PROBABILlTY
AND
MATHEMATlCAL STATISTICS
THE GENERALIZATION OF THE KAC-BERNSTEIN 'EIEBRETM
BY
M. B. QUINE
AND
E. SENETA (SYDNEY)
Abstract. The Skitovich-Darmois Theorem of the early 1950's
establishes the normality of independent X,, X,, . .., X, from the independence of two linear forms in these random variables. Existing
proofs generally rely on the theorems of Marcinkiewicz and Cramkr,
which are based on analytic function theory. We present a self-contained real-variable proof of the essence of this theorem viewed as
a generalization of the case n = 2, which is generally called Bernstein's
Theorem, and also adapt an early little known argument of Kac to
provide a direct simple proof when n = 2. A large bibliography is
provided.
Key words: independence; characterization; normality; Bernstein's theorem; Cramkr's theorem; Marcinkiewicz's theorem; characteristic function; Laplace transform; real-variable; real function; moments; cumulants.
The Skitovich-Darmois Theorem asserts that if n 2 2 is fixed, X I , ..., X,
are independent, and Y, =
aj Xj is independent of Y, =
bjXj for
some constants (aj), (bj) with aj bj # 0,j = 1, ..., n, then each Xjis normally
distributed. This theorem implies Cramtr's Theorem (Cramkr [6]) through
a simple application of the case n = 4 (Linnik [25]). On the other hand, proofs
of the Skitovich-Damois Theorem are not self-contained in that they require
(1) the use of Cramir's Theorem (at the very least to cover the case where
for some j # k, aj/bj = ak/bk, since both sums then contain a multiple of
b j X j+ b, Xd, and
(2) the proposition that if a characteristic function 4 (t) = E(eitx),t real,
has the form exp(~(t)),where P(t) is a polynomial, then the degree of the
polynomial is not greater than 2.
This last is a form of Marcinkiewicz's Theorem, which is in terms of
a complex variable z instead of t. The complex-variable version is easier to
prove directly (e.g., Linnik [26], p. 65); the real variable version is quite long
x;=,
x;=,
442
M. P. Quine and E. Seneta
and difficult (Lukacs [28], pp, 213-221; Bryc [ 5 ] , p. 351, although the jump
from real t to complex variable z is sometimes made rather cursorily. As regards (I), the proof of Cramkr's Theorem depends on a deep result from the
theory of entire functions, Hadamard's factorization theorem, which is stated
but not proved in probability monographs (e.g., Linnik [26]). Thus proofs of
the Skitovich-Darmois Theorem to a large extent depend on external theorems, whereas an essentially self-contained proof, not heavily dependent on
results from entire function theory, for the most part in real variable terms, and
avoiding use of the proposition about polynomial exponents, is desirable from
a didactic viewpoint.
The essence of the Skitovich-Darmois Theorem is to view it (Darmois
[lo], p. 6) as an extension of Bernstein's Theorem (the case n = 2) by putting
aside the possibility that aj/bj = a,/bk for some j # k. This enables us to produce, in Section 2, a self-contained proof of the kind desired. Naturally, this
proof borrows and interrelates a number of clever arguments to be found in the
works of authors such as Skitovich, Lancaster, Lukacs and King, and Dugu6,
when they address the Skitovich-Darmois setting. There are also novel elements, such as the proof of Lemma 4, and the switch from characteristic functions to Laplace transforms following Lemma 5, in Section 2.2.
In Section 3, which deals with the case n = 2, we adapt the largely overlooked real-variable argument of Kac [17] to prove Gnedenko's [16] generalization of Bernstein's Theorem [3]. Our overall treatment in both Section 2 and
Section 3 rests heavily on Lemma 2, which is due to Lancaster [22].
The paper includes a large bibliography which, whilst not complete, seeks
to illuminate the early published history on this topic, disrupted as it was by
World War 2 and its aftermath.
Z THE SWITOVICH-DARMOIS THEOREM
We state our result before proceeding (A restricted version was the purpose of Marcinkiewicz [31].)
