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A property of the Fourier transform of probability measures on the real line related to the renewal theorem.

1. Introduction and the main result. Let F be a probability measure on R, [F.sup.n*] be its n-fold convolution. We assume m = [[integral].sup.[infinity].sub.-[infinity]] x F (dx) [member of] (0, [infinity]) since it is the most interesting case in the renewal theory. We denote the Fourier transform [[integral].sup.[infinity].sub.-[infinity]] [e.sup.izx] F(dx) by [phi](z).

If A [subset] R is a Borel set and x is a real number, the sets -A, xA, and x + A are defined in the obvious way by symmetry, expansion (or contraction), and translation. We say that F is periodic with the period [omega] > 0 if [omega] is the greatest positive number such that F is supported on [omega]Z. If such [omega] does not exist, we set [omega] = 0.

Let [{[X.sub.n]}.sub.n=0,1,...] T be a sequence of independent random variables with the common distribution F and set [S.sub.0] = 0, [S.sub.n] = [[summation].sup.n.sub.k=1] [X.sub.k]. Thus [{[S.sub.n]}.sub.n=0,1,...] forms a transient random walk on R going to +[infinity]. We also set, for any interval I, U(I) = [[infinity].summation over (n=0)] [F.sup.n*](I), which is the 0-resolvent measure for the random walk {[S.sub.n]}.

As the renewal theory (see [5], [1], [4], [2]) reveals, there are following cases: If [omega] > 0, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for any interval I where [absolute value of I] denotes the length of I. In any case, [lim.sub.x[right arrow]-[infinity]] U(x + I)= 0.

For this, Feller and Orey [6] give a rather short proof, which is based on the symmetrized measure V defined by V(I) := 1/2 (U(I) + U(-I)). Let us review very briefly their method in the case [omega] = 0. They prove

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and make use of transience of {[S.sub.n]}. The proof of (1) relies on the following weak convergence (2) of a family of finite measures. Let [m.sub.s](d-z) = [1/[1 + [z.sup.2]]] [Real part] (1/[1 - s[phi](z)]) dz and m(dz) = [[pi]/m] [[delta].sub.0] (dz) + [1/[1 + [z.sup.2]]] [Real part] (1/1 - [phi](z)) dz, a mixture of a point mass and an absolutely continuous one. It is shown in [6] that

(2) [m.sub.s](dz) [??] m(dz) as s [right arrow] 1 - 0

if [omega] = 0,where [??] indicates weak convergence.

Remark 1.1. It holds [Real part](1/1 - s[phi](z)) [greater than or equal to] 1/2 and [Real part] (1/1 - [phi](z)) [greater than or equal to] 1/2. Indeed, w = [1/1 - z] maps the unit disc {z [member of] C| [absolute value of z] [less than or equal to] 1} conformally to {[infinity]} [union] {w [member of] C [Real part]w [greater than or equal to] 1/2}. An extreme example can be found in Example 2.1 in Section 2, although in the case [omega] > 0. As we make s [right arrow] 1 - 0, the density [1/[1 + [z.sup.2]]] [Real part] ([1/[1 - s[phi](z)]]) of [m.sub.s](dz) produces an acute thorn, which will form a point mass of m(dz). Some examples of thorns are observed in Examples 2.1 and 2.2.

Remark 1.2. At every z such that [phi](z) = 1, we can prove [phi]'(z) = im, whether [omega] = 0 or [omega] > 0. Hence 1/[1 - [phi](z)] has only isolated singularities, which forms a negligible set, so that the measure [1/[1 + [z.sup.2]]] [Real part] (1/[1 - [phi](z)]) dz is well-defined. The set of singularity/ is 2[pi]/[omega] Z if [omega] > 0 while z = 0 is the only singularity if [omega] = 0.

Remark 1.3. In many cases, [Real part] (1/[1 - [phi](z)]) behaves rather mildly near a singularity a:If [[integral].sup.[infinity].sub.-[infinity]] [[absolute value of x].sup.1+[delta]] for some [delta] [member of] (0,1), then [Real part] (1/[1 - [phi](z)]) = O([[absolute value of z - a].sup.-1+[delta]]) as z [right arrow] a. This is an exercise involving the expansion [phi](z) = 1 + im(z - a) + O([absolute value of z - a].sup.1+[delta]]).

In this note, we are motivated to understand (2) deeper and aim to establish the following result which includes also the case [omega] > 0.

