theory of probability and its applications

Volume XXIII

T H E O R Y OF P R O B A B I L I T Y A N D ITS A P P L I C A T I O N S

Number 3

1978

ON ESTIMATES OF THE STABILITY MEASURE FOR DECOMPOSITIONS OF PROBABILITY DISTRIBUTIONS INTO COMPONENTS R. V. YANUSHKYAVICHIUS

{Translated by W. M. Vasilaky)

1. Introduction Soon after the appearance of H. Cramer’s and D. A. Raikov’s theorems on the decomposition into components of the normal and Poisson laws, which are representative of typical theorems on the characterization of distributions, the well-known theorem of N. A. Sapogov (see, for example, [1], Chap. VIII, § 2) appeared. The result of N. A. Sapogov appears to be the first in the theory of stability models characterizing distributions, which greatly developed the range of probability theory. The problem of the stability of decompositions into components was investi­ gated by a number of authors from the qualitative as well as the quantitative point of view. In its general form this problem can be formulated in the following way. Let there be two metrics fi and v in the space of probability distributions = {P}. For the distribution Q e & denote by K q = {O'} the set of all the components of the distribution Q. It is natural to expect (in light of the aforementioned theorems of H. Cramer and D. A. Raikov, as well as the much later results of Yu. V. Linnik and his successors) that the closeness of the distributions P = P' * P" and Q = O' * Q" will guarantee the closeness of the components P' and O', if the decomposition of O is of the required form. More precisely, fi(P, Q ) ^ e implies 0(P') = inf M P ', O'): O' 6 K q }< 8, where 8 is small, if e > 0 is sufficiently small. The results of this form are called stable decompositions of distributions into components. A measure of stability, introduced by V. M. Zolotarev [2], can be the quantity o ) = sup {©(P'): P' eK p,PeB „(e, O)}, where P^(e, 0 ) = {P: p(P, Q ) ^ e } . Considered most often is the case fi —v, wherein the notation /L,**.(£, O) = O) is used. Here the most important achievements were the following results (see [2 ]). 507

508

R. Y. YANUSHKYAVICHIUS

Theorem (Yu. V. Linnik and V. M. Zolotarev). For any metric /jl, realizing the weak topology (for example, the Levy metric L), and for any law Q e 8P, as e 0 , /M e>Q )“*0. Numerical estimates of the stability of decomposition of the normal law were investigated by N. A. Sapogov, V. M. Zolotarev, S. G. Maloshevskii (for the results of these investigations see [1]), G. P. Chistyakov [5], V. V. Senatov [7]; the Poisson law by O. V. Shalaevskii, Yu. Yu. Machis (see [1]), G. P. Chistyakov [5]; convolutions of the normal and Poisson laws by G. P. Chistyakov [8]; the binomial law by B. Ramachandran, Yu. Yu. Machis (see [1]), and O. Kallenberg [4]. Numerical estimates of the stability of components of simple indecomposable laws generated by law E and the binomial law B with generating function q+pz (for q = 1 we have B —E) were obtained by Yu. Yu. Machis in [3], [6 ], where it was established that PP(e, B) = (m - Vra2 -4 e )/2 , where p is the uniform metric, 0 ^ e ^ m 2/4 , m = min (p, q). Yu. Yu. Machis [6 ] also proved that for 0 ^ e ^ 1 the estimates e/2 ^ /M e, E) = 2e hold. The author [9] has determined that for equality (3) holds (see below). Taking this opportunity, we note that the stability estimates of decompositions of the degenerate law in the uniform metric were determined by Yu. Yu. Machis [6 ]; in [ 10] they are derived by a simpler and shorter method. Investigations of the stability of special type distributions were conducted by O. Kallenberg [4], where he established, in particular, that the decompositions of lattice laws with finite spectrum are stable in the uniform metric. The work of O. Kallenberg [4] has some inaccuracies, which are corrected in Section 5 of the present paper. The author expresses his deep gratitude to V. M. Zolotarev for posing the problem and for valuable advice regarding its solution. 2. Main Results and Commentary A natural question arises: in what way is /3r( •, G) dependent on G for some fixed metric r? Possibly it is stipulated by the decomposability of G, the extent to which the “tails” of distribution G diminish, etc. With the aim of analyzing this dependency, we examine the asymptotic behavior as e -> 0 of the stability measure of indecomposable laws G, with some restrictions on the “tails”, where the metric r is either the Levy metric L or the uniform metric p. We recall some concepts which are used in the theory of probability (see, e.g., [1]). The point x is called a point of increase of law F (x), if F(x + e ) —F(x —e ) > 0 for arbitrary e > 0. The set of all points of increase of the law F is called the spectrum S(F) of the law F. The set of all the discontinuity points of F is called the discrete spectrum D(F) of F. Denote lext F = inf S(F),

rext F = sup S(F).

