# 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 , , , ) 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  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  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 . 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  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  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

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

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

 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.

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

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

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

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

 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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Author: Printer friendly Cite/link Email Feedback Isozaki, Yasuki Japan Academy Proceedings Series A: Mathematical Sciences Report 9JAPA Nov 1, 2012 3490 Defect zero characters and relative defect zero characters. A note on linear independence of polylogarithms over the rationals. Convergence (Mathematics) Fourier transformations Fourier transforms

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