Note on pseudorandom sequences

Note on pseudorandom sequences

1) is considered, where (al denotes the A SEQUENCE znCl = lMx,I, x, (0, fractional part of the number a and M > 1 is a natural number. It is shown tha...

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1) is considered, where (al denotes the A SEQUENCE znCl = lMx,I, x, (0, fractional part of the number a and M > 1 is a natural number. It is shown that when this sequence

is used as a pseudorandom sequence

in calculations

by the

Monte-Carlo method, the central limit theorem holds for a fairly wide class

of the

functions integrated. In a number of papers (see, for example, 11, 21) the necessity for a theoretical justification of the algorithms for obtaining pseudorandom sequences has been mentioned. One important question which arises in this connection is the question of the appli~abili~ of the central limit theorem when estimating the remainder of the integration. In the present paper this question is discussed

in the case where a simple

Linear recurrence procedure is used to obtain the ~seudoraadom sequence.

Although

by [3, 41 the results presented below may be obtained with somewhat less rigid assumptions, a more elementary approach is used in this paper. It was then essential to use some of the results of IS] which are briefly formulated below (Theorem 1). We denote by D, a unit s-dimensional

hypercube and assume that the

function f(x), X = (xx, *. . I xsf is Riemann integrable in the strict sense in D, We specify the natural large numbers M,, . . . , M,_, and associate with f, a function of a single variable, +$(x> = f(x, ~M~~~,~~~*~~~~,. . . , 84, f 0.Ms_pD* where (aj denotes the fractional

part of the number a,

fJ1

Below we denote by M the

least of the numbers M,, . . . , M,,r. As usual, we say that f(x) satisfies schitz eondition with exponent y (0 < y 6 1) and constant L > 0 in f), , if ____~.__,_

l_--_-__l--“---.

* Zh. u’jchisl, Mat. mat. Fiz.,

__-.

12.4,

x_.--_--~

~~77-~~~2, 1972.

_-.-...-

-.~.-_--.-

-..-.

a Lip-

_.

.I.‘

308

S. M. Ermakov

We formulate as a theorem the results of

If 4,(x) is Riemann integrable than some M,:

(2)

if this f(X) satisfies

[sI required

below.

in [O, 1) for all M1, . . . , M,_i if M is greater

the Lipschitz

condition (2), we have for M > M,

We note that the additional condition imposed by the requirement of the integrability of 4(x) is not too burdensome. It is satisfied in any case if f(X) is continuous in D,. We also require a number of fairly simple statements which will be formulated as lemmas.

Lemma1 For any integer k > 0 we have 4 (31

s

f(lM”lf,...,

;, if the integrals

{irP’+*-*z) )dx =

j&x, 0

dt,

= M, = . , . = Iv,-* = Y, on the right and left sides exist.

The proof is by a method used repeatedly in [51. The interval (0, 1) is divided into iWkequal parts and the integral on the left side of (3) is correspondIn the j-th term of the resulting sum the change ingly divided into Mk integrals. of variable Mkx = j + y is then made, and this leads to the required result. Lemma

2

Let f satisfy

in DS the Lipschitz

the function of hs variables

f”l(X,)

condition (2) and @ I-L=xzi

. . . fm”(Xk) satisfies

i!(x) /* Then 8 in the ks-dimensional

on pse~oraRdom

Note

a Lipschitz

unit cube D,,

tct mt +***+mk-I. integers,

condition

with exponent

309

y and constant

(m, + . . . + mk) x

number and mi, 1~ a‘< k are non-negative

Here k >, 1 is a natural

not simultaneously

sequences

zero.

The proof for k = 2 follows

from the equation

f”“,(X,‘) f”‘: (X,‘) ‘- f”l (X,“)fl%‘(XZ”) = = Pn*(X,‘) (fl,(X*‘) - f”l(X,“)) + f”‘ifX,“) (fmZ(X?‘\ - f”‘z(X,“)). The general

case

is proved by induction.

Now let m,, . . . , mk be natural fying the condition ip+s
p-1,

PfI ’

numbers

satis-

. . . . k-1.

(41

I f”‘,( (&f$x}, . . , {Jft,+A-1,}) x . . . X p”k [{fkf~~~),. . . , {itl’~i”-‘)) s 1)

Ixlml,*..,mJ= As before,

and i,, . . “, i, natural

numbers

the existence

of this integral

is assumed

for all M greater

~3.r

than some

M,.

