*On leave from the Centre National de la Recherche Scientifique.
UNBIASED DIE
RO~~ING
WITH A BIASED DIE
by
P. Camion*
Department of Statistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 920
April, 1974
UNBIASED DIE ROLLING
~'!ITH
A BIASED DIE
by P. Camion
C. N. R. S.
1.
IntT'oduation:
&
University of North CaroUna
We will here use some algebraic methods lectured by M.
P. SchUtzenberger at
T~ulousetmiversity in
1965, some of these were
intro~
duced in [6].
X* be the monord freely generated by a finite alphabet
Let
p:X+JR
l
for which
+
p(x)
=1
X.
is defined and may be interpreted
X€X
as a probability distribution.
A word in
X* may be viewed as a finite
sequence of trials corresponding to the tossing of a die that
finite number of times.
length
n by
p(u)
=
p
IT
p(a.)
j€[l,n]
l j
first aim is to build up a set
E
= E(1)
+ .•. + E
'0,'
Yi)
U€E\
n
~
is extended to the set
E
c
where
u
= ai
of words of
... at
1
The
n
X* which is partitioned into
( k,)wlth~
.
for k the cardinality of X :
p(u) =
l(j) p(U) ,V
u€E
n
i,j
€
[l,k] ,V n
€
N
..;B-
'W~ F
n
=F
n r
, for any
F
X*
c
He must have, for whatever
L
(2)
=0
lim
p(u)
n-+oo ~€r\EX*
Such an
.
E has been built up by von Neumann for
lar construction is extended here.
p
X
= {O,l}
[8].
That particu-
The generating series of probability for the-
values under consideration in (2) is given, for any
Card (Xn \EX * )
proved and the generating series for
X and any
p.
(2) is
is found explicitJ;yy .
A
computable formula for the mean delay is given as well.
The
adv~~tage
of this procedure is that only a small memory capacity and a
Actually the number of comparisons required is
few computations are required.
less than the value of the mean delay.
But the efficiency is very poor.
In
the next paragraph we give a procedure with high efficiency and a reasonable
amount of computation.
2.
Von Neumann sequences in the set
A classical solution for
X
{O, 1}
= {O,l}
is to take
with even length having a0right factor in
{01,10}
E to be the set of words
and without a proper left
factor with the same property (i.e. the property of having even length and
having a right factor in
{01,10}).
Here we write
(4)
where
is the set of words in
of words in
E ending with
1.
E
ending with
Clearly, the mapping
by
a(uOl)
0
= ulO
,
and
E(2)
is the set
a : E(l) ~ E(2)
define~
is one to one and since
p(ulO)
= p(uOl)
is the set of von Neumann sequences.
L
p(u)
= (1
, for every
p,
(1) is verified.
E
Clearly (2) is satisfied since
- 2p(O)p(1))n •
uEt\EX*
Moreover (4) may be easily computed for all
set
p
be.
However the
E here described is not the best possible, as proved by W. Hoeffding and
G. Simons
3.
n, whatever
[5] who define several other sets with better values of (4).
An extension of von Neumann sequences in the case where
X has more than
two symbols
3.1 Construction of the set E of sequences produaing the output syrribols
Let
We define in
X*
Card X
a prefix code
=k
.
C, that is a subset of
C n CXX *
= {</>}
X*
for vnlich
•
We write the partition
(6)
where
C.
of words
C.
~
is the set of words of length
~
u
= aa
n
in
C.
C
1
=
</>
, that is the words with two equal symbols.
is the set of all words in
C of length
sJrobols and no left factor in any
equal symbols, all
ot~ers
Cj , j < i •
are distincts.)
(5) for a prefix code is met.
i
C
2
is the set
Now, recursively
having at least two equal
('l'hus,
C.
~
has exactly two
Then, by definition, the requirement
Now let
7l. <X> , that is the ring over the rational inte-
R be the ring
*
gers of the monoid X.
To every subset
X*
F of
corresponds a polynomial
L u € ZZ <X>. For notational simplicity, we just write F for such a polyu€E
nomial. We now consider R[t] and as well the ring of formal series R[[t]] •
1 -
L
belongs to
R[t]
Its inverse is
l:;;;i:;;;k
L
1 +
i2:0
and the monomials that we find in the
in
C.
in
C has a unique factorization in
(5).
are all possible products of words
Those monomials all have coefficient one since every word factorizable
Now, the mapping
into a morphism of
71.