THEOREM
1. Let n 2 2 be fixed, XI, ..., X, be non-degenerate and independently distributed random variables, and suppose that
n
Y'=CXj
R
and
. j=l
&=xbjXj
j= 1
are independently distributed, where the constants {bj) satisfy bj # 0, bj # b,,
j # k. Then each Xj is normally distributed.
2.1. Real variable arguments. As a first step to a proof of Theorem 1 we
follow Skitovich [37] by symmetrizing. Let (X;,
..., Xb) be an independent
replica of (XI, ..., X,) and define
n
Y;=
EXj
j= 1
and
Y;= i b j ~ > .
j= 1
Generalization
Then
yj = Xj-Xi,
01the Kac-Bernstein
Theorem
443
j = 1, ..., n, are independent and
n
n
are also independent. The characteristic functions of the symmetrized variables
are of course real-valued, but the independence of the linear forms gives more:
Then 0 < J j (t) < 1 for all- -real t.
p r o o f (based on Skitovich 1371). The independence properties can be expressed as L(u, u)=R(u, v) for - m < u < a,-a < v < a,where
"
If the lemma is false, then, by continuity and since L (0, 0)= R (0, 0) = 1, there
exists a number w such that
R(u, v) > 0 for lul < Iw1 and Ivl < Iwl, and R ( w , w ) = 0.
(2.3)
This entails either & (w)= 0 for some k or Jk (b, w) = 0 for some k. In the first
case, let u, = (1 - bk/c)w and v, = w/c,where c is chosen so that IcI > max(1, Ibkl)
and b,/c > 0. Then we have u, + bkvl = w and
L(ul, ~ 1=)
n $j(ul+bjvl)'6k(ulf
bkvl) = O,
j#k
so R(ul, vl) = 0. This contradicts (2.3), since lull < Iwl and Ivl] < Iwl. On the
other hand, if Jk(bk w) = 0 for some k, then taking U, = bi W / C and
v, = (1-b,/c) w, with c chosen such that Icl > max (1, bz) and bk/c > 0, we arrive
at the same contradiction.
Lemma I implies that for j = 1, ..., n the second characteristic function Gj(t)= logJj(t) is uniquely defined as a real-valued function for
- co < t < ca. The following lemma guarantees that we can differentiate gj(t)
any number of times (see, e.g., Feller [13], XV.4, Lemma 2).
LEMMA2. FOP j = 1, ..., n, E ~ z ~ <l ' co for any r 2 1 .
P r o o f (after Lancaster [22]). Let a = mini lb, 1 and /?= maxi Ibi1. Take
0 < E < 1 and choose A so that
P(IX,I>A)<c
for i = 1 , 2 ,..., n.
Put y
(2.4)
= (2n--l)jj/ol>
2n-1 (23 since n 2 2). Then
(l-~)"-lP(lX~l> yA) < P(IXjI > yA, lXil < A for all i # j )
444
since lXjl < I YII
M. P. Quine and
E. Seneta
+xi,lXi1 gives
and Ib,Xjl d [Y,l +zi,j(biX,( gives
Now
by Boole's inequality, and
It follows from (2.4H2.6) and the independence of Y, and Y, that, for
j = 1, 2, ..., n ,
Writing the right-hand side of (2.7) as E' we have shown that, for y as defined
above,
P((Xj(> A) < E implies P (IXj(> y A) < E'.
If we take E < n-3, then it follows from Bernoulli's inequality ((1+x)" 2 1 +ax
for x > - 1 and a = 1 , 2 , ...) that n ( l - ~ ) " - l > l for n 2 2 . Thus
E' < n3 cZ <
s 2 1 , we have proved
Then, if we put 6 , = E, E, = n3
that, with E,, < n - 3 and k 2 0,
on putting c = n - 3 and g = n3eO,so 0 < g < 1. Finally,
Genera~zationof the Koc-Bernstein Theorem
445
Since the ratio of the ( k + 1)-st to k-th terms in this sum is y r g 2 k - i 0,
D'AlembertYstest shows that the sum is finite.
