Abstract. Let X be a random variable with probability distribution PX concen- ... The second part is devoted to an example of X with a singular distribution such.
ON SMOOTH BEHAVIOUR OF PROBABILITY DISTRIBUTIONS UNDER POLYNOMIAL MAPPINGS 1
1
tze , Yu. V. Prohorov , V.V. Ulyanov F. Go
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Universitat Bielefeld Steklov Mathematical Institute, Moscow Moscow State University Abstract. Let X be a random variable with probability distribution PX concentrated on [?1; 1] and Q(x) be a polynomial of degree k 2. The characteristic function of a random variable Y = Q(X ) is of order O(1=jtj1=k ) as jtj ! 1 if PX is suciently smooth. In comparison for every " : 1=k > " > 0 there exists a singular distribution PX such that every convolution PXn? is also singular while the characteristic function of Y is of order O(1=jtj1=k?" ). While the characteristic function of X is small when "averaged" the characteristic function of the polynomial transformation Y of X is uniformly small.
1. Introduction
Let Q(x) denote a polynomial of degree k 2:
Q(x) = k xk + k?1 xk?1 + + 1 x:
(1-1)
Let X be a random variable with a probability distribution PX concentrated on [?1; 1]. The characteristic function gY (t) of a random varable Y = Q(X ) , i.e. gY (t) = E expfitY g is of order O(1=jtj1=k ) as jtj ! 1 when PX is suciently smooth (see Remark 1.3 below ). In the rst part of the paper we extend this result (see Theorem 1.1 and its Corollaries) and obtain in particular that under some conditions on the characteristic function gX we have
jgY (t)j = O(1=jtj k1 ?" ) as jtj ! 1;
(1-2)
where 0 " < 1=k. The second part is devoted to an example of X with a singular distribution such that any nite number of convolutions with itself remains singular (see Lemma 3.3). Furthermore, since (1-2) holds for characteristic function of Q(X ) (see Corollary 4.3), its behaviour is similar to the case where X has a smooth distribution. 1 Supported 2 Supported
by SFB 343 at Bielefeld, FRG. by SFB 343 at Bielefeld, FRG, and Ministry of Education, Science and Culture, Japan. Typeset by AMS-TEX 1
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In the following we shall introduce some notions concerning the smoothing of distributions. We introduce a smoothing random variable V independent of X with probability density
sin(x=2) 2 1 pV (x) = 2 x=2 ; ?1 < x < 1;
and characteristic function
gV (t) = Fix any T 1 and put Then
1 ? jtj; 0;
jtj 1; jtj > 1:
WT = X + V T ?1 :
gWT (t) =
gX (t)(1 ? jtj=T );
jtj T; jtj > T:
0; Since WT converges weakly to X as T ! 1 , we have for any real t
E eitQ(WT ) ! E eitQ(X ) as T ! 1:
(1-3)
Moreover for any positive number we get
P fjX + V T ?1 j > 1 + g P fjV j > T g ! 0 as T ! 1: Hence
Z jxj>1+
eitQ(x) pWT (x)dx ! 0 as T ! 1;
(1-4)
where pWT (x) is a probability density of WT . Combining (1-3) and (1-4) we conclude that in order to prove jgY (t)j A(t) it suces to show lim lim jJ (t; T )j A(t); (1-5) !0 T !1
where
J (t; T ) =
Z
jxj1+
eitQ(x) pWT (x)dx:
2. General Results and their Proofs Recall that gX (t) = E expfitX g. Put for T > 0 X (T ) =
ZT
?T
jgX (t)j dt:
3
Theorem 1.1. Let X be a random variable such that jX j 1. Assume that there exist a non-decreasing positive function (t) on (0; 1) and a constant " : 0 " < 1=k, such that for all t 1 and b 1 X (bt) b" (t): (2-1) Then for jtj 1 we have jgQ(X ) (t)j C (jtj) jtj?1=k ;
(2-2)
where the constant C does not depend on t (see (2-5) and (2-10)).
Proof . By the remark in (1-5) it is enough to prove lim lim jJ (t; T )j C (jtj) jtj?1=k : !0 T !1 We write
(2-3)
!
Z
ZT 1 itQ ( x ) J = J (t; T ) = e e?i x gX ( )(1 ? j j=T ) d dx 2 jxj1+ ?T ! Z Z 1+ T 1 eitQ(x)?i x dx d : gX ( )(1 ? j j=T ) = 2 ?(1+) ?T (2-4)
The inner integral in the right-hand side of (2-4) can be estimated by the following
Lemma 1.2. (Vinogradov (1985), sec. II, ch. 2, Lemma 4, p. 199). be a polynomial of degree k 2 with real coecients R(x) = rk xk + rk?1 xk?1 + + r1 x and Then
r = 1max jr j: j k j
Z 1 2 iR ( x ) e dx min(1; 32 r?1=k): 0
We put
I = and
Let R(x)
Z (1+) ?(1+)
eitQ(x)?i x dx
= 2max j(1 + )j j j; j k
where j are the coecients of the polynomial Q(x) (see (1-1)).