Ai
nt ]]
¢l: X
ZZ<X>
-+
71.
into
C , as is well known, a consequence of
defined by
</l(x)
=1
, V X E X
71. and further into a morphism of
extends
R[[t]]
into
Then
.
L
i2:0
or
\
i)-l
( 1 - £c.t
1
where
c
denotes
i
previous remark.
*=
C
u
i2:0
by
= Card
Thus
Ci
is L
'>0
1-
,
1
and a'I
denotes
¢lAi
number of words of length
P
X -+ IR , into a morphism of
+
Ai ' by our
in
i
7l. <X>
and for
A(t)
=
into lR[[t]]
R[[t]]
denote
¢l A.
P 1
One has, for example,
(8)
= Cerd.
We will also consider the natural extension of
Ai
defined by
</lC i
= L a,.t i
L
c. t i)-l =
lp
'>0
1-
&.
1p
t
i
= a.
P
(t)
by
SJ••
1p
-+
Denote
IR,
and since a"'"ip
= ~pA.~ =
p(u) , the formal"5serie (8) gives, for every
the probability for a word of length
Now let
the
k
P be the
distincts letters in
X into k
of
subsets.
X.
X
Let
C*
to be in
*
words of
kl
E(X)is the subset of
where
i
i
which is the set of all sequences of
E(t)
= A(t)Pt k
E ending with the letter
•
E(l)
Denote
by
E .
(9) is a partition
x.
For any distribution of probability
L
p(x)
=1
,
XEX
we see that if
U€E
)
p(u)
tx
)
~X)
denotes the subset of words in
does not depend on
E(x)With length
n ,
x.
n
3.2 The generating series
Property 1
(10)
c
l
One has
=0
, ci
= k(k
- 1) •.• (k - i + 2)(i - 1) , 2 SiS k •
This is a straightforward consequence of the definition of
that
c
k
= k(k
- 1) ••• 2(k - 1)
say that a finite prefix code
of a word in
F
F , every word in
(k - l)k!
c,
(11)
F.rlSiSn
L
~
n
has a left factor in
prefix codes satisfy the polynomial equality
i
= 1? .
We observe
Following M. P. SchUtzenberger, we
is complete when, if
:.ll
x
C.
is the largest lengtb
F •
Then complete
Property 2: CuP is a compZete prefix code.
Proof:
If a word of
if
has no two equal symbols, it belongs to
(C u p)X* n (C u p)XX *
CuP verifies
If it
c.
has two equal symbols, it has a left factor in
Also
P.
=~
We write in place of
(11)
(12)
Property 3:
For every probabiZity distribution p
f(tJ = 1 ....
poZynomiaZ
c.
1,~p
t
The reciprocal polynomial of
companion matrix is non-negative.
i
an of the roots of the
have a modu Zus larger than one.
f(t)
is a Frobenius polynomial, i.e. its
Thus its largest positive root has the
largest absolute value among all its roots.
From (12) by using the morphism
c.
lp
=1
~
,one obtains
p
- kIn p(x)
xeX
\
k-i has a real root ~ in lO,l[, since it
c. t
l:Si:Sk lp
takes a negative value for t
0 • On the other hand, it does not have any
which proves that
l..
=
other positive root since its
by
t -
C*P .
Let
quot~ent
~
is a polynomial with ncn-
negative coefficients.
In (9) we have defined
E as
the set of words with no left factor in
E.
for a word of length
We have
n
THEOREM 1
to be in
1 -
D
n
L
2sisk
and
2
neE
d
n
converges.
c.lp t
i
X*\EX* , that is
D be the set
We denote by
- kIn p(x)t
xeX
k
d
np
the probabilitJ
-.. . . " •• >,
Let
D(t)
be the formal serie corresponding to the set
D.
One has
= L
(1 - E(t»-l D(t)
Xit i
idN
~
Applying the morphism
(1 - kl
(16)
IT
XEX
then by (8),
l;'e
~
, we get, with
defined in (8),
(t)
p
= (1
p(x)tka (t)-l d (t)
P
_ t)-l
obtain (14).
Then by (13), the numerator of (14) has
1
as a root so that the denomina·
tor of (14) reduces to
=1
f(t)
L
-
i
c. t
2::;i::;k
~p
.
Hence property 3 completes the proof.