ON ESTIMATES OF THE STABILITY MEASURE

509

Let ©^ be the class of probability laws G with bounded spectrums, such that if X G is a random variable with distribution function (d.f,) G, then m = min {P(XG = lext G), P(XG = rext G)}. The set of indecomposable laws in ©%we shall denote by © m. The class of all laws G e @m, m > 0, for which the points lext G and rext G are isolated points of 5(G), we shall denote by &m. Obviously, !&i = ©i. We note that the range of the values of m is the set [0 , §] U {1}. We shall give an example of indecomposable laws which belong to the class @m. Any probability law with a bounded spectrum, satisfying the condition D(G) = {lext G, rext G}, is indecomposable. The classic arcsin law is indecom­ posable (see [1], p. 73), and hence is an element of class @0. By Lemma 3.3.4 in [1] it is easy to show that for m > 3 the relation ©m = ©^ holds. A. M. Zubkov, the editor of this article, was able, by a simple device, to improve upon this assertation and prove that ©m = ©^ for m > 4 . Indeed, if @m 7* ©L then there exists a random variable X with d.f. H and non-degenerate random variables X \ and X 2 with d.f. Hi and H 2, respectively, such that H = Hi * H 2. Denote P(Xi = lext Hi) = x,

P(X2 = lext H 2) = y.

Since X i and X 2 are non-degenerate random variables, P(Xi = rext Hi) ^ 1 - jc,

P(X2 = rext H 2) ^ 1 - y.

From the relations P(X = lext H) = F(Xi = lext H i )P(X2 = lext H 2) = xy, P(X = rext H) = P(Xt = rext H i )P(X2 = rext H 2) ^ (1 - x ) ( l - y) follows the estimate m ^ min {xy, ( l- x ) ( l- y ) } . We note that the quantity min {xy, (1 —jc)(1 -y)} as a function of y is maximum for xy = (1 —x)(l —y), i.e., for y = 1 —x. Consequently, min {xy, (1 - x)(l - y)} ^ x (l -

jc).

Since x(l —x ) ^ \ , we have m ^ 4. It is not difficult to show that ©m is a proper subset of ©m for 0 < m ^ \ , i.e., for any m e (0, 4] there exists a law B m, such that B m e @m\©m- Indeed, let B m ’ be a two-point law with discontinuities z and 1 - z at points —1 and 0, and Bm be a law with discontinuities 1 —z and z at points 0 and 1. If B m = B m r * Bm, and Y is a random variable with d.f. B m, then, obviously, p (y

= - i ) = p ( y = i) = 2(1 - z),

P(Y = 0) = l - 2 z ( l - z ) . Solving the equation z (l —z) = m, we find that in order to satisfy the condition B m e ©m\@m it is sufficient to put z = (1 —Vl —4m)/2.

510

R. V. Y ANU SHK Y AVI CHIUS

Let us investigate the stability of the decompositions of the laws G e @m. It is interesting to note that there exists a G e @0, for which 0p(e ,G )S ?

e e (0,1),

for

i.e., the decompositions of G are unstable in the uniform metric. Indeed, let G be a continuous probability law from class @0- Let, in addition, Fnl(x) = G(x), and fO for x e (—o o ,- l /n], Fni(x) = \ \ for x s ( - l / n , l / n ] , [ l for x e ( l / n , oo), if n ^ 1. Obviously, L(Fn2, E) -» 0. Later on we shall need the following lemma. Lemma (V. M. Zolotarev). For any d.f. {Pk} and {Qk} the following inequality holds: L ^ ( * Pk, * \fc=i k=\

I L(Pk, Q k). fc=i

/

This lemma is derived, for example, in [1], Chap. VIII, § 1. Using Zolotarev’s lemma, we note that L(Fm G )^ L (F ni, G) + L(Fn2, F)-»0

for

n-+oo,

where Fn =Fn l %Fn2. It is known that convergence in the Levy metric is equivalent to convergence in the uniform metric, if the limiting distribution is continuous. Consequently, p(Fn, G)-»0. Furthermore, inf {p(Fn2, G '): G f e K G} = b The estimate @p(e9G) ^4 for e e (0,1) is obviously valid for any non-degenerate law G, whose set of components K G consists of continuous non-degenerate distributions. For m > 0 we were not only able to establish the existence of the stability of the decompositions of laws G e @m in the uniform metric, but also to obtain numeri­ cal estimates of the stability measure pp. Theorem 1. J/G e @m, m > 0, then there exists a constant e0 = eo(G) > 0, such that for 0 ^ e ^ eo ft>(e, G) ^ \(m - Vm2—4e),