LEMMA3. For j = 1 , ..., n there exists a polynomial Fj(t)with real coefficients and of degree at must n , such t h ~ t
t,&j(t)=Fj(t),
-m<t<co.
P r o of [ideas similar to Lukacs and King [29], pp. 391-392; see also Bryc
[ 5 ] , pp. 76-78). The equality of (2.1) and (2.2) gives
It follows from Lemma 2 that each GJhas at least n derivatives. Differentiating
(2.8) r times, 1 < r < n, with respect to u and setting v = 0 gives
M
where rZj, = ( -iyII,pl(0) is the r-th cumujant of gj (Laha and Rohatgi [21],
p. 223). If we integrate (2.9) with respect to u, we get
Integrating a further r - 1 times with respect to u, at each stage using the
identity @(0) = i'fc, we obtain
If we denote the right-hand side of (2.10) by d,(u), it follows that dr(u) is
a polynomial of degree r in u, with real coefficients on account of the present
symmetric case with gj(u) = G j ( - u ) in which fjs = 0 for odd integers s. Thus
in the matrix form (2.10) becomes
where $(u) = (GI (u), ..., gn(u)): d(u) = (dl (u), ..., d, (4'and
b;
b",..
446
M. P. Quine and E. Senata
Since the bj's are all unequal, B must be non-singular, so it follows from (2.11)that
LEMMA4. For j
=
1, ..., n , gj
-
JV (0, 2~:).
Proof. Taking r = 2 in (2.10) we obtain
bfgj(t)=-ct2,
(2.13)
.
j= 1
where c = xbj2 D;, 03 being the variance of Xj. It follows from (2.13) and
Lemma 1 that for -each j
In order that (2.14) be consistent with Lemma 3, it is necessary that the degree
of the polynomials pj(t) be at most 2 and normality of the zj's follows. r
The normality of the Xis themselves could now be deduced from Crarnkr's
Theorem, as is done at this point by Kac [17] and Skitovich [37]. Of course, if
it were known that the X,'s had symmetric distributions, then the arguments of
Section 2.1 could by applied directly to the Xi's themselves. We now show how
to establish the normality of the Xj's themselves from Lemma 4, without direct
use of Crarnkr's Theorem,
-
2.2. kaplrmce transforms. Lemma 2 is clearly true in terms of the original
Xis, and since Xj-Xi
N ( 0 , 2 4 ) from Lemma 4, where Xj and Xi are
independently and identicaIly distributed with characteristic function $j satisfying t$j(t)4j( - t) = exp (--a;t2), it follows that 4j(t) # 0 for any real t, and
dj(t) has at least n derivatives. We put IClj(t) = log 4j(t), where log refers to the
principaI branch (since #j(t) may be complex valued even though t is real), so
$j(0) = 0. The following lemma implies that the Xi'shave at most n non-zero
cumulants:
LEMMA5. For j = 1, 2, ..., n there exists a poZynornia2 Pj(t) of degree at
m s t n, such that
j(t=Pj(t),
-co<t<oo,
where q)(O) = rcj,, the r-th cumulant of Xj.
Proof. We need only mimic the proof of Lemma 3, replacing Gj (t) by
Ifij(t), with minor adjustments for non-symmetry. ta
The remainder of our derivation is in terms of the Laplace transform
which the next lemma shows is finite.
6. For j = 1, ..., n,
LEMMA
447
Generalization d the Kac-Bernstein Theorem
-
P r o of. According to Lemma 4, X j - Xi JV ( 0 , 2 0 3 . Clearly, we can
assume without loss of generality that Xj has zero median, that is,
P ( X j < 0) $ < P ( X j < 0 ) . Then the distribution function F j of Xj satisfies
6 + P ( X j < x)+ P ( X , - X i
< x).
Writing
1
"
fi
-m
@ ( x )=-
j exp(-u2/2)du,
we obtain
As x + CQ ,f
-F j ( x )is similarly bounded. This means we can integrate by parts in
to get
-m < A j ( v ) =
0
m
-m
0
1 e - V x d F j ( x ) + [ e - u X d ( ~ j ( x ) - l <)
CQ.