Remark:
For
= p(y)
p(x)
we then have
i
L
ndN
Card D t
n
n
n
is the cardinal of D
n
and since
being given by (10), one has
C
1 -
(18)
, V x,y EX, knd
L
k(k - 1) ... (k - i + 2)(i - l)t
i
- kklt
k
i<k
= ----------------------(1 -
L
k(k - 1) ... (k-i+2)(i-l}ti)(1-kt)-1
i::;k
If we write as well
t n,p
= ~PEn
.
E tn
ndN n
= L
E(t)
By theorem 1 we have
Xm _ D , and, from applying
m
~
p
and t (t)
p
Lt
ndN n,p
:
l t
Q::;n::;m n,p
=1
- d
m
=1
= L
t
tn, where
nE 1" n,p
since
L
E ~-n
Q::;n::;m n
=
-8-
.e.np
may be interpreted as the probability that the sequence of symbols obtaine,:
at time
n
be a word in
E.
If
Ivl
denotes the number of symbols in the
W01'
.e.
is the distribution of the Ivl for V EO E . By the
Ivl,p
same argument as that one used by M. P. SchUtzenberger in [6] 111.7 page 1209
v we see that
it can be proved easily that this distribution is dominated by an exponential
distribution and then has moments of every order.
fact that the;' exists a word
occurs, since, if
property.
v
u
EO
X*
is any word in
X*u c EX *
such that
= v2
P , u
This argument relies on the
This actually
is easily seen to have this
Hm;ever, we give a proof which is valid for any prefix code
C,
CuP being a complete prefix code. (*)
The distribution
THEOREM 2:
Zlvl,p
Ivl
of the
v
for
E has moments of
EO
every order.
(*)
Gordon Simons shows that such a word
under consideration.
Let
THEOREM:
and let
u
always exists in the situation
Here is his statement and. proof.
P be a complete finite prefix code for a finite alprabet
A be a nonempty subset of
P.
There exists a 'mrd
u
EO
X*
X
for vrhi.C{
* * cPAX
* *
XuX
P is complete, tbere exists a finite set
Since
Proof:
X*
for which
exists a
u
EO
1
= P*S
X*
and
P* n S :.: {e}
for '\;hich
,
where
* *
P *slul c PAX
e
Then
P*s u u
j l 2
P*sju
C
u j c P*AX*
P*AX* for each
,j
,
j
, and
= {sl' ...
is the empty word.
, and, in turn, a u 2
Proceeding recursively, one
for which
S
ca~
obtain
EO
=X
There
X* for
ul ,u2 ' •.• u
1
= 1, ... ,n . Let u = u1u 2
* * = PSuX
* * P*AX *
XUX
C
,s }
n
u
n
-9:d'en'~\tllt.ttCl"
The
of the rationa.D. fraction giving
.e.
p
(t) verifies Property
3 as well as its powerswhich will appear in the derivatives.
Then all series
under consideration converge.
Example:
k = 3
Then, knowing the first three
Card D. , 1
~
~
i
~
3 , we may compute the follow-
ing by the recurrence relation
(20)
Card Di ; 3 Card D. 2 + 12 Card D. 3 .
~~-
We finally obtain the array.
(See page 9a.)
Since f *.( t) has a real root
roots
b
-1
and
c-
1
a-I = 0,367392
and two complex conjugated
, we know from (18) that the first member of (18) has the
form
( 21 )
and since
- 1t )-1 + x (l-c
- 1 t )-1
xl ( l-a -1)
t -1 + x 2 (l-b
3
Ib -1,
~
a -1 ,
Ic -1 I·;~
a -1 , one has
We have a Vandermond system of linear equations:
x
1
+x
2
-'.. x
3
=1
ax + bX + cX
l
2
3
2
2
2
a xl + b x 2 + c x
from which
=3
=9
3
-9e.i
3
i
1
Card D./3
1
Card D.1
i
(x1+2I x 2 1)a /3
1
3
3
1
1,1834
2
9
9
1
1,0737
3
27
21
0,777
0,9741
4
Sl
63
0,777
0,8838
5
243
171
0.703
0,8020
6
,7;29
441
0,605
0,7276
7
2.187
1269
0,580
0,6601
8
.6.561
3.375
0,514
0,5990
9
.-19~683
9.099
0,462
0,5434
10
59.049
25.353
0,429
0,4930
11
177.147
67.797
0,382
0,4473
12
531.441
185.247
0,3!J.8
0,4058
13
1.594.323
507.327
0,318
0,3682
14
4.782.969
1.368.405
0,286
0,3340
15
14.348.907
3.742.2)+5
0,260
0,3031
16
43.046.721
10.185.039
0,236
0,2750
17
129.139.163
27.647.595
0,214
0,2495
18
387.417.48x
1162252',4-.102
- 75462057
0,194
0,2264
20516324x
0,176
0,2054
5.581~5.131X
0,160
0,1863
15.210.34'4 .10 2
41. 364.307.10 2
0,145
0,1690
0.132
0.1534
11. 260.990.10 3
0,120
0,1392
30.661.704.10 3
83.420,13.1.06
0.108
0,1262
0,098
0,1146
19
31+86757,2.10 221 1. 0.46.0271,10 2
2
22
31380813.10
2
23
"94142439.10
~ 28242731.10 3
25 8472819193.10 3
20
{1_31/3 3 )i/3
i
0,777
0,605
0,470
0,366
0,285
0,221
0,172
0,134
-10-
(a-3){b-3)
( a ~c Hb-c) ,
from which, after computing,
a
= 2,721892
= 1,189301
Xl
,
a
D = X*\EX* , E
2
(1 - k!/kk)i/k , i
=
u
ndN
= 0,367392.
