On the discrepancy of Markov-normal sequences

M. B. L

EVIN

On the discrepancy of Markov-normal sequences

Journal de Théorie des Nombres de Bordeaux, tome

8, n

o

_{2 (1996),}

p. 413-428

<http://www.numdam.org/item?id=JTNB_1996__8_2_413_0>

© Université Bordeaux 1, 1996, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

On

the

_discrepancy

of

Markov-normal

_sequences

par M.B. LEVIN

RÉSUMÉ. On construit une suite normale de Markov dont la

dis-crépance

est

O(N-½ log2 N),

améliorant en cela un résultat don-nant l’estimation

_{O(e- c(log}

N)1/2).

ABSTRACT. We construct a Markov normal sequence with a dis-crepancy of

O(N-½ log2

N).

The estimation of the

discrepancy

was

_previously

known to be

_O(e-c(log

N)½).

A number a E

_{(0, 1)}

is said to be normal to the base q, if in a _q-ary

expansion

of _a,

each fixed finite block of

_digits

of

_{length k}

appears with an

_asymptotic

frequency

of

_q-’‘

_along

the _sequence Normal numbers were

intro-duced

_by

Borel

_(1909).

Borel

_proved

that almost _every number

_(in

the

sense of

_Lebesgue

measure)

is normal to the base _q. But

_only

in 1935 did

Champernowne give

the

_explicit

construction of such a

_{number, namely}

obtained

_{by successively}

_{concatenating}

all the natural numbers. Let P = be _an irreducible Markov transition

matrix,

the

stationary probability

vector of P

_{and p}

its

_probability

measure.

A number a

(sequence

is said to be Markov-normal if in a _q-ary

expansion

of a each fixed finite block of

digits

_{appears with} an

asymptotic

frequency

Of

According

to the individual

_ergodic

theorem

_{71- ost}

_{all sequences}

(num-bers)

are normals.

Markov normal numbers were introduced

by

Postnikov and

Piatecki-Shapiro

[1].

They

also

_{obtained, by}

_generalizing

_{Champernowne’s method,}

the

_explicit

construction of these numbers. Another

_Champernowne

con-struction of Markov normal numbers was obtained in

_{Smorodinsky-Weiss}

Markov normal numbers

using completely

uniformly

distributed sequences

(for

the

_definition,

see

[5])

and the standard method of

modelling

Markov

chains. In

_[6]

Shahov

_proposed

_using

a normal

_periodic

_systems

_of

dig-its

_(for

the

definition,

see

[5])

to construct Markov normal numbers. In

[7]

he obtained the estimate of

_discrepancy

of the sequence be In this article we construct a Markov normal _sequence with

the

_discrepancy

of _sequence

_equal

to

O(N-1/2Iog2

_N).

Let be a _sequence of real

numbers, p -

measure on

[0,1).

The

quantity

is called the

_discrepancy

of

The _sequence is said to be

_{p - distributed}

in

_{[0, 1)}

if

_N)

- _0.

Let the _{measure /.i} be such that

where

_{(0, 1, ..q - I) ,}

1~ =

1, 2, ....

It is known that if and

_only

if a is Markov normal

_number,

the _sequence

is

_{it-distributed.}

The

_discrepancy

satisfies

_{D (,u, N ) -}

for almost all a.

The

_following

facts are known from the

theory

of finite Markov chains

[8,9]:

Let a Markov chain have d

_cyclic

class

C1, ...,

Cd.

We enumerate the

states el, ..., eq of the Markov chain in such a _way, that if _ei E

_Cm,

_ej E

Cn

and i

_{&#x3E; j,}

then

_{m &#x3E;_n.}

Here matrix P_ has

d 2

blocks where

Pi,j =

0

_except

for Matrix

Pd has

a

_{block-diagonal}

structure. Let

_{Pl , ..., Pd}

be the block

_diagonal

of matrix

pd.

There exists

a number

_l~o

such that all the elements of matrices

Pk°

(i

=

1,

...,

d)

are

greater

than zero

[9,

ch.

4].

Let () be the minimal element of these

_matrices,

and the

_{i j}

element of matrix

_P~,

_{k = 1, 2, ....}

It is evident that

where we choose minimum values for

_{i, j}

so that _{ez, e~} are included in the same

_cyclic

class.