(1)

where this estimate cannot be improved in class (£m. Theorem 2. I fG e £>m, m > 0, then there exists a constant e\ = e \(G) > 0 and a constant C > 0 (depending only on m), such that for 0 ^ e ^ e\ (2)

/3jl(s , G)^imax (3, m~x)e + Ce2. In the particular case G = E we have Theorem 3. For e e [0,1] the following equality holds: n

0 is stable in the uniform metric. Lemma 1. Let G e @m, m > 0. I f enl 0, n foo, then PP(en, G )i0

for

rcf00-

P roof of the lem m a . The stability measure p p(e9 G) obviously is mono­ tone non-decreasing as a function of e . Assume that p p{em GyiO for £„ jO, i.e., that decomposition of the law G e 8 m is unstable in the uniform metric. Since G is indecomposable, that means that there exist d.f. Fn with the property

(4)

p(Fn, G )^ e „ 10,

nf°°»

and d.f. F'n e KFn with the properties (5)

p(F'n,Er)7* o,

p(F'n, G 2) J 0,

nt°o,

for any real r and s, where E r{x) = E ( x - r)9 Gs = G(x - s). We shall prove that in this case (5) is satisfied for d.f. F„, satisfying the condition F'h * F 'n = Fn. Indeed, let X m X'n9X"n be random variables with d.f. Fn, F ’n, Fn, respectively. Note that if p(F£ F ro)-> 0 as «t°° for some r0, then p(F'n9 G -ro)-+0. Then in fact P(X" = r0) ^ l (rcfoo). Therefore F(X'n = X n- r 0)-* 1 (nf 00), and consequently p(F„, G_ro) -> 0 (nt°o), but this contradicts relation (5). If p(F„, Gs)-» 0 (nt°o), then for all sufficiently large n P (Xn = lext G + s) ^ m/2,

P(AT^ = rext G + s ) ^ m/2.

Since eM^ F(Xn < lext G) ^ P (X'n < - s ) P(X" = lext G + s)9 en ^ P(ATn > rext G) ^ P(X'n > - s ) V ( X l = rext G + j), then P(X„ < - s ) ^ 2en/m,

P (X'n > - s ) ^ 2e„/m,

i.e., p(F„, F _s)-»0 as nfoo, which is inconsistent with condition (5). Since (5) holds for any real values r and s9instead of F'n(x) we can consider F'n (x +med F'n). Thus the following assumption does not diminish the generality of the arguments. Let (4) hold, and let there be sequences {F„i}, {Fm2}, such that for any real numbers ru si, r2, s2 the following relations hold (nfoo): (6 )

p(F „i,F ri) > 0 ,

p(Fn2, GSl)> 0,

p(Fni, Gr2) > 0 ,

p(Fn2, F S2) > 0 ,

where Fni * Fn2 = Fn, and the median of Fni is equal to zero. To prove the falsity of premises (6) we require some additional notation and relations. Recall that the arithmetic sum of sets A and B of real numbers is the set A + F ={x: x = jci + *2, x\G A, jt2eF}.

512

R. V. YANUSHKYAVICHIUS

Denote F({x}) = F(x + O )-F(x),

D C(F) = {x: F({x}) £ c}.

We shall prove that for 0 < c S l D e(F) s D c/2{Fi )+ D c/2(F2) c D c*/4(F).

(7) Indeed,

F({x}) =

F1({s })F2({x - s}).