= Ix 3 1 = 0,05703.
= 1,3043071
Ix 1
Ix1 1 + 21x21
We also compute
-1
E:
IN, which is the ratio
(Card D. )!k i
~
wner
x3~ , which gives a natural extension of Von Neumann
s2quences but is inferior to the one described here.
3.3 Computing the mean de Zay
the construction of (9), the probability of producing an output symbol
By
at the arrival of the i
th
input symbol is the coefficient of degree
series
v.'here
=
7T
IT
p(x) , f (t)
p
XIZX
=1
-
c. t
i
This coefficient is
1];1
end the mean delay is:
g·~(l) = k!7Tf-1 (1)(k - r-1 (1).r-1(1»
p
(24)
since
g' (1)
f (1) = k!7T , by (13).
p
p
=k
r! (1)
- _P,"-:--
k!7T
p
i
of the
1.0
* *
-
009
0.8
~
1
0
-
....
•
0.7
-
0
i
o
(x +2 x )a / Si
.x
i
Card D./3
1
•
Cl_31/e 3)i/3
1
2
Using Th. 3, one may approximate Card D by (3-a)an
n
)(
"*
<>
o
-r
0.6
x
0
.....
0.5
0
~
•
-*
0
-¥-
0.4
0
~
0
•
~
0.3
0
~
0
..¥-
<>
•
-l-
o
-*-
0.2
o
&
.....
---
(I
~
o
~
0.1
1
2
3
4
5
6
7
8
9
10
(
11
12
13
14
15
16
17
18
19
20
-11THEOREM 3:
L
The mean de Zay is
d
L
d
nE]N np
•
converge since the denominator of the second member of (14) is
nEIN np
actually a polynomial whose roots are outside the unit circle.
f (t) - k!~fk
P
= (1
- t)h(t) , we have by derivation h(l)
If we write
= kk!~
- fl(l) •
P
~lUP
by (14), (13) and (24)
L
d
= h(l)/f (1) = h(l)/k!~ = g'(l) •
ne:!N np
p
4.
T01JJard an effiaient easiZy avmputabZe p:t'oaedure
4.1
The efficient aonst:t'uation
P. Elias [2] published a procedure (independantly obtained, he says, by
J. A. Lechner and J. Gill) lrhich is proved to approach the best possible efficie
(= the expected number of output digits per input digit) which is
Hk(p)
=- 2
P(x)LogkP(X)
where
is the cardinality of X .
k
X€X
We describe the procedure in the binary case.
factorized ir.to words of length
n.
Now a mapping
The sequznce of input is
xn
0 :
~ X*
is defined
and the imA.ges of the factors of the input sequence are concatenated.
( n,)
1.
x.~ with i
words of
symbols
1
and
n - i
symbols
The
form a set in
0
w'hich any two ',fOrds are equiprobal)::l.e, If the binary vriting of (~) is
1.
jl
j2
js
jl
j2
jS)
(n,)
and the one2- + 2 + •• , + 2
, then Card (X
u X u
u X
=
l.
to-one image under
j
XS={R.}
word.
1:.
y
of those words is
jl
x
u
u
Js
x
If
j
s ::
°,
the empty wcrd', .and any word of this set may be mapped on the empty
We ccmpute the best possible efficiency
respectively
p
= 25%;
40, 62%; 55, 18% '
p
for
n
= 2,4,8.