It is readily seen that Aj(v) has continuous derivatives of all orders r 2 I ,
with
m
bAj(v)/dvr= (- l y
j~'e-~~dF~(x).
-m
By Lemma 5, the cumulant generating function g j ( v ) = logAj(v) exists for all
v since Aj(v) # 0, and thus has continuous derivatives of all orders. It is clear
that 9'f)(0) = ( - 1)' ujr, where K ~ ,is the r-th cumulant of X j .
We are now in a position to prove Theorem 1. Recall for the sequel that
q, = EXj and ~ c =~ VarXj
,
= a.; It follows from Lemma 5 and the mean
value theorem of order n + 1 that
(2.15)
The fact that
(2.16)
9 7 (v) 2 0
follows for instance by noting that 9 y ( v ) is the variance of the conjugate
distribution
e-'' dFj ( x )
dGj(x) =
cO e-ux ~ F ~ ( X ) '
S-
448
M. P. Quine and E. Seneta
(This is suggested by an argument of Dugub [11], p. 56; see also Linnik [26],
p. 62.)
Lemma 4 implies
2Yj(v)+Pj(-v)
= -.j
2 2
v
from which it follows by taking derivatives at v = 0 that all even cumulants
higher than the second are zero, so that (2.15) reduces to
From- this 'and (2.16) we obtain for n 2 3
The right-hand side of (2.17) is an odd function of v, and hence will be large and
negative, for either large positive v or large negative u, if xjq2,,,+ is non-zero for
any m = 1, ..., [(n- 1)/2]. But this would contradict the lower bound in (2.17).
It follows from Lemma 5 that
glrj(t) = iujl
Kj2 z
t-- 2 t ,
that is,
-
Xj N(lcjl, xj2). ra
3. ON FORMS OF BERNSTEIN'S THEOREM
In conclusion we indicate a simple direct proof of
THEOREM
2. Let X1 and X , be non-degenerate and independently distributed random variables and suppose that
Yl = pXl + q X 2
and
I.; = a x , - bX,
are independently distributed, where p, q , a and b are all real and non-zero. Then
XI and X, are each normally distributed.
The reader will recognize this as the case n = 2 of Theorem 1. The case
p = q = a = b = 1 is known as the celebrated Bernstein's Theorem (after Bernstein [3], who assumed also that X, and X, had finite, equal variances and
positive densities). Bernstein's Theorem was generalized by Gnedenko [16],
who proved Theorem 2 in full generality, taking (without loss of generality)
p = q = a = 1, b # 0, - 1. For a modern proof, see Quine [34], Theorem 1.
Our proof, in which passage to logarithms is unnecessary, borrows a little from
this, but shows that the Bernstein case is rather special and requires extended
treatment. However, such treatment is shown to have already been available, in
elegant and simple real variable terms, in Kac [17].
Kac's paper precedes even Bernstein's. From its received date, shortly after
his arrival just before World War 2 at Johns Hopkins University on a Polish-Jewish (Parnas Foundation) Fellowship to the U.S., the paper was written
Generalization of the Kac-Bernstein Theorem
449
by Kac largely in Lwbw (then in Poland, now L'viv, in Ukraine; Russian name:
L'vov); see Kac [IS]. It is possibly due to ongoing disruptions in scientific
communications caused by the war, and partly due to its own apparent restrictiveness, that Kac's paper has not received its due within the very large literature emanating from Bernstein's Theorem. There is no mention of it in the
Russian papers, or the French sources (Frkchet [ 1 4 ] ; Darmois [7]-[lo]; Dugut [Ill, [12]) which deal with the topic in terms of characteristic functions.