Theorem 1.1. For any [alpha] > 0 and 0 [less than or equal to] s < 1, let [m.sup.([alpha])] (dz) = 1/1 + [[absolute value of z].sup.[alpha]+1] [Real part] (1/1 - s[phi](z)) dz.

Then the family of finite measures [m.sup.([alpha]).sub.s] (dz) converges weakly, say, to [m.sup.([alpha])] (dz):

(3) [m.sup.([alpha]).sub.s] (dz) [??] [m.sup.([alpha])] (dz) as s [right arrow] 1 - 0.

Moreover, if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The proof will be given in Section 3.

Theorem 1.1 gives an explanation for the roles played by the assumption [omega] = 0 and the factor 1/(1 + [z.sup.2]) in (2). Moreover, if we make [alpha] [less than or equal to] 0 in the expression of [m.sup.(a).sub.s] (dz) and [m.sup.([alpha])] (dz), we easily deduce that they are infinite measures from Remark 1.1. In this sense, the statement of Theorem 1.1 is exhaustive concerning the value of a that enables weak convergence.

2. Examples. In this section, we investigate several examples of F and [phi]. Let [alpha] > 0.

Example 2.1. If [omega] > 0, [phi](z) is a periodic function with the fundamental period 2[pi]/[omega]. The simplest case among them is F(dz) = [[delta].sub.m] (dz): the unit mass at m = [omega] > 0. In this case, [phi](z) = [e.sup.imz] and [Real part] (1/[1 - [phi] (z)]) = 1/2. The limit measure is hence [m.sup.([alpha])] (dz) = [[summation].sub.n[member of]Z] [pi]/m[(1 + (2[pi][absolute value of n]/m).sup.[alpha]+1]) [[delta].sub.2[pi]n/m] (dz) + 1/2(1 + [[absolute value of z].sup.[alpha]+1], dz. Next let us observe how [m.sup.([alpha]).sub.s] (dz) produces a series of acute thorns at each point in 2[pi]/m Z. We have

[Real part](1/[1 - s[phi](z)]) = [Real part] (1/[1 - s[e.sup.imz]]) = [1/2] + [(1 - [s.sup.2])/2]/[(1 + [s.sup.2]) - 2s cos(mz)].

Here the first term corresponds to the absolutely continuous part of [m.sup.([alpha])] (dz). In a neighborhood of z = 2[pi]n/m, where n is an integer, it holds

cos(mz) = cos(m(z -- 2Ln/m)) = 1 - (1 + o(1)) [1/2] [m.sup.2][(z - 2[pi]n/m).sup.2]

and hence

[(1 - [s.sup.2])/2]/[(1 + [s.sup.2]) - 2s cos(mz)] = (1 + o(1)) [1 - s/[(1 - s).sup.2] + [m.sup.2] [(z - 2[pi]n/m).sup.2]

as s [right arrow] 1 - 0. The last term is very close to a scaled/ translated version [1/1 - s] f ([z - 2[pi]n/m]/[1 - s]) of a function f (x) = [1/1 + [m.sup.2][x.sup.2]], approximating a point mass v[[delta].sub.2[pi]n/m] with v = [[integral].sup.[infinity].sub.-[infinity]] f(x)dx = [pi]/m

Example 2.2. If [omega] = 0 and F is not singular with respect to the Lebesgue measure, (3) follows from (2) in a straightforward manner as follows. To begin with, we note that [sup.sub.[epsilon]<[absolute value of z]<[infinity]] [absolute value of [phi](z)] < 1 for any [epsilon] > 0 and hence [Real part](1/[1 - s[phi](z)]) converges to [Real part](1/[1 - [phi](z)]) uniformly on {[epsilon] < [absolute value of z] < [infinity]}. In view of (2), [1.sub.[-1,1]](z) [Real part](1/[1 - s[phi](z)])dz converges weakly to [pi]/m [[delta].sub.0] (dz) + [1.sub.[-1,1]](z) [Real part](1/[1 - [phi](z)])dz as s [right arrow] 1 - 0, which convergence can be traced back to [3]. For [absolute value of z] > 1, [sup.sub.0<s<1] (1/[1 - s[phi](z)]) < [infinity]. It is then immediate to deduce (3) since 1/1 + [[[absolute value of z].sup.[alpha]+1]] is an integrable function. Among Example 2.2, the exponential distribution is the most remarkable case: F(dx) = [1/m] [e.sup.-x/m]dx. In this case, [phi](z) = 1/[1 - imz] and [Real part](1/[1 - [phi](z)]) = 1. The limit measure is hence [m.sup.([alpha])] (dz) = [pi]/m [[delta].sub.0] (dz) + 1/[1 + [[absolute value of z].sup.[alpha]+1]] dz. Next let us observe how [m.sup.([alpha]).sub.s] (dz) produces an acute thorn at z = 0. We have

[Real part](1/[1 - s[phi](z)]) = [Real part] (1/[1 - s/(1 - imz)]) = 1 + s [[1 - s]/[[(1 - s).sup.2] + [m.sup.2][z.sup.2]]].