I s g D ( F i ): x —s g D ( F 2)

Further, since E

F i ({s})F2({x - . s} )< c/2,

s £ D c/ 2 ( F i )

E

F i ({s})F2({x —s}) < c/2,

s : x - s £ D c/ 2 (F2)

for x e D c(F), F ^sym u -s})

E s6

D c/ 2 ( F 1) : x -

sg

D c/ 2 ( F 2 )

S F ({ r} )-

Fl({s})F2( { x - s } ) -

E s £ D c/ 2 ( F i )

E

F i ({s})F2({x —s}) > 0.

s : x - s £ D c/ 2 (F2)

Consequently, x e D c/2(Fi ) + D c/2(F2). And, conversely, if x g D c/2(Fi ) +£>c/2(F2), then F({x}) ^ Z F i ({s })F2({x - s}) £ c2/ 4, s e D c/ 2 ( Fi ) : x —s e D c/ 2 (F2)

i.e., jc e D c2/4(F). The validity of (7) is proved. Since p(Fn, G ) ^ L ( F n, G), the theorem of Linnik-Zolotarev (see the intro­ duction) implies that relations (8)

min {L(Fnh F), L(Fnh G)}

0 as n -» oo

hold for all / = 1,2. From the sequence {Fnj}, j = 1, 2, we take two subsequences {Ffc;} and {F%} in the following way: F kl i =Fnkj if min {L(Fnki, E), L{Fnkj, G)} = L(Fnkj, E) and F \ = Fnii if min {L(Fnij, E), L(Fn,h G)} = L(F„„, G). Note that any term FPh j - 1,2, of the sequence {Fn/}, / = 1,2, is either an element of the subsequence {Flj} or of {F%}, j = 1, 2. We shall prove that if (8a)

min {L(Fku E \ L(Fku G)} = L(Ffcl,F )

for some sufficiently large fc, then for this k the relation min {L(Ffc2, F), L(Fk2, G)} = L(Ffc2, G) holds. Using relation (8) we see that, for any r > 0, there exists N r such that (8b)

min {L(Fny, F), L(Fn/, G )}^ r,

/ = 1, 2,

for ft ^iVT. Assume our assertion is wrong, i.e., assume that, for some s0> N r, L(FSq2, G )> F (F So2,F ). Then (8a) and (8b) imply that L(FSol, F ) ^ r ,

L(FSo2, E) < T.

513

ON ESTIMATES OF THE STABILITY MEASURE

Using Zolotarev’s lemma we obtain (8c)

L(FSo, E ) ^ L ( F SoUE )+ L(FSo2, E) < I t.

Since clearly for n ^ N T, en ^ r, we would have L(FSo, G) ^ r. The last relation and estimate (8c) are incompatible if G ^ E and r > 0 is sufficiently small. From the assertion just proved it follows that, for any sufficiently large n, there exists a kn such that F nl l * F 2n 2=Fkn.

(8d)

If, however, equation (8a) does not hold, i.e., if min {L(Fku E), L(Fku G)} = L(Fku G) for some sufficiently large k, then it is not difficult to prove that for such a k the relation min {L(Ffc2, E), L(Fk2, G)} = L(Fk2, E) holds. Indeed, in order to do this it suffices to assume that this last equation does not hold for some large k0 and to use estimates (4), (8) and L(Fk0, G * G ) ^ L { F koU G)+L{Fko2y G ). Consequently, for any sufficiently large fc, there exists an lk such that F h * F i 2 =Flk.

(8e)

As already pointed out, the sequence {Fnj}, y = 1,2, is completely given by the elements F nl h i - 1,2. Therefore, using relations (8d) and (8e) we note that instead of (8) it is sufficient to investigate one of the following two relations (since the other is treated in exactly the same way): L (F h , U F lu G ) - * 0,

L(F2m ,G)-*0, L (F \2, E ) ^ 0,

as n -> oo. Assume the first relation is satisfied. In order to simplify notation we shall investigate instead of the subsequence { F h } the sequence {F„i} itself, and instead of {F2n2} the sequence {Frt2}. Then L(FnU E) ^ 0,

(8f)

L(Fn2, G)

0.

The sets T)m/4(Fm), / = 1,2, consist of a finite number of points for any c > 0 (we exclude the trivial case D m/4(Fni) = 0 ) ; therefore we denote &ni

min D m/ 4 (F^),

bni max D m/ 4.{Fn*),

i

1,2.

We note that D m/ 2 (Fn) ^ 0 , (9)

a = min D m/2{Fn) = min D m2/16(Fn), b = max D m/2(Fn) = max D m2/16(Fn),

for sufficiently large h, where a = lext G, b= rext G. From the right-hand side of relation (7), for c = m i l we conclude that anl + an2 ^ a, bnl + b2n ^ fc, and from the left that ani + an2 ^ a , b nl + bn2^ b. Consequently, for sufficiently large n ( 10)

ani + an2 —a,

bni + bn2 = b.

