Equivalence of two conditions on singular components

STATISTICS& PROBABILITY LETTERS ELSEVIER

Statistics & Probability Letters 40 (1998) 127-131

Equivalence of two conditions on singular components 1 M.S. Sgibnev* Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk 90, 630090 Russia Received September 1997; received in revised form February 1998

Abstract

We consider random walks with drift and establish the equivalence of two general conditions on the singular components of the underlying distribution and of the distribution of the first positive sum. (~) 1998 Elsevier Science B.V, All rights reserved

Keywords: Random walk with drift; First positive sum; Singular component; Renewal measure

Let {Xk }k~:~=lbe a sequence of independent, identically distributed random variables with common distribution F. We set So---0, S, = ~--~=l Xk, n ~>1. Suppose the random walk {Sn} drifts to - ~ , i.e. with probability one S, ~ - e ~ as n ~ c ~ . Define r/=min{n>~l: S , > 0 } , F+(A)=P(S, EA, r / < ~ ) , r/_ =min{n~>l: Sn~<0}, and F_(A) =P(S,_ EA). Let Fm*, m>l 1, denote the m-fold convolution of F and let F °* = E be the measure of mass 1 concentrated at the origin. In renewal theory and boundary-value problems related to random walks one oRen needs to use certain conditions on the absolutely continuous (or singular) component of either the underlying distribution F or the distribution F+ of the first positive sum. Specifically, it is well known that the distribution, D, of the random variable Mo~ = SUPn>~0Sn coincides, up to a constant factor, say c, with the renewal measure U+ = x--,~ Z-~m=0Fro* + generated by the defective distribution F+ of the first positive sum S,: D = ¢ U+; this can be deduced from the Wiener-Hopf factorization theory (Feller, 1966, Sections XVIII.3, XVIII.5 and XVIII.6). This fact enables us to apply numerous renewal theoretic results to the study of the asymptotic properties of the distribution of M ~ . A number of renewal theorems were obtained under the assumption that some convolution power of the underlying distribution has an absolutely continuous component (condition (F) below), which made it possible to get nice total variation bounds for the remainder terms. As pointed out by Rogozin (1976), condition (F) is even necessary if we wish to estimate the total variation of the remainder. Now, when such renewal theorems are applied in order to investigate the asymptotic properties of the distribution of the supremum, an inconvenience immediately arises, stemming from the fact that in this case the indicated

* E-mail: [email protected]. 1 This research was supported by Grant 96-01-01939 of the Russian Foundation for Fundamental Research. 0167-7152/98/$ - see front matter (~) 1998 Elsevier Science B.V. All rights reserved PII S 0 1 6 7 - 7 1 5 2 ( 9 8 ) 0 0 0 7 4 - 1

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M.S. S#ibnev I Statistics & Probability Letters 40 (1998) 127-131

absolute continuity condition is inevitably formulated in terms of the unknown F+ rather than in terms of the underlying distribution F (see Lotov, 1985, Theorem 2). In this note we prove the equivalence of two conditions of general type on the absolutely continuous components of F and F+. The absolutely continuous part of an arbitrary distribution G will be denoted by Gc and its singular component, by Gs: Gs -- G - Go. For a complex-valued measure x, we shall denote by ~(2) its Laplace transform: ~(2) = fR exp(2x)x(dx) for appropriate values of 4. In particular (Fm*)s(r) will stand for the Laplace transform at point r of the singular component (Fm*)s of the m-fold convolution Fm*. By the factorization theorem (Feller, 1966, Section XVIII.4),

1 - ~ ( 2 ) = [1 - i _ ( 2 ) ] .

[l - ~ + O ) ] ,

~2=0.

(i)

We now formulate two conditions on the distributions F and F+. Condition (F). For some convolution power m = re(F)/> 1 the distribution F m* has an absolutely continuous part, i.e. is spread out. Condition (F+). For some convolution power m = m(F+ ) ~>1 the distribution F_~* has an absolutely continuous part. Theorem 1. Suppose the random walk {Sn} drifts to - e c , 1 < i f ( r ) < c ~ , and ff(2)¢ 1 on {2 E C: ~R2= r} for some r > 0 . Then condition (F) with (Fm* )s(r)< 1 for some integer m =m(F)~> 1 is equivalent to condition (F+ ) with (F~_* )s (r) < 1, where m = m (F+ ) >>.1. Proof. Suppose (Fm*)s(r)>.l. By the analyticity of if(2) in the strip / / ( r ) = { 2 E C: 0<~N2~
H=L/a + Z

E BjkSjk* + R,

(2)

j=l k=l

where L is the restriction of Lebesgue measure to [0, co), a = fRx G(dx), Bjk are some constants whose exact values are irrelevant for our purposes, and the remainder term R satisfies the inequality f0 °~ exp((r

q)x)lRl(dx) < cxz,

where [RI stands for the total variation of R. We have (G m* )s (r - q) -- (F m*)s (r) < 1. Let H+ = ~m°°__0G~_*. Note that H_ is a finite measure. Under the hypotheses of the theorem, relation (1) holds for 0 ~<~2 ~
M.S. Soibnev I Statistics & ProbabilityLetters 40 (1998) 127-131