Here the first term corresponds to the absolutely continuous part of [m.sup.([alpha])] (dz) and the second term is very close to a scaled version 1/[1 - s] f (z/[1 - s]) of a Zunction f (x) = 1/[1 + [m.sup.2][x.sup.2]], approximating V[[delta].sub.0] with v = [[integral].sup.[infinity].sub.-[infintiy]] f(x)dx = [pi]/m

Example 2.3. The case [omega] = 0 and F is singular is the most troublesome one. To be specific, let a > 0, b > 0, and 0 < c < 1 be such that b/a is an irrational number and set F = c[[delta].sub.a] + (1 - c) [[delta].sub.b]. Its Fourier transform [phi](z) = c exp(iaz) + (1 - c) exp(ibz) satisfies lim [inf.sub.z[right arrow][+ or -][infinity]] [absolute value of (z) - 1] [less than or equal to] lim [inf.sub.k[member of]Z,k[right arrow][+ or -][infinity]] [absolute value of [phi](2[pi]k/a) - 1] = 0. Indeed, [phi](2[pi]k/ a) = c + (1 - c) exp(2[pi] b/a ki) and the sequence {exp(2[pi] b/a ki); k [member of] Z} runs densely over the unit disc in C. Hence it holds lim [sup.sub.z[right arrow][+ or -][infinity]] [Real part](1/[1 - [phi](z)]) dz = [infinity] and, for any fixed s [member of] [0,1), lim [sup.sub.z[right arrow][+ or - ][infinity]] [Real part](1/[1 - s[phi](z)]) = 1 /(1 - s ). So one can not expect a priori bound C[(1 + [[absolute value of z].sup.[alpha]+1]).sup.-1] for the density of [m.sup.([alpha]).sub.s] on {[absolute value of z] > 1} as in Example 2.2. Still Theorem 1.1 implies that [m.sup.([alpha]).sub.s] converges weakly.

3. Proof of Theorem 1. Since the random walk [{[S.sub.n]}.sub.n=0,1,...] is transient, we have U((-h, h)) = V((-h, h)) < [infinity] for any h > 0.

Define a family of measures [V.sub.s] for 0 [less than or equal to] s < 1 by

[V.sub.s](I) = [1/2] [[infinity].summation over (n=0)] [s.sup.n]([F.sup.n*] (I) + [F.sup.n*](-I)).

Each [V.sub.s] is a finite measure on R. As s [right arrow] 1 - 0, [V.sub.s] ((-h, h)) [??] V((-h, h)) < [infinity]. The following statement is given in [6] but we prove it here for the sake of reader's convenience. Let Fg(z) = [[integral].sup.[infinity].sub.-[infinity]] [e.sup.izx]g(x)dx and [F.sup.-1] [gamma](x)= 1/2[pi] [e.sup.-ixz] [gamma](z)dz = 1/2[pi] - F[gamma](-x) for x integrable functions g(x) and [gamma](z).

Lemma 3.1. For any function g(x) [member of] [L.sup.1] (R) such that Fg(z) [member of] [L.sup.1] (R), we have, for any y [member of] R,

(4) [[integral].sup.[infinity].sub.-[infinity]] g(y - x) [V.sub.s] (dx) = [1/2[pi]] [[integral].sup.[infinity].sub.-[infinity]] [e.sup.iyz] Fg(z) [Real part](1/[1 - s[phi](z)]) dz,

Proof. The Fourier transform of [V.sub.s] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The equation (4) follows from the Parseval identity or the Funibi theorem.

In the next lemma we prove the existence of a function with a crucial property.

Lemma 3.2. Let 0 < [alpha] < 1 and [tau](z) = [((1 - [absolute value of z]) [disjunction] 0).sup.2], [[delta].sub.[alpha]](z) = exp(- [[absolute value of z].sup.[alpha]]), and [[psi].sub.[alpha]](z) = [tau](z)[[delta].sub.[alpha]](z). We also set t = [F.sup.-1][tau], [d.sub.[alpha]] = [F.sup.- 1][[delta].sup.[alpha]], and [p.sub.[alpha]] = [F.sup.-1] [[psi].sub.[alpha]].