514

R. V. YANUSHKYAVICHIUS

We turn to the direct proof of the falsity of premises (6). We shall prove that uni + bn 2 < b for sufficiently large n. In fact, the median of Fn\ is equal to zero, and therefore 8n S P (Xn < a) £ V(Xnl S 0)P(Xn2 < a) P(Xn2 b ) § P(X„, S 0)P (*„2> b ) ^ \ n X n2>b), where X nU X n2 are random variables with d.f. FnU Fn2, respectively. Consequently, an2 = a, bn2^ b . From (10) we conclude that artX^ 0 , bnl^ 0 . If, therefore, anl + bn2 = b, i.e., (by (10)) an\ = bnu then ani = bnx = 0. Hence an2 = a, bn2 = b. Therefore, (11)

^ P (Xni < 0)P(Xn2 = a),

Y{Xni < 0) ^ 4sn/m,

(12)

en ^ P(Xnl > 0)P(Xn2 = b \

P(Xni > 0) ^ 4sn/m.

Taking rx= 52 = 0 in relation (6), we find that (6), (11) and (12) are inconsistent. Denoting ani = SnU a n 2 = a + b, i.e., 0 , then (9) and (13) are inconsistent. Consequently, an\ + bn2< b, i.e., 5wX+ yn2< 0 for sufficiently large n. In accordance with (8), ani-»0, bni-*0, n -» 00, and from this and (10) we obtain that an2-»a, Z>n2-* 6. Therefore, for (14) On the other hand, d.f. G is continuous from the left, and, therefore, for any A > 0 there exists a k > 0, such that F ( b - K ^ X G< b ) ^ \ , where X G is a random variable with d.f. G. Choose A = m 2/ 32. The k corresponding to it we shall denote by *m. By (14) there exists an N, such that |S„i + yn2| ^ K m for n ^ N . Relations P (b - K m ^ X G< b ) ^ m2/ 32,

P (b - Km^ X n< b ) ^ m 2/ 16

and (4) are obviously inconsistent for n ^ N . The lemma is proved. We turn to the construction of numerical estimates of the stability measure 13P. Let p(F, G ) ^ e , F = Fi* F 2. We introduce the following notation: Fi( 0) = z u F2(a) = z4,

1 - Fi(0 +) = z2, F2(a+0) = u —z 5,

1■ ~ F2(b +0) = z3, F2(b) = l - v + z 6.

Since p(F, G) ^ e, the following chains of inequalities hold: l - « + c ^ l - F ( a + 0 ) ^ ( 1 - F i( 0 ) ) ( l- F 2(a+0)) = (1 -Z i)(l-W + Z 5), e ^ F ( a ) ^F i(0 )F 2(a + 0) = Zi(u - z5). We obtain two more systems of inequalities analogously:

515

ON ESTIMATES OF THE STABILITY MEASURE

In fact, it is sufficient to note that 1 - v + e ^ F 1(0+)F2(6),

e M 1 - F^O +M l ■-F2(b)),

e ^ (1 - F 2(b + 0))(1 - F 1(0)),

£ ^ F 2(a)Fi(0+).

Obviously, z, 1^0 for i = 1, 2, 3, 4. Solving the systems, we obtain (15)

Zi = z* = 2(m-

y/u^-Ae),

z2^ z* = i { v -

—4e),

or (16)

Zi ^ 2(w + V«2-4 e),

z2^ 2(u + y/v*-4e).

By Lemma 1, we conclude that (16) does not hold for sufficiently small e > 0. From estimates (15) we obtain that p(Fu E ) ^ 2(m - \ l m 2~4e) and, furthermore, (17)

F2( a ) ^ z t

1—F 2(ft + 0 )^ z * .

Since G( jc) + e ^ F (x )^ F i(0 + )F 2( x ) ^ ( l - z * ) F 2(x), for a < x (18)

we obtain

F2(x )S G (x ) + e + Z* (- t- --- = g(Jc) + Z2*. 1“ *2

Analogously, (19)

F 2( * ) a G ( * ) - r f .