Let

_{f ( j )}

be the number of

cyclic

class

_{states ej}

_(ej

E =

_0,

According

to

_{[9, ch.4]}

we obtain

We

_have,

from the

_{irreduciblity}

of matrix

_P,

that

We use matrices

P,,

= with the rational elements

and we choose as follows:

Let i be fixed and be

_greater

than zero. Then we denote

It is evident that

If

kl

is

_{sufficiently large,}

then

_using

₍₃₎

and

_(6)-(8),

we obtain

where we choose minimum values for

_{i, j}

so that

_belong

to the same

cyclic

class.

It is evident that

_Pn

_(n

&#x3E;

_kl)

is an irreducible matrix with

_{a d-cyclic}

class.

Applying (3),(4)

and

₍₉₎

we obtain

where n &#x3E;

_ki,

and is the

stationary

probability

vector of

Pn.

According

to

_{[7, 10]}

there exist

_integers

&#x3E;

_0,

such that

If

k1

is

_sufficiently

_large,

then

_{applying (5)-(8)}

and

_(11),

we obtain

Let the measure _J-tn on

[0, 1)

be such that

where the O-constant

_{depends only}

on P.

Proof. It follows from

_{(2), (5)}

and

₍₆₎

that

We

_apply

_{(2), (14)}

and obtain

=

0,

then

PCiCj = 0 and

_according

_to

(5), (7), (8), (11)

_we

have

_(n)

=

0 (pn)

and

. , , . _..

and vk =

ck+1 or b.

On the basis of

_{(5), (7), (8)}

and

₍₁₁₎

we deduce that

where If

2qB /p’ .

It is easy to

_compute

that

Hence and from

_{(17) - (20)}

we obtain

and formula

₍₁₅₎

is

_proved.

Statement

₍₁₆₎

is

_{proved analogously. o}

We obtain the Markov normal number a =

dld2 ...

by

concatenating

blocks

_an

=

(aI,

...,

aA2n),where

ai E

(0, 1 , ..., q -1~,

i =

1, 2, ...

We choose the numbers ai as follows:

Let

We set ao =

i,

if

bo E Si, i

=

0,

...,

q -1.

If we choose the numbers

We choose cvn

(and

consequently

an(cvn))

such that

Proof.

_(To

follow

_later.)

Let

Every

natural N can be

represented uniquely

in the

following

form with

integers k

For

_Q

= 0 we use the

_symbols

M)

and _a,

M) .

THEOREM 1. Let the number a be

defined

by

(21), (23), (2l~)

and

(27).

Then a is Markov-normal and the

following

estimate is true:

where the O-constant

_{depends only}

on P.

Proof.

_Using

_{(29), (30)}

and

_(31),

we obtain

According

to

_{(21), (24)}

and

₍₂₅₎

we have

It is evident that

We

_apply

₍₃₁₎

and obtain

On the basis of

_(26)-(28),

Lemma 1 and Lemma 2 we deduce that

According

to

₍₃₄₎

we have for M

A.,,. -

r

It follows from

₍₃₁₎

that

It is evident from this that statement

₍₃₅₎

is valid both for M

_{A2,. -}

2r

as well as for M E

_{[A2, -}

Substituting (35)

into

₍₃₃₎

and

_bearing

in mind

₍₃₀₎

we deduce

Using (29), (30)

and

₍₅₎

we obtain

Hence and from

_{(1), (31)}

the statement of the theorem follows. N We denote

Proof.

_According

to

₍₃₆₎

we have

Using

(37),

we obtain

From

₍₃₉₎

and

₍₄₀₎

we

_give

the assertion of the lemma. m

LEMMA 4. Let 0 u2

A2,,, M &#x3E;-

0

i, j

=

0, ..., q -1,

n &#x3E;

k1-Then

where the constant in

_symbol

0

_{depends only}

on P.

Proof. Let

_Nl

=

(ul/d,

N2

=

[U2/d].

We

change

the variable x =

dy

_{+ z}

and obtain

_according

to

₍₁₃₎

Let

Applying

(13), (10),

we obtain

where

_1,El

_1,

z, =

f (j) - f (a).