(2-5)
4
According to Lemma 1.2 we have
Z 1+ Z 0 + jI j 0 Z 1 ?(1+) = (1 + ) expfitQ((1 + )y) ? i (1 + )ygdy 0 Z1 + (1 + ) expfitQ(?(1 + )y) + i (1 + )ygdy 0 ! 1=k
where
jtj
1
(1 + ) D min
; ; 1=k j ? 1tj1=k
(2-6)
1=k
D = 64 (2jtj1)=k :
We proceed now to bound J (see (2-4)). Put d = + j 1j: Let J;0 be the part of the integral J for values of satisfying
j j 2 d jtj; and J;m, m 1, be the part of J for values of satisfying 2md jtj < j j 2m+1d jtj: Then by (2-6) we get 2 jJ;0j
!
Z j j2djtj
jgX ( )j d (1 + ) D ?1=k
= (1 + ) D ?1=k X (2djtj) and for m = 1; 2; : : : we have 2 jJ;mj
Z
(2-7)
!
j g ( )j d m (11=k+ ) D X (2 ? 1) ( + j 1j)1=k 2m djtj " > 0 and integer
we have where = 0:027.
m m(") = ?1 ln(2=(3" ? 1))
(4-1)
'm (T ) 53em T " ;
(4-2)
Corollary 4.2. Under the conditions of Theorem 4.1 we have for all T 0
ZT
?T
jLm(t)j dt 55 em T " :
(4-3)
Proof of Corollary. Inequality (4-3) immediately follows from the obvious estimate
ZT
?T
jLm(t)j dt 'm(T + 0:5) 'm (T ) + 2:
Corollary 4.3. Let Q(x) be a polynomial of degree k 2 and " : 0 < " < 1=k. Let
X denote a random variable with singular distribution and characteristic function Lm(t) for any m satisfying (4-1). Then
jgQ(X ) (t)j = O(jtj"?1=k ) as jtj ! 1:
Proof of Corollary 4.3 immediately follows from Theorem 1.1 and (4-2).
We shall give a proof of Theorem 4.1 right after Lemma 4.4. First, we x an integer N 1 and consider a probability space ( ; F ; P ) with
consisting of all sequences = (N ; N ?1 ; : : :; 0), such that j assumes values 0; 1 and 2. Furthermore, F denotes the set of all subsets of and for each sequence we put P fg = 3?N ?1 .
Lemma 4.4. The random variable Z on ( ; F ; P ) de ned as the number of changes
in a sequence N ; N ?1 ; : : : ; 0 has a binomial distribution with parameters N and 2=3, i. e. N 3?N 2k ; k = 0; 1; : : :; N: P fZ = kg =
k Proof of the Lemma 4.4. follows by elementary calculations.
Proof of Theorem 4.1. Let us x any N 2. We represent any integer j : 0 j 3N +1 ? 1, in the form (cf.(3-6a)) j = N 3N + N ?1 3N ?1 + + 0 ; (4-4)
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for i = 0; 1 or 2. Let 1 (j ) denote the number of changes in a sequence N ; : : :; 1 in (4-4) and let (j ) be de ned as before (see Corollary of Lemma 3.5; recall that (j ) is de ned for j 9). It is clear that for j : 9 j 3N +1 ? 1
(j ) 1 (j ) (j ) + 1:
(4-5)
By (4-5) and Corollary of Lemma 3.5 we derive for m 1
Lmj em(1?1 (j)) :
(4-6)
Since for 0 j < 9 we have 1 (j ) 1, inequality (4-6) holds for all integers j : 0 j 3N +1 ? 1, and m 1. It follows from Lemma 4.4 and (4-6) that for N 2 we have
'm (3N +1 ? 1) 2 em
+1 ?1 3NX
j =0
e?m1 (j)
1 03N+1?1 X e?m1 (j) 3?N ?1 A : = 2 em 3N +1 @ j =0
(4-7)
Assigning to each j : 0 j 3N +1 ? 1; a probability 3?N ?1 we may consider a sequence N ; N ?1 ; : : :; 0 as a sequence of independent random variables to which we can apply Lemma 4.4. Hence 1 (j ) has a binomial distribution with parameters (N ? 1; 2=3). Therefore the right-hand side of (4-7) can be rewritten in the following form 2 em 3N +1
NX ?1
e?mr N ? 1
?N +1 2r
3 r r=0 = 18 em (1 + 2e?m )N ?1 :
(4-7a)
For given " : 1 > " > 0, we de ne a real number m" so that 1+2e?m" = 3". Thus, for m m" = ?1 ln(2=(3" ? 1)) we get from (4-7a) that
'm (3N +1 ? 1) 18 em 3(N ?1)" : If then
3N +1 ? 1 T < 3N +2 ? 1;
'm (T ) 'm (3N +2 ? 1) 18 em 3N" < 18 em (T=2)": (4-8) Recall that (4-8) has been proved for N 2, i. e. T 26. For 1 T < 26 we have 'm (T ) 1 + 2 T 53 T ": Combining (4-8) and (4-9) we get (4-2).
(4-9)
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Finally we would like to note that this paper was motivated by the classical work of Wiener and Wintner (1938).
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