From estimates (17)-(19) it follows that for sufficiently small e > 0 p(F2,G ) g |( m - V m 2-4 e). The validity of relation (1) is proved. In order to prove that the estimate of ($p(e, G) cannot be improved, it is sufficient to prove the existence of a law Q e (£m, such that PP(e, Q) = \ ( m - ' J m 2-Ae). This follows from a simple example derived by Yu. Yu. Machis in [3]. Indeed, let Q be a two-point law with discontinuities m and M at points 1 and 0, respectively, where m Furthermore, let Fi) = (1 —z*) + z*t,

il/(t;F2) = (M + Z o ) + ( m - z * ) t,

where z* = (m —Vm2~4e)/2; these are the generating functions of Fi and F2. It is not difficult to ascertain that the best approximation of Fi and F2 by laws from class K q is p(F\, E) = Zo, p(F2,Q ) = z t We note also that p(Fi * F 2, Q) = e, i.e., Fi *F2e B p(e, Q). Theorem 1 is proved in full. 4. The Proofs of Theorems 2 and 3 First we shall prove the validity of estimate (2). Let L(F, G) ^ e, F = Fi * F 2, and let the median of Fi be equal to zero. Furthermore, let X, X u X2 be random variables with d.f. F, F u F2, respectively. From the Linnik-Zolotarev theorem it

516

R. V. YANUSHKYAVICHIUS

follows that there exist non-decreasing functions 5(e), e e (0,1), 5(e)j0 as e jO, such that Fi(-8(e)) ^ 5(e), 1 -F i(5 (e)) ^ 5(e), (20 ) P(X2 - 5(e)) i?i>-5(e). The following chain of inequalities is valid: e ^ V ( X > b + e ) ^ P ( X 1>8(e) + e)F(X2^ b - 8 ( e ) ) , i.e., P (X i> 5(e) + e )^ e /(t; —5(e)), G(x + e) + e ^ (1 —e/(v —8(e)))F2(x —5(e) —e). We have, therefore, proved that (21)

G{x + 5 -f 2 e) + v —5 F2(x) = G(x + 5(e)-f2e) + e~ v —5 —e

From the conditions of the theorem follows the validity of the relations (22) (23)

P ( - f i( e ) S X ,S 0 ) s £ - S ( « ) , G{a) = G(a'),

G(b) = G(b')

for some a' > a, b' < b, where by a and b we denote as above lext G and rext G. Consequently, P (—S(e ) S X j g 0)P(Z>' + S(e) + e ^ X 2< b - e ) ^P (b' + e S X < b - e ) ^ 2 e , P (b' + S(e) + e ^ X 2< b - e ) i

4e 1 -2 5 (e )

From this, recalling (21), we have for sufficiently small e > 0 P(X2 g b - s ) S v -

v —8

1—2S

e.

Consequently, the following estimate is valid: (24)

, , l + 4 u -7 S + 2S P (X2^ b - e ) ^ v — ;-----— — r r ^ e (® -S )(l-2 « )

The fraction in (24) we shall denote by h(e, 8, v). Further we note that (25)

P ( 0 S X i< a ( e ) ) s J - 5 ( e ) ,

F2(a + 8 ( s ) ) ^ u -8 (e ).

Just as from the last estimate, it follows that F i(-5 —e + 0) ^ e/(u —5), then l - G ( x - £ ) + £ £ l - F ( x ) § = ( l ----£ - ) ( l - F 2(x + a + e)). This brings us to the inequality (26)

u —8 —e

ON ESTIMATES OF THE STABILITY MEASURE

517

From (23), (25), (26) we have F2(a + e ) ^ u -eh (e, 8, u).

(27)

Taking into account (24) and (27), we obtain £ = (1 - F & e + 0))(1 ~ F 2(b - e ) ) ^ ( 1 - F i ( 2 e +0 ))(v-eh(e, 8, u)),

e ^ F i ( —2e)F2(a + s ) ^ F i ( —2e)(u —eh{e, 8, w)).

This means that (28)

F t(-2 e )^

u —eh(e, 8, w)’

l- F x ( 2 e +0):

v - e h ( e , 8, u)

The explicit expression of the stability measure for G e £>i, i*e., the case m = 1, is considered in Theorem 3. If O C ra ^ l, then from the given arguments it is apparent that L(FU E ) ^ e / ( m -eh (e, 8, m)).

(29)

Having (20), we have obtained (28). Denoting 8 in estimates (20) by 8m from estimates (28), we conclude that 8n+1 = e /( m - h ( e , 8n, m)). We shall not investigate in this paper the convergence of the sequence {8n} (in the case of its being monotone non-increasing) nor seek the limit 8 = lim 8n. We only note that in the last case 8 must satisfy the equation 8 = e/(m —eh(e, 8, m)).

(30)

Let 50(e ) be a root of this equation, 8o(e)^(l + 5m)e/m for sufficiently small £ > 0 . If such a root does not exist or the sequence {£„} is not monotone non-increasing, then we shall choose 80(e) = (l + 5m)e/m,

0o(e, m) = h(e, 80, m).