The following

D.9

lim 13~[ml,. . ‘, tn.jJ = 31-tm

(5)

holds

equation

for any function

also satisfies

I>!

G(m,

Proof.

f(X) Riemann

the Lipschitz

I

(6)

f”j(X)dX..

, . . +

J

f”“l (X)dX..

By Lemma

m#(ks

~~~X)~~.

in the strict

sense

in 0,.

If f

(2) in D, , we have for M > M, .

Jf”‘k(WdX I<

Da

Ds +

J Da

integrable

condition

[m,.. . . , m,$]-

.

-

l)~“r+-.,*“‘k-*jC2-Y.

1 we can put ZM[m,, . . . , m,] in the expression for i, = 0. 4(x) corresponding

After this it is easy to see that the i_ntegrand is a function to the function

of ks variables

f”l(X,)

. . . f”,(X,),

powers of the number M. By Theorem 1 this implies follows from Lemma 2 and Theorem 1. We also consider the sequence ca = x is some number of (0, 1) and f n+, - IM+

where M,, . . . , Mks_r are (5).

The inequality

& = Ed, c2, . . . of numbers

(6)

of (0, 1) such that

n = 0, 1, . +. , M 32, an integer.

(7)

310

S. M. Ermakov

By means

of the sequence

6 we form the sequence

Yo, Y,, . * * of points in D,

by the rule i=o,

vi = (‘is’ Ci***> * * * 9 Q+lfs_,), and investigate

the behaviour if,%I]=

(8)

of the remainder N-l

1

RN

I, *..,

-+f=O

as a function

of the quantity

It is shown in

[sI that

co = x. if as c4 we choose

an absolutely

normal number,

we

have lim lim RN &, SO]= 0. iM-+rn N-em It is known that a set of absolutely

normal numbers

has on (0, 1) a measure

equal

to 1. We also consider

the integrals 1

1~ =

s

NV [f, 4) p dx,

P = 1,2,. . . .

0

If

we

write

N--i

we have Rx If, .z] = -+

c

Ff Yi)

i=*

and ,,=iv-p~(~l(K+~” 0

f-0

what follows it will be convenient to establish the relation between I, and the moments of some random variables and use some known facts of probability In

theory. Let a be a random variable

uniformly

and consider the corresponding sequences RN&, al obviously has moments identical is bounded,

the I, define F&)

=

a distribution

distributed

in (0, 1).

We put tD = a

(7) and (8). The random quantity with I,. On the other hand, since function

[email protected], al< yl= mes Ix: R,Ef, xl < yl.

f

Note on pseudorandomsequences unique (apart from a set of measure zero).

311

(Carleman’s condition 161, p. 212, is

easily verified.) We will also consider the s-dimensional random quantity 5, uniformly distributed in D,, and the random variable v = I, = N-p

(9)

CtJk

z

fyo. Since

PI mi!

i

. . , mk! s

m,+...+mp-p it #

i2 #

O
. . . # ih,

on the assumption that f satisfies by Lemma 3 iim Zp -_ N-P M-ro.

x

Jf*k(x)dx

Cd

c

=

Dr.

r =

1,.

k = 1,. . . . D..

. . , k,

all the constraints imposed above, we obtain

Vl,+...+??lk=p

...

f”l( yi,) . . . Ifmk( yi,) dx,

0

P! mi! . . . rnk! s

relax

x ...

D8

0

.($q.) p, i=O

Da

where the 17iare independent samples of the random variable Q. Therefore, re, as M + = every 1, is identical with the p-th moment of the arithmetic mean of the independent samples of the random variable qr with zero mean and variance f2(X)dX--

Oz. =

J

(If(X)q2~

DS

the nature of the convergence being determined by the central limit theorem subject to the condition that all the moments of the random variable r, exist (see, for example, [61, p. 229). For finite M and N the function FN(y) can obviously

be represented by two

terms FNW

=?&)

+GMW,

where pN (y) is the distribution function of the random variable G, (y) converges weakly to zero as M + 00. If f(x)

satisfies

pLp-’

X -

iif?

the Lipschitz

< LOP-‘p @s -

N--J N-l

c

,=O

condition (2), we have by (6) and (9)

1) M-y.

ni, and

S. M. Ermakov

312

Or, since L,< p, we have 1 f xpd GM(x) 1 < p$‘(ps - 1)M’Y. This estimate enables us to judge the dependence of the moments of the function GM on the dimension of the domain of integration and the number M. The result may be formulated as the following theorem. Theorem 2 If f(X) is Riemann integrable in the proper sense in I), and for some M, for M > M, all the 4((x) are Riemann integrable in (0, I), we have

liinJly.lles { eo:~RNU,eol< I/}) =-&ew(-%) But if f(x)

satisfies

in DS the Lipschitz

condition (2), for fixed N and M > M,

This theorem implies that for every function f(X) satisfying for sufficiently inequality