We nave
-12It seems that for high efficiency
the mapping
n
has to be large and the decoding by
0 will need some computations.
4.2 Decoding with
pe~utation gro~s:
We first remind the reader that if k
then
(~)
0 modulo k
2
i ~ 0 , n.
for
is a prime and
Also for
i
l
n
k,
a power of
+ ••. + i
k
=n
(1)
if at least one of the
J
is not
or
0
n.
, o f words of length n with it repetition of
1. 1 ' ... ,1. k
symbol of X, t € [l,k] may be partitioned into subsets with respec.,
So the set
the
i
tth
S,
j
:tiTe:'cardinalities
j
k 1,
,k's , with
best possible partition for defining
J
l
~ ..• ~ Js
is given by the procedure given above,
Jl
. Js
{jl' ... ,js} with Card S1.'
. i =k +. '.'+ k
however any set of integers
0
I!'''' k
O.
will allow the definition of a suitable
of permutations on
Q, Card Q
define such a partition.
=n
, with
r
G
u
Gu
is the subgroup of
the orbit of
u
under
U E:
r
=k S
eX8~ple,
any group
(any power of
G
wllJ
k)
G, with some notational
induced by
.
,find
graphic ordering of
u
G
==
n
G.
wU.l' ,bEL:partitionc
X
Card G
G in which the word
G , we see that
then allows a suitable e
Given a
G
For
G and since
Card G Card u
where
Card
Let us denote again by
abuse, the group of permutations on
into orbits under
Actually the
> 0 •
Card u
G
u
is fixed and
divides
Practically, the problem reduce
, then determine the location of
u G and define
k
.
S
u
G
is
This partition
to the follmri:cg.
u
for the lexico-
eu as the corresponding word in
J
Xu ,
-13where
ju = LogkCard u
order
k
2t-l
G
G that we will use is the subgroup of
The group
71.
of the affine group of
t
,n = k
n
,k
a prime.
Denote the
permutation
i .... > i + 1 mod n
by
and the permutation
0
i .... > a i mod n
by
a ,a being a unit of
in
G has the form ao
invariant subgroup of
of
G.
j
71.
with order a power of
n
k.
Then every element
(0)
since we know that the cyclic group
But we may also write
G
This means that every permutation in
oja
for the generic element
G may be obtained by first
applying a permutation of type (3) and afterward a cyclic shift.
the cyclic subgroup of order
with type (3).
n
of
G
u
H
Denote by
and by K the subgroup of pcr!llUtationa
K
. has exactly one wcrd
G=HK=KH
in each orbit of the set
is an
G
under
H •
Then the decoding algorithm will be the following.
1-
For every word
phic order.
order of
If u'
u'
v
of
u
among all
.
VI
k
has
i digits.
2.
Determine the order of
t
, find the v'
€
H
v
with the highest lexicogra-
is the word "'ith the highest order in
which is a divisor, say,
P
K
i
This is an integer between
k t-l
of
.
l.n
u
H
5u
, determine
and
Card u
tt.c
K
,
k
,
5u.
Since
Card uH , say,
,we find an integer which will be written with
with the known left factor of
0
H
This integer written in basis
It is the left factor of
u
u
to form
Qu.
j
divides
digits and concatenated
-14Here is an example.
H
of the orbits
v
k
, for
v
tioned into orbits under
of
(k) •
=2
n
= 8.
We have determined all leading words
having not more than four ones.
The set is parti-
G (which are found by letting operate the elements
We find a class of four leaders, with cyclic order 8, which means
that a corresponding
u
will be decoded into a word of five digits.
1000000018
1010100018
1100000018
10010000
1111000018
11010010
1010000018
·11101000
11100010 8
11010100
11001010
10001000/4
1110010018
1110000018
10100100
1101100018
1101000018
11000010
1100110014
1100100018
11000100
1010101012
The figure gives the cyclic order of the elements in the corresponding class.
The best possible efficiency of the procedure is
p
= (5.2 5 +
,4
9.4.2
+ 8.3.2
328
+ 3.2.2 + 2)\8.2
= 46,58%
,
which is 86% more than the efficiency of the procedure of von Nevmann.
When
k
is an odd prime, the group
group of a cyclic group [7].
K is known to be cyclic, as a sub-
This makes the computation easier.
However sup-
pose for example that
form
with
a i b.1
s
a
€
u
K has two generators. Every element in K has the
2
s
a
a
a
We compute u , u
; where s is the smallest integer
u
.