We have for a + 3e ^ x ^ b - 3e G(x + s) + e ^ F l(2e+0)F2( x - 2 e ) ^ ( l ----- W x - 2e), \ v-eO o' F2(x ) = G(x 4- 3e) + e/(v —(0o + 1)£), 1 - G(x - e) + e

(1 —Fi(—2 e ) ) ( l- F 2(x+2e)) s ( l ------e— \ l - F 2(x + 2e)),

F2{x ) ^ G(x - 3e) ~ e/(u - (0O+ l)e). Since e ^ F ( a - e)^Fi(0+ )F2(a - e ) ^ 2F2( a - e ) , £ = (1 Fx(0))(l - F2(b + £ + 0)) ^ |(1 - F2(b + £ + 0)), F2(a —e ) ^ 2e, 1 —F2(b + e + 0) ^ 2£.

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Thus we have shown that L(F2, G) ^ max (e/(m - (0O+ l)e), 3e). From this and (29) follows estimate (2). The proof of Theorem 2 is complete. We do not investigate in this article the convergence of {Sn}, nor examine the solution of equation (30), but limit ourselves to relation (2). This is not only because the calculations are inconvenient. Indeed, if for m e (0, mi) we obtained some estimate, then an ra2 > 0 exists such that for m e (0 , m2) a more accurate estimate is valid. For the proof of this assertion it is sufficient to set in (28) 8 = 80 and note that P(—2s = X i ^ 2e) ^ 1 —2e/(m —sh(s, 80, m)). Using this estimate instead of (22) and (25) in arguments (23)-(27), we obtain that for small s > 0 (31)

Fi(—4 e ) ^ s/(m

— e K( e ,

m)),

1 —Fi(4e + 0 ) ^ s / ( m —s k ( s , m)),

where k ( s , m )< h(s, 80, m). For m < \ , (31) is obviously a better estimate than (28). Repeating this process again, for m < iw e obtain an estimate more accurate than (31), etc. We turn to the proof of Theorem 3. If 0 is the median of the law F ' KF, then, obviously, L(F'(x),E(x + @))mh and, consequently, ^ l = 2 - Since (3L(- ,E ) is a non-decreasing function, (3L,(e ,E ) ^ 2s (see the Introduction), it is sufficient for us to prove that @L(s, E ) ^ 2 s for 0 ^ e ^ 4. Consider the independent random variables X \ and X 2 (for e e [0 , 4]): P(Xi = -2 e ) = P(X1= 2e) = 2e,

P(Xt = 0) = 1 - 4e,

P(X2 = ~e) = P(X2 = e) = l If Fi and F2 are d.f. of the random variables X \ and X 2, then it is not difficult to be convinced that inf {L(FU E '): E 9e KE} = 2e,

inf {L(F2, E'): E' e KE} = e,

i.e., Pl (s 9E) ^ 2s, where L(F, E) = e, if F = Fx * F2. The theorem is proved. 5. A Note on Kallenberg’s Paper [4] In [4] there are some insufficiently justified conclusions on the upper estimate of the stability measure f$r(e, G), where G is a lattice distribution with a finite spectrum. Let 0(F ') = inf {r(F\ G'): G ' g K g }, yr(e, G,&) = sup {©(F'): F ' e K F, F e & D B r(e, G)}, where r is a Levy metric or the uniform metric, and & is a uniformly bounded family of real-valued lattice probability laws. In the case G e & 9from the results of O. Kallenberg [4], the estimates of yr follow automatically; nevertheless, for estimates of the stability measure an additional argument is required, showing that the class 3F is sufficiently large in the class of probability laws B r(e, G). We

ON ESTIMATES OF THE STABILITY MEASURE

519

shall prove this in Lemma 2. Later, it will be expedient to make use of the notation u —G({0}). We note also that the condition (32)

^ £ { H :S ( H ) £{0,1, •••,/* } , {0,N} 0

pL(e, G )^-yL{(X2 + u)e/u, G, &) + 6e/u, PP(e, G) ^ yp((S(N +1) + u)e/w, G, 2F) + 6e/u. P r o o f . A s above, let L ( F G ) ^ e, F = Fi * F 2; random variables X, X i, X 2 have d.f. F, Fi, F 2, respectively. We note, recalling (32), that F ( - e ) ^ e , 1 F (N + e + 0) ^ e. We translate the random variable X u such that