Oatline of proof for 'Ileorem 2. Using our Lemma 2 (which does not
require symmetry of the Xi's)
we obtain EIXjlr < GO, r 2 1. Then, assuming
without loss of generality E X j = 0, we obtain, as in Lemma 2 of Quine [34],
the equality
Y,) = pa at -bq &,
Further, since Y, and Y, are independent, 0 = Cov (Yl,
where CT;
= Var Xj> 0, and since
z = pb + aq # 0 (otherwise Yl would be a multiple of Y2). Inverting the matrix
in (3.2) gives
zXl=bYl+qY2
and zX2=aYl-pYz.
Taking characteristic functions, we obtain
Without loss of generality, let us put p
b = a?/a; > 0. Hence from (3.3) we get
=q =a =
1, so z = b f 1 , where
= 1 implies the existence of E > 0
The continuity of # j ( s ) together with 1$~(0)
such that ($j(s)( > 0 for - E < s < E , j = 1, 2. Hence from (3.4) we obtain
$ j ( s ) # 0 for any s, -a < s < a,j = 1, 2.
Returning to the general formulation, from (3.1) we infer that
( 4 (sP))
~ =~(, ~
4Yq(sq)
d -
'is
which leads to
450
M.
P. Quine and
E. Seneta
If we now write
&=(-b)X2+aX1
and Yl=qX,-(-PIX,
and apply (3.5) mutatis mutandis, we obtain
#2
( t ) = #bJ'I(lrq)
1
(-tLZlb)
from which and ( 3 3 , putting y = bp/(aq), we get
13-61
Thus, if y2 > 1, we have
#5(t)= &(-t/IJ).
which is the characteristic function of N ( 0 , at). If 0 < y2 < 1, put 6 = l / y in
(3.6) to obtain
#i2 ( - t/S) =
$2
and proceed as for y2 > 1.
When y2 = 1, the case y = - 1 has already been dismissed since it corresponds to z = pb + aq = 0. The case y = 1 corresponds to Bernstein's formulation, and (3.6) (and the analogous equation for #, (t)) gives
that is, the distributions of X1 and X, are symmetric about 0, with real characteristic functions 4, (t) and 4, (t). Now, Kac [I71 initially assumes that X, and
X, are independent and symmetrically distributed about 0, and that
Yl = (COS
8) X1 + (sin 8) X2 and
Y , = (sin j?) X1-(cos 8)X,
are independent for every j?, and deduces that X I and X2 are identically normally distributed, as was to be, later, Bernstein's conclusion. In fact, his proof
uses the independence assumption only at 8 = n/4 and /?= 3 4 4 to show that
X, and X2 have the same (real) characteristic function # which satisfies
Using the continuity of 4, I# (5)1 < 1and # (0) = 1 , and the Cauchy method used
to deal with the f d i a r functional equation IC, ( 2 4 = t,b2(x), - cc < x < cc ,Kac
deduces 0 < 4 (t)< 1, and then # (5) = exp (kt2) for some k < 0.
We remark that the cases /3 = n/4 and jl = 371/4 are not in fact different,
since both assert the independence of X,+X2 and X, -X,, and hence to treat
our special case y = 1, one may 'tap in' directly to Kac's brief argument, condensing it a Iittle more. m
Another early paper (Lukacs [27]) also relates to Bernstein's Theorem,
although it is concerned with the characterization of the normal distribution
function from the independence of the sample mean R =
Xi/n, and sample variance SZ =
(Xi-B)'/(n - I), where Xi, i = 1, 2, ..., n, are inde-
z:=,
zl=,
.
,
Generalization of the Kac-Bernstein Theorem
pendently and identically distributed with finite variance. This characterization
was established under more stringent moment conditions by Geary [IS].
Quine [34] showed that the present Lemma 2 can be combined with Lukacs'
approach to prove the characterization with no moment assumptions whatsoever. In the case n = 2, if we write as with Bernstein, Yl= XI+X, and
Y, = X,-X,, we see however that S = Y,/2 and S2 = Y2'/2, SO that in this
case the characterization amounts to Bernstein's Theorem under the restrictive
initial-condition that X, and X, are identically distributed.
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[Zy R. G. L a h a w d V.
School of Mathematics and Statistics
University of Sydney
NSW 2006, Australia
Received on 12.1 .I999