H
U
S
has to be a power of 2 (maybe 2
0
)
.
This means
(as) c K H
u
-15We find similarly
and then
5'
(as, b
sl
)
= K H'
(This is true because
u
H
is an invariant subgroup of HK
taniously and as soon as
Remark:
S'
= G ).
Elements· of
u
K
are computed simul-
is determined, they are all known.
As a final remark we observe that for
m a prime and whatever
k
an
easily computable procedure with poor efficiency consists in the following.
First factorize the given sequence into words of length
repeated symbol, the
m cyclic shifts of each word
u
m.
shifts.
u
is not a
are distinct.
is the set of input symbols,which may be linearlyordered,
an integer in
If
u
If
X
may be assigned
[I,m] corresponding to its lexicographic rank among its cyclic
This integer is the decoding symbol of
u.
This procedure is also an
extension of the procedure of von Neumann.
Acknowledgement.
I am grateful to Gordon Simons who introduced me to this
problem and helped me find the references.
a
~endix.
See N. Bourbaki
"Polynonus et fraction rationnelles" for more details.
Some aZgebraic justifications.
For
R any ring with unity, we denote by
L
series of the form
i€ Ii
L
a t
a.t
i
,
where
L
b.t
a.
J.
J.
i
....... i
J.€ .II.~:
+
. IN
J.€
i
J.
€
R[[t]]
R ,
l;J
i
€
the ring of all formal
IN .
We have
=
and
The formal derivative
D
L
i;;::O
D is defined by
a.t
i
J.
= L
i:2:l
. t
J.a.
J.
i -l
and the property
l;J
u ,
is easily verified.
Now suppose
coefficients).
of
Thv
to
= vDu
v,~
Suppose
R[[t]] , D(u.v) = uDv + (Du)v ,
From. now on, R is assumed to be commutative.
are polynomials (i.e. formal series with
It is known that
is a unit of
+ lillY
V €
R.
v
is invertible in
R[[t]]
aL~ost
iff the coeffici(
For simplicity, let this coefficient be
and, consequently
Du
= v- 2 (vDw
- wDv) .
Then
+ ••• •
all zero
1.
w
= vu,
b
D , which may be directly computed from the definition of
u
obtained by multiplying this series on the right by
that
D is also
vDw - wDv.
Suppose now
R is the ring of real ntunbers and that we have the situation
q = gh
where
h
is a polynomial and where the series of coefficients in
Then, if in
ge~eral,
v
i
= L\ v..]. t
g
converges.
,
iElN
L
OS:iS:n
q.].
= L
L
OS:kS:n OS:iS:k
~ .h.].
-j{-].
=
=
Hence
Lqi
see that
converges.
L
OS:iS: oo
If
g
is the series
h-
l
and if
Lgi
converges we
r,
g. = l/h(l) •
Applying this to the case where
q=Du,g=v
].
h = vDw - wDv , we see that if the series
Lg
i
converges we are allowed to
write, with the usual notation for derivatives
if only we knm,r that the series of coefficients of
be the case in our paper since no roots of v
v
-2
converges.
This will
are in the closed unit circle.
-Co
Bibliography
[1]
Dwass, Meyer "Unbiased coin tossing with discrete random
variablesl~,
Ann.
Math. Stat., 1972, 860 - 864.
[2]
Elias, Peter "The efficient construction of an unbiased random sequence",
Ann. Math.
[3]
Stat.~1972,
Vol. 43, 865 - 870.
Lechner, James "Efficient techniques for l..Ulbiasing a Bernoulli generator"
(abstract), Ann. Math. Stat.~ 1971, page 2171.
[4] Bernard, Jacques and Letac, Gerard "Construction d'evenements equiprobabler:
et coefficients multinomiaux modulo pn ll , IZlinois J. ofMath.~ 1973,317332.
[5]
Hoeffding, Wassily and Simons, Gordon IiUnbiased coin tossing with a biased
coin", Ann. Math.
[6]
Stat.~
1970, Vol. 41, No.2, 341 - 352.
Schutzenberger, M. P. "On a special class of recurrent events", Ann. Math.
Stat.~
1961, Vol. 32, 1201 - 1213.
[7]
Albert, A. "Fundamental concepts of Algebra", Chic. Univ. Press.
[8]
von Neumann, John (1951), nVarious techniques used in connection with
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36 - 38, U. S. National Bureau of Standards, Washington, D. C.
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