), no. 8, 1098–1103.
MR2803750 (2012f:60051)
[2] Benjamin, Arthur T.; Plott, Sean S.; Sellers, James A. Tiling proofs of
recent sum identities involving Pell numbers. Ann. Comb. 12 (2008), no.
3, 271–278. MR2447257 (2009g:05017)
[3] Calkin, Neil J.; Davis, Julia; Delcourt, Michelle; Engberg, Zebediah; Jacob, Jobby; James, Kevin. Discrete Bernoulli convolutions: an algorithmic approach toward bound improvement. Proc. Amer. Math. Soc. 139
(2011), no. 5, 1579–1584. MR2763747 (2012d:05049)
[4] Erdös, Paul. On a family of symmetric Bernoulli convolutions. Amer. J.
Math. 61, (1939). 974–976. MR0000311 (1,52a)
[5] Erdös, Paul. On the smoothness properties of a family of Bernoulli convolutions. Amer. J. Math. 62, (1940). 180–186. MR0000858 (1,139e)
[6] Hosking, J. R. M. Moments of order statistics of the Cantor distribution.
Statist. Probab. Lett. 19 (1994), no. 2, 161–165. MR1256706 (94k:62086)
[7] Garsia, Adriano M. Arithmetic properties of Bernoulli convolutions.
Trans. Amer. Math. Soc. 102, (1962), 409–432. MR0137961 (25 #1409)
[8] Lad, F. R.; Taylor, W. F. C. The moments of the Cantor distribution.
Statist. Probab. Lett. 13 (1992), no. 4, 307–310. MR1160752
[9] Kershner, Richard. On Singular Fourier-Stieltjes Transforms. Amer. J.
Math. 58 (1936), no. 2, 450–452. MR1507167
[10] Kershner, Richard; Wintner, Aurel. On Symmetric Bernoulli Convolutions. Amer. J. Math. 57 (1935), no. 3, 541–548. MR1507093
[11] Morrison, Kent E. Cosine products, Fourier transforms, and random
sums. Amer. Math. Monthly 102 (1995), no. 8, 716–724. MR1357488
(96h:42001)
[12] Morrison, Kent E. Random Walks with decreasing steps, 1998,
http://www.calpoly.edu/~kmorriso/Research/RandomWalks.pdf
502
Pawe÷J. Szab÷owski
[13] Jessen, Bµrrge; Wintner, Aurel. Distribution functions and the Riemann zeta function. Trans. Amer. Math. Soc. 38 (1935), no. 1, 48–88.
MR1501802
[14] Salem, R. A remarkable class of algebraic integers. Proof of a conjecture of
Vijayaraghavan. Duke Math. J. 11, (1944). 103–108. MR0010149 (5,254a)
[15] S. F. Santana and J. L. Diaz–Barrero, Some properties of sums involving
Pell numbers. Missouri Journal of Mathematical Sciences 18(1), (2006),
33–40
[16] Solomyak, Boris. On the random series $nsumnpmnlambdansp n$ (an
Erd½os problem). Ann. of Math. (2) 142 (1995), no. 3, 611–625.
MR1356783 (97d:11125)
[17] Peres, Yuval; Solomyak, Boris. Absolute continuity of Bernoulli convolutions, a simple proof. Math. Res. Lett. 3 (1996), no. 2, 231–239.
MR1386842 (97f:28006)
[18] Szab÷
owski, Pawe÷J. A few remarks on Euler and Bernoulli polynomials and their connections with binomial coe¢ cients and modi…ed Pascal matrices. Math. Aeterna 4 (2014), no. 1-2, 83–88. MR3176129 ,
http://arxiv.org/abs/1307.0300
[19] Wintner, Aurel. A Note on the Convergence of In…nite Convolutions.
Amer. J. Math. 57 (1935), no. 4, 839. MR1507117
[20] Wintner, Aurel. On Convergent Poisson Convolutions. Amer. J. Math. 57
(1935), no. 4, 827–838. MR1507116
[21] The On-Line Encyclopedia
http://oeis.org/.
Received: June, 2015
of
Integer
Sequences R
(OEIS R ),