129

Hence Eqs. (1) and (2) imply /4+ = ( E -

G _ ) , H = K + ( E - G _ ) * R = K +R1,

(3)

where K is a g-finite absolutely continuous measure and

/0

exp((r - q)x)lRt [(dx) ~<[1 + G _ ( r - q)]-

/0

exp((r - q)x)lR[(dx)
(Actually, R = R [ ( _ ~ , 0 ) + R I t 0 , ~ ) - - r l + r2, the convolution G_ • r l is concentrated on ( - c o , 0), while only rl and G_ • r2 influence the behaviour of Rl on [0,oc).) We shall prove by contradiction that (F_~*)s(r)< 1 for some m = m(F+)>~1. Suppose (G~_*)s(r- q ) = (F~*)~'(r) ~> 1 for all m >/1. Choosing sets Am of Lebesgue measure zero such that (G_~*)s(Am) = (G~*)s(R), we obtain that, on one hand, (3O

JU ~

e(r-q~rn+(dx)~

,.=1.4.,

Z(G'~*)~[(r- q ) = e o m=l

and, on the other, by Eq. (3),

f

elr-q)xH+(dx)=

U.,C, A .,,

L

/0

e(r-q)XRl(dx)<~ e(r-qlxlRll(dx)
.,C,A o

This contradiction shows that there exists an integer m = m(F+)~>1 such that ( G +m* )s~ ( r- q ) -~ = t :F'* + ) ] ' ( r ) < 1. Conversely, let ( F ~ * ) s ( r ) < 1 for some m=m(F÷). Then, by the same Theorem 1 of Sgibnev (1991), we have l

m~

H+=L/a+ + Z Z B ~ k * + R+,

(4)

j=l k-I where a+ = f~ x G+(dx), B~ are some constants whose exact values are irrelevant for our purposes, and the remainder term R+ satisfies the inequality 0 ° exp((r - q)x)lR +I(dx) < co. Relations (1) and (4) imply

H = H _ *H+=KI +H_ *R+ =K1 + R z ,

(5)

where K1 is a a-finite absolutely continuous measure and f ~ exp((r - q)x)lR21(dx)<<.H_((-oo, O]). f0°~ exp((r - q)x)lR+l(dx)~ 1

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130

for all m ~>1. Choosing sets Bm of Lebesgue measure zero such that (Fm*)s(Bm)= (F m*)s(R), we obtain that, on one hand,

fU

e(r-q)XH(dx) >/ ~ mcc-,Bm

m=l

f[

e(r-q)xG~*(dx)

Um~l Bm

O<3

= Z(Gm*)s(r - q)=oc m=]

and, on the other, by Eq. (5),

fUZlBme(r-q~H(dx)=fuz, Bme(r-qlxR2(dx) .N
q)x

IR2i(dx) < oo.

The contradiction shows that there exists an integer m=m(F)>_.l such that (Gm*)s(r- q)-=(Fm*)s(r)< 1. This completes the proof of the theorem. [] We complement Theorem 1 with a similar result for random walks with positive drift. Theorem 2. Suppose EXI E (0,co). Then conditions (F) and (F+) are equivalent. Moreover, tf 1 <,ff(r)>.O, then the inequality (F m*)~(r) < 1 for some positive integer m = m ( F ) is equivalent to (F~*)s(r)
1 v(2)= [1 - F-~_---(2)J ' v+(2),

~R2=0.

Since F_ is an improper probability distribution, 1/[1 - i f _ ( 2 ) ] is the Laplace transform of some finite measure. Applying the aforementioned result of Rogozin (1976), we prove that (F)ce~(F+). As for the remaining assertion of the theorem, we only need to repeat with some modifications the reasoning of the proof of Theorem 1. We only note that in this case the set ~ does not contain any real roots of the characteristic equation; moreover, it may be empty. Remark 1. We have also proved that condition (F) with (F m*)s(r) < 1 is equivalent to the following, at first sight more restrictive condition: }-'~m~__l(Fm*)'Z(r)< ee. This observation is consistent with the fact established by Stone (1966) that if condition (F) is fulfilled, then the renewal measure U = z-.,m=0 x'-'°° F m* can be represented as a sum U' + U", where U' is absolutely continuous and U" is a finite measure, whence it follows that the singular part Us(R) is finite. Remark 2. Assertions similar to Theorems 1 and 2 are also valid for the distribution of the first non-negative sum.

Acknowledgements I wish to express gratitude to the referee for helpful comments and suggestions.

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References Feller, W., 1966. An Introduction to Probability Theory and Its Applications II. Wiley, New York. Van der Genugten, B.B., 1969. Asymptotic expansions in renewal theory. Compositio Math. 21, 331-342. Lotov, V.I., 1985. Asymptotics of the distribution of the supremum of consecutive sums. Mat. Zametki 38, 668-678 (in Russian). Rogozin, B.A., 1976. Asymptotics of renewal functions. Theory Probab. Appl. 21, 669--686. Sgibnev, M.S., 1991. Exponential estimates of the rate of convergence for the higher renewal moments. Siberian Math. J. 32, 657-664. Stone, C.J., 1966. On absolutely continuous components and renewal theory. Ann. Math. Statist. 37, 271-275.