Then [[psi].sub.[alpha]] is bounded, nonnegative, supported on a compact set; [p.sup.[alpha]] is bounded, strictly positive, and [p.sub.[alpha]](x) [??] [1/[[[absolute value of x].sup.a+1]]] [conjunction] 1,where '[??]' means that the ratio r(x) between both sides satisfies 0 < [inf.sub.x[member of]R] r(x) [less than or equal to] [sup.sub.x[member of]R] r(x) < [infinity]. In, particular, [[psi].sub.[alpha]] and [p.sub.[alpha]] are both integrable and continuous.

Moreover, the functions that appear here are even and real-valued.

Proof. It follows from the formula I.2.4 in [7] that t(x)= 4/[x.sup.2] (1 - [sin x/x]) [??] [1/[[[absolute value of].sup.2]] [conjunction] 1.

It is known that [d.sub.[alpha]](x) is the density of a symmetric [alpha]-stable law. As such, [d.sub.[alpha]] (x) is infinitely differentiable (see, e.g., [8,exercise 1.5 (p.49)]), strictly positive, and satisfies [d.sub.[alpha]](x) [??] [1/[[absolute value of x].sup.[alpha]+1]]] [conjunction] 1.

Let '*' denote the convolution of two functions. Then [p.sub.[alpha]] (x) = [F.sup.-1]([tau][[delta].sub.[alpha]])(x) = (t * [d.sub.[alpha]])(x), from which follows [p.sub.[alpha]](x) [??] [1/[[absolute value of x].sup.[alpha]+1]] [conjunction] 1. The other statements can be deduced easily.

Proof of Theorem 1. For h [member of] (0, 1), set

[g.sub.h](x):= h[[psi].sub.[alpha]](x/[h.sup.1/[alpha]]).

Since [[psi].sub.[alpha]] is an even function, [1/2[pi]] F [g.sub.h](z) = [F.sup.-1][g.sub.h](z) = [h.sup.1+1/[alpha]]/[p.sub.[alpha]]([h.sup.1]/[[alpha].sub.z]). Thus it holds supp([g.sub.h]) = [-[h.sup.1/[alpha]], [h.sup.1/[alpha]]], [[parallel][g.sub.h][parallel].sub.[infinity]] = h, and F[g.sub.h](z) [??] [1/[[absolute value of z].sup.[alpha]+1]] [conjunction] [h.sup.1+1/[alpha]]

Choosing g = [g.sub.h] and y = 0 in (4), we obtain

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

On one hand, there exists a positive constant [C.sub.0] (depending on [alpha]) such that

F[g.sub.h](z) [[absolute value of z].sup.[alpha]+1] > [1/[C.sub.0]]

if [absolute value of z] > [h.sup.-1/[alpha]]. We have from (5) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for any h [member of](0, 1) and s [member of] [0, 1).

On the other hand, if we fix h 2(0,1), then there exists a positive constant [C.sub.1](h) depending on h (and [alpha]) such that

F[g.sub.h](z) > [1/[C.sub.1](h)]

for any z [member of][-[h.sup.-1/[alpha]], [h.sup.-1/[alpha]]]. Hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for any s [member of][0, 1).

These bounds imply that {[m.sup.([alpha].sub.s])(dz); s [member of][0, 1)} is a tight family of finite measures on R and there exists a finite measure [m.sup.([alpha])](dz) such that (3) holds.

If [omega] = 0, the density [1/[1 + [[absolute value of z].sup.[alpha]+1]]]R(1/[1 - s[phi](z)]) converges uniformly to [1/[1 + [[absolute value of z].sup.[alpha]+1]]]R(1/[1 - [phi](z)]) as s [right arrow] 1 - 0 in every compact interval excluding the origin. Hence [m.sup.([alpha])](dz) = v[[delta].sub.0](dz) + [1/[1 + [[absolute value of z].sup.[alpha]+1]]]R(1/[1 - [phi](z)]) dz where v [member of] [0, o) is the mass assigned to the origin by the limit measure. To be consistent with (2), we must have v = [pi]/m.

