A Remark on Random and Equidistributed Sequences 1 Strongly

A remark on random and equidistributed sequences

Nicolas BOULEAU

Lab oratoire de Mathematiques Appliquees UA CNRS

ENPC La Courtine NoisyleGrand Cedex

If is an equidistributed sequence on the sdimensional cub e the average of

the values of a function f on the sequence converges to the integral of f for f Riemann

integrable We study here the quite natural fact that a slight random p erturbation of the

sequence allows to integrate more irregular functions The notions used are dened in

and

Strongly distributed sequences

Let b e a probability measure on s IN equipp ed with its natural top ology

Except what envolves the convolution pro duct the sequel could b e extended to any lcd

space

Let X b e a sequence of random variables with values in dened on a probabi

lity space A IP Almost surely as means IPalmost surely If with probability

the sequence X is distributed on ie if almost surely

f C IR f X f d

then X will b e said to b e almost surely distributed

Now the example of iid random variables X with common law shows that

some random sequences have the following stronger prop erty

Denition A random sequence X is said to be strongly distributed if for every

bounded measurable function f from into IR

f X f d as

That a strongly distributed sequence X b e almost surely distributed comes

from the fact that if D is a countable dense subset of C IR such a sequence satises

f X f d as f D

which implies easily

narrowly as

Prop osition Let X be a sequence of random variables with values in If for

every open set A in it holds

X A as

A n

then X is strongly distributed

We shall use the following lemma

Lemma Property for open sets implies property for continuous f s

Pro of Let f b e continuous We may supp ose f Then

k k

G nG f

k k

n n

for an increasing sequence of op en sets G Approximating uniformly up to the function

f by

with A G nG f

k A k k k K

the fact that

f X f d as

K n K

gives

Z Z

N N

X X

f d lim f X f X f d lim

n n

N N

n n

which shows the lemma

For the prop osition consider a b ounded measurable function f By the Lusin pro

p erty for every sequence decreasing to zero there exists a sequence of op en sets G

k k

and a sequence of continuous functions f such that

k G f f outside G k f k k f k

k k k k k k

Let us write

N N N

X X X

f X f X f f X

n k n k n

N N N

n n n

By the lemma the following inequalities hold as

f X f d k f k G lim

n k k

f X f d k f k G lim

n k k

hense as lim f X f d

Remark The prop erty that b e satised for continuous functions f is weaker than

the prop erty for op en sets It is indeed equivalent to the almost sure distribution

of X And there exist almost surely distributed sequences which are not strongly

distributed consider a deterministic equidistributed sequence take X

n n n

and take f x

fxn xg

Prop osition Let be an equidistributed sequence on and V be a sequence

n n

of iid random variables with absolutely continuous common law on IR Then the

sequence X f V g is strongly distributed on the cube Here fxg means the

n n n s

fractional part of x component by component and is the Lebesgue measure in dimension

Pro of Let us dene absolutely continuous probability measures on the cub e by

A f xgd x

n A n

for every measurable subset A of

Lemma For every f Lebesgue measurable and bounded on the cube

f f lim

n s

Pro of The function

g y f fy xgd x

s s

is continuous and b ounded on IR as convolution pro duct of a function in L IR by a

function in L IR Hence

Z Z

g g y dy f y dy

s s

Now to achieve the pro of of the prop osition we use a classical argument

Let S f X f

N n n

a It holds S as

Indeed

X X

f f j IPN m jS

n n

N m

k f k

thus j as lim jS

b It holds S as

Indeed if M N M

jN M j M

j jS S k f k N k f k

M NM M

So by the lemma

f X f d

n s

Extension

The preceding prop osition fails if is no more absolutely continuous as seen when

is a Dirac mass But if do es not charge p olar sets it is p ossible to integrate with X

quasicontinuous functions and even a little more For conveniency we work now with the

s s

torus T IRZ instead of the cub e

Prop osition Let be an equidistributed sequence on T and V be a sequence

n n

of iid random variables with values in T with common law which does not charge

sets of Newton capacity zero Let X V Then

n n n

f X f d as

n s

for every f from T into IR with the fol lowing property

u v quasicontinuous and bounded such that

v ud u f v and

Pro of

a Let us supp ose rst that the probability b e a measure of nite energy integral

for the classical Dirichlet structure on T and that f b e a quasicontinuous version of an

element of H T Then approximating f in H by functions f in H and continuous

and denoting the translation by x give

1 1

C k f f k j f f j C k f f k

n x x n x x n

H H

and that implies that the convolution pro duct f is a continuous function on T

b Let us supp ose now f b e quasicontinuous and b ounded and the probability mea

sure do not charge p olar sets Then there exist probability measures of nite energy

integrals such that

p p p p p

b e the energynorms of the s Let C k U k

p p p

Let For each p let us choose a continuous function f and an op en set G such

p p

that

k f k k f k f f outside G C apG

p p p p

We have

f f f f

p p p p p

but

jf f j k f k e G k f k C apG C k f k

p p p p p p

where e G is the equilibrium p otential of G

p p

It follows that f is uniform limit of continuous functions and therefore is continuous

c Under the hypotheses of part b putting

it holds therefore by the equidistribution of

f d f

s n

then the same argument that in the pro of of prop osition shows that

f d as f X

s n

d At last if f satises prop erty we have

N N

X X

ud lim uX lim f X

s n n

N N

n n

N N

X X

f X v X v d lim lim

n n s

N N

n n

almost surely Hence

f X f d as lim

n s

The functions satisfying can b e shown to b e the functions which are b ounded and

nely continuous at almost every p oint of T

REFERENCES

L Kuip ers and HN Niederreiter Uniform distribution of sequences Wiley

M Fukushima Dirichlet forms and Markov pro cesses NorthHolland