Substituting

this formula into

_(41),

we obtain

_according

to

_(13),

that

where

It is known that

Using

(40)

we

_get

Hence and from

_(42-44)

the assertion of the lemma follows We consider further that ai,

i =1, 2, ...

is the

sign

of the number

It follows from

_(25),

that

where

Then

Proof. It follows from

₍₄₆₎

and Lemma 3 that

Changing

the order of summation and

_applying

the

_{Cauchy inequality}

we obtain that

We

_change

the variable b to cr and assume, on the

_right-hand

_side,

the summation on _cz, i =

_1,

_{..., r -} 1.

We denote

_by

_S(cvn)

the

_right-hand

side of formula

_(52).

It is evident that

_S(wn)

does not

_depend

on M and In.

Applying (26),

we obtain

and

Changing

the order of summation and

_using

_(51),

we obtain

Hence and from

_(47)-(49)

we deduce formula

(50).

LEMMA 6. Let n &#x3E;

kl.

Then

Proof.

_{Applying (49)}

and

_(22),

we

_get

According

(23),

we obtain

It is obvious that

Hence and from

₍₅₅₎

we

obtain,

changing

the order of summation

According

to

_{(53), (36)}

and

_(22),

we have

Applying (7)

and

₍₁₁₎

_,we obtain

On the basis of

₍₅₄₎

and

₍₁₄₎

the lemma is

_{proved. m}

Proof. Let _y &#x3E; x.

Applying (48)

and

_(22),

we obtain

As in the

_proof

of Lemma

_6,

we

_get

As in

_(56),

we have

Hence and from

_{(59), changing}

the order of

_summation,

we obtain

Using

(53), (36)

and

_(22),

we

_get

Applying (7)

and

_(11),

we obtain

It follows from

_(58),

that

Similarly

for x _y.

_{According (14)}

the lemma is

_proved

LEMMA 8. Let n &#x3E;

_kl,

_{)y -}

r. Then

Pro of.

_{L et y &#x3E;}

x .

As in the

_proof

of Lemma 6 and Lemma

7,

we

_get

It follows from

_(12),

that

Similairly

for x y.

According

to

(14)

the lemma is

proved.

where

Let

According

to Lemma

8,

(12)

and

₍₁₄₎

we obtain

It follows from Lemma 7 that

Changing

the variable

_{y to}

_{yl = y - x - r and}

_applying

Lemma

_3,

we obtain

Similarly

estimate is valid for

B3.

Hence and from

_(61)-(62)

we obtain the assertion of the lemma.

Proof of Lemma 2.

_Substituting

₍₆₀₎

into

₍₅₀₎

and

_bearing

in mind

_(5),

we

deduce

Lemma 2 is

_proved.

Remark.

_By

a similar method and the method in

[12]

a Markov normal vector for the multidimensional case can be be constructed.

_By

the method

in

_[12]

one can reduce the

_logarithmic

_multiplier

in

₍₃₂₎

to

_0(log

_N3/2).

To

reduce the

_{logarithmic multiplier}

forther see

[15].

Problem.

_According

to

_[12-14]

the Borel and Bernoulli normal numbers

ex-ist with

_discrepancy

_0(~V’~~).

It would be

_interesting

to know whether Markov normal numbers exist with

_discrepancy

_O(N-C)

where c &#x3E;

_1/2.

Acknowledgements.

I am _very

_grateful

to Professor David Gilat and to

Professor Meir

_Smorodinsky

for their

_hospitality

and

_support.

REFERENCES

[1]

A.G. Postnikov and I.I.

Piatetski-Shapiro,

A Markov _sequence of

_symbols

and

a normal continued

_fraction,

Izv. Akad. Nauk

_SSSR,

Ser.

_{Mat., 1957,}

v.

21,

501-514.

[2]

M.

_Smorodinsky

and B.

_Weiss,

Normal sequences for Markov shifts and

intrin-sically ergodic

subshifts,

Israel Journal of

_{Mathematics, 1987,}

v.

59,

225-233.

[3]

A.

_{Bertrand-Mathis,}

Points

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de

_Champernowne

sur certains _systems

codes; application

aux

03B8-shifts,

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Th. &#x26;

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Sys.,1988,

v. _8, 35-51.

[4]

N.N.

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Pseudorandom numbers for

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Markov _chains, U.S.S.R.

Comput.

Maths. Math.

_{Phis., 1967,}

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Imitation of

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