If [omega] > 0, then [phi](z) = 1 if and only if z [member of] [2[pi]/[omega]] Z. It follows that [1/[1 + [[absolute value of z].sup.[alpha]+1]]]R(1/[1 - s[phi](z)]) converges, as 1 - 0, to [1/[1 + [[absolute value of z].sup.[alpha]+1]]]R(1/[1 - [phi](z)]) uniformly on any compact set K such that K [intersection] [2[pi]/[omega]] Z = 0. Hence the limit measure can have point masses only at points belonging to [2[pi]/[omega]]Z. It is straightforward to verify

m([alpha])({2[pi]n/[omega]}) = [m.sup.([alpha])]({0})/(1 + [(2[pi][absolute value of n]/[omega]).sup.[alpha]+1])

by periodicity.

To prove [m.sup.([alpha])]({0}) = [pi]/m, we introduce [[??].sub.[epsilon]] = F * N(0,[epsilon]), where '*' denotes the convolution of two measures and N(0, [epsilon]) is the normal distribution with mean 0 and variance [epsilon] [member of](0, [infinity]). It is absolutely continuous and Theorem 1.1 (the non-periodic case) is applicable.

Since [[??].sub.[epsilon]] is the probability distribution of the sum of [X.sub.1] and an independent centered normal random variable,

(6) [[integral].sup.[infinity].sub.-[infinity]]x[[??].sub.[epsilon]] = m

The Fourier transform [[??].sub.[epsilon]](z) of [[??].sub.[epsilon]] is given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then this family converges weakly to, say, [m.sup.([alpha];[epsilon])](dz). In particular, [m.sup.([alpha];[epsilon])]({0}) = [pi]/m by (6).

We denote the Radon-Nikodym density [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We define the error terms R(z) and I(z) in the expansion [alpha](z) = 1 + imz + R(z)+ iI(z) so that [absolute value of R(z)] + [absolute value of I(z)] = o(z) as z [right arrow] 0 and R(z) and I(z) are real valued.

For all [epsilon] [member of](0, 1/3) that is sufficiently small, we can find a neighborhood [U.sub.[epsilon]] [subset] (- 1/2, 1/2) of z = 0 such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [absolute value of I(z)] [less than or equal to] [epsilon] [absolute value of z], and [absolute value of R(z) [less than or equal to] [epsilon] z for any s [member of] [1 - [epsilon], 1) and z [member of] [U.sub.[epsilon]]. Moreover, it follows that R(z) [less than or equal to] -1/2 [(m - [epsilon]).sup.2][z.sup.2] < 0 from [absolute value of [phi](z)] [less than or equal to] 1. We set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It is elementary but tedious to prove that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

using the above estimates. We omit its proof. By the definition of [m.sup.([alpha];[epsilon])], we have

[C.sub.1]([epsilon])[m.sup.([alpha];[epsilon])]({0}) [less than or equal to] [m.sup.([alpha])]({0}) [less than or equal to] [C.sub.2]([epsilon])[m.sup.([alpha];[epsilon])]({0}).

Since [epsilon] is arbitrary and [m.sup.([alpha];[epsilon])]({0}) = [pi]/m, we have [m.sup.([alpha])]({0})= [pi]/m.

doi: 10.3792/pjaa.88.152

References

[1] D. Blackwell, A renewal theorem, Duke Math. J. 15 (1948), 145 150.

[2] D. Blackwell, Extension of a renewal theorem, Pacific J. Math. 3 (1953), 315 320.

[3] K. L. Chung and W. H. J. Fuchs, On the distribution of values of sums of random variables, Mem. Amer. Math. Soc. 6 (1951), 1-12.

[4] K. L. Chung and J. Wolfowitz, On a limit theorem in renewal theory, Ann. of Math. (2) 55 (1952), 1-6.

[5] P. Erdos, W. Feller and H. Pollard, A property of power series with positive coefficients, Bull. Amer. Math. Soc. 55 (1949), 201 204.

[6] W. Feller and S. Orey, A renewal theorem, J. Math. Mech. 10 (1961), 619 624.

[7] F. Oberhettinger, Tables of Fourier transforms and Fourier transforms of distributions, translated and revised from the German, Springer, Berlin, 1990.

[8] G. Samorodnitsky and M. S. Taqqu, Stable non-Gaussian random processes, Stochastic Modeling, Chapman & Hall, New York, 1994.

2000 Mathematics Subject Classification. Primary 60K05; Secondary 60G50.

By Yasuki ISOZAKI

Department of Mathematical and Physical Science, Kyoto Institute of Technology, Matsugasaki, Sakyo-ku, Kyoto 606-8585, Japan

(Communicated by Masaki KASHIWARA, M.J.A., Oct. 12, 2012)
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