sup {y: Fi(y)^Ve} = 0. From this F(X\ ^ 0) ^ Ve; consequently, sfl P(X2< - e ) S P(Xi S 0)P(J*T2< - « ) S F ( - g ) S g , « - e S P ( X i < 0 ) + P(X2S e ) s=2Vg + P (—e = X 2= e). Therefore P(—e = X 2= e)> u/2 for sufficiently small e > 0 . Analogously, u —e = P{Xx< - 2 e ) + P ( - 2 e ^ X 1^ 2 e ) + V(X2< - e ) ^ 2>J~e+ P (-2 e ^ X x^ 2 e ) . Let us introduce into our examination integer-valued random variables X*, X *, X 2 with d.f. F*, F * , F*, respectively, connected by the equality X * = X * + X and defined by the formulas x * J [ X i + 3e] ’ l 0

if - 3 e £ X , £ N + 3e, otherwise;

by [a] we mean the integer part of a. Further, let n - 3e < x S n +1 - 3e, n an integer. Then P(Xf < r ) S P ( X | < n +1 - 3e) (33)

3SP(X? S n ) + P (X i< -3 s ) S P ( X f < x + 3e) + 2e/u,

since kuP(Xi < —3 e )^ P (A T < -e )S e . Assuming i # j, we have for all integers ngo V (n + 3 e < X i< n + l - 3 e )P (-2 s Si X, =S2e) S F (n +1 —e) —F(n +e + 0 )S 2 e, i.e., P(n + 3e N + e ) Si e, P ( X > N + 3 e )^ 2 e /u .

520

R. V. YANUSHKYAVICHIUS

Using these estimates, it is not difficult to prove that F}(jc) ^ F ? (x —3e) —6e/u. From this and (33) it follows that L(Fh F f ) ^ 6e/u, i = 1,2. By Zolotarev’s lemma we obtain that L(F, F*) ^ 12e/u. It remains, therefore, to consider the case of the uniform metric. We define the random variable X f , j = 1, 2, as follows:

As in the case of the Levy metric, it is not difficult to prove the validity of relation p(Ff , Fj) ^ 6e/u, j = 1, 2. Noting that p(F, F*) ^ P(X * X*) ^ P(Xi 5 * Xf ) + P(X2 * X * )

it is easy to obtain the estimate p(F, F*) ^ 8(N +1 )e/u. The lemma is proved. Using the estimates of yr (see [4]) and Lemma 2, we obtain without difficulty, the estimate of the stability measure fir. Received by the editors June 24, 1977

REFERENCES [1] [2] [3] [4] [5]

[6]

[7] [8]

[9]

[10]

Yu. V. Linnik and I. V. O stro vsk ii , The Decomposition of Random Variables and Vectors, American Math. Soc., Providence, R.I., 1977. V. M. Z o l o t a r e v , On the problem of stability of the decomposition of the normal law into components, Theory Prob. Applications, 13 (1968), pp. 697-700. Yu. Yu. Ma c h is , On the stability of the decomposition of a two-point distribution law , Lith. Math. Trans., 13, 4 (1973), pp. 131-138. (In Russian.) O. Ka l l e n b e r g , Factorization stability for polynomials and finitely supported probability measures, Chalmers Inst. Techn. and Univ. of Goteborg, 7 (1971), preprint. G. P. Ch is t y a k o v , On the accuracy of estimates in a theorem on the stability of decompositions of the normal distribution and the Poisson distribution, Theory of Functions, Functional Analysis and its Applications, 26 (1976), pp. 119-128. (In Russian.) Yu. Yu. Ma c h is , On the stability of decompositions of the unit distribution function, Theory Prob. Applications, 14 (1969), pp. 688-690. V. V. Se n a t o v , On the refinement of estimates of stability for H. Cramer's theorem, Notes of Scientific Seminars LOMI, 61 (1976), pp. 125-134. (In Russian.) G. P. Ch is t y a k o v , On the numerical stability of decompositions of some probability distributions, Abstract of Reports of 2nd Vilnius Conference in Prob. Theory and Math. Statistics, Vilnius, 1977, vol. 2, pp. 210-211. (In Russian.) R. V. Y a n u s h k y a v ic h iu s , On the stability of decomposition of the unit probability law in the Levy metric and the n-dimensional Poisson distributions in the uniform metric, Lith. Math. Trans., 17, 3 (1977), pp. 200-202. (In Russian.) R. V. Y a n u s h k y a v ic h iu s , Estimates in a theorem on the stability of decompositions of an n-dimensional Poisson distribution, Student Scientific Papers, no. 2, Vilnius National Univ. Pub., Vilnius (1977), pp. 1-13. (In Russian.)