du.

its conditions

great M and N, the measure of the set of those Q, for which the

(RNif, &Cdl < ;, is satisfied,

is 1 - sfy, M, N), where

s(!/.M,N)=11(2/rc)

[ox,

(++6(M,iv)

Jr

s(n~,N)-+O

If M and N are fixed S(M, N) = 6, an4 for two arbitrary distinct satisfying the conditions estimate (10) is satisfies

npu M,iV-ccxs. functions

f

of Theorem 2, the measure of those +, for which the simultaneously, will be at least 1 - 26. And generally,

if we denote by [al the integral part of the number a, the following corollary of Theorem 2 can be obtained.

For any [8 “1 distinct

functions satisfying

the conditions

eDcan be found such that the inequality (10) is satisfied

of Theorem 2, an

for each of them.

Therefore, these results may serve as a basis for the use of the sequences (7) as pseudorandom sequences and for the use of the inequality (1) to estimate the error in the Monte-Carlo method. In this connection

the following remark may be made.

Note on pseudorcndomsequences Remark.

In the literature

313

on‘the study of the convergence

to zero of R,[f,

t,l

instead of the points Y,, points Fi formed by the rule

are often considered.

It is easy to see that in this case Lemma 3 cannot be used,

because condition (4) will not be satisfied.

Nevertheless

in this case an analogue

of the central limit theorem for s-dependent random variables can be obtained. Therefore, convergence to the normal law holds in this case also, but the variance u7 will be different. In calculations by the Monte-Carlo method pseudorandom sequences are often obtained by formula (7), with the essential difference in this case that the calculations are carried out with a fixed number of places. This question was discussed in IS]. Some additional remarks on this topic are made below. Let +, be some number of (0, I), which ensures for each function of the given set that (10) is satisfied, and that ep,e, . . . is the representation of co as a fraction to base M. It is obvious that the i-th element of the sequence 6 will be of the form

In any case in practical

calculations

a finite number of places is used.

If M is so

great that the quantity 0.5M -r can be neglected, it is for our purpose sufficient to put ti = eiM-‘. If, as usual, we have for the representation of c0 a fixed number of binary digits, the approximation of t, must be such that each of some sufficiently large number of its first M-ary ‘digits” the same as the corresponding Therefore,

must be at least approximately

“digit” of the exact E,.

the attempts at an empirical selection

not without justification, bound to be suitable

although each specific

for the integration

of the “best” cc and M are

pair E,, and M is not in general

of all the functions satisfying

the condi-

tions of Theorem 2. In this paper, in order to obtain concrete tions satisfying

a Lipschitz

obviously be no difficulty

results,

we have considered

condition with exponent less than unity.

func-

There would

in obtaining similar results for other classes

of func-

tions, if we obtained for them an estimate of the quantity *

Is

v(t)dx--

0

as a function of M.

J/w

1

JJs

Translated

by J. Berry

314

S. M. Ermakov REFERENCES

1,

FRANKLIN, 17, 2859,

2.

CHENTSOV,

J, N. ~e~rministic 1963. N. N.

Pseudorandom

Mat. mat. Fizz., 7, 3, 623843, 3.

simulation

of random processes.

numbers for modelling 1967.

Markov chains,

Cornput.,

Zh. uychisl.

LEONOV, V. P, Same Applications of Leading Seminvariants to the Theory of Stationery random Processes (Nekotorye primeneniya starshikh aeminv~iantov teorii statsionaruykh slnchainykh protsessov). ‘Nauka”, Moscow, 1964.

4.

IBRAGIMOV, I. A. Asymptotic distribution of the values Ser. matem., mekhan. i astron., 1, 13-24, 1961.

5.

SOBOL’,

I. M.

An approach

Computing and Applied 100-112, 6.

aunt.

Tashkent,

of some sums. Vestn.

to the evaluation of multiple integrals, in: Problems fVopr. vychisl. i prikl. matem.), Vol. 38,

~~kemat~cs

1970.

PROKHOROV, YU. V. and ROZANOV, YU. A. veroyatnostei). “Nanka”, Moscow, 1967.

Theory

of Probability

(Teoriya

k

LG’T.

of