Tail probability of random variable and Laplace transform

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1 Applicable Analysis Vol. 84, No. 5, May 5, Tail probability of random variable and Laplace transform KENJI NAKAGAWA* Department of Electrical Engineering, Nagaoka University of Technology, Nagaoka, Niigata 94-88, Japan Communicated by S. Saitoh (Received 3 April 4; in final form April 4) We investigate the exponential decay of the tail probability PðX > xþ of a continuous type random variable X. Let ðsþ be the Laplace Stieltjes transform of the probability distribution function FðxÞ ¼PðX xþ of X, and be the abscissa of convergence of ðsþ. We will prove that if < < and the singularities of ðsþ on the axis of convergence are only a finite number of poles, then the tail probability decays exponentially. For the proof of our theorem, Ikehara s Tauberian theorem will be extended and applied. Keywords: Tail probability of random variable; Exponential decay; Laplace transform; Tauberian theorem Mathematics Subject Classifications (): Primary 44A; 6A99. Introduction In various engineering problems, it is often important to evaluate the tail probability of a random variable. For instance, the tail probability of the waiting time in a queue is used for the design of buffer size and call admission control in communication networks. Further, the error probability of an error correcting code is also evaluated by the tail probability. In large deviations theory, one considers the probability that the average of a sequence of random variables exceeds some value and discusses the exponential decay of the tail probability when the number of random variables becomes infinite. In this article, we consider how the tail probability PðX > xþ of one random variable X decays as x tends to infinity. We say that the tail probability PðX > xþ decays exponentially if PðX > xþ e x, <, * nakagawa@nagaokaut.ac.jp Applicable Analysis ISSN 3-68 print: ISSN X online ß 5 Taylor & Francis Group Ltd DOI:.8/

2 5 K. Nakagawa or, more precisely, x log PðX > xþ ¼, < : ðþ x! is called the exponential decay rate. We give a condition on which the tail probability decays exponentially. We consider the Laplace Stieltjes transform ðsþ of the probability distribution function FðxÞ ¼ PðX xþ, and investigate the decay of the tail probability PðX > xþ based on analytic properties of ðsþ. Even in a case that we cannot obtain the explicit form of PðX > xþ, we sometimes can obtain ðsþ explicitly. For example, in queueing theory, we may calculate the Laplace Stieltjes transform ðsþ of the waiting time by the Pollaczek Khinchin formula. In such a case, if ðsþ is inversely transformed easily, we could obtain the tail probability; however, inversion is difficult in general. Since our purpose is to study the exponential decay of the tail probability, it is not exactly essential to know the explicit form of the tail probability itself. Rather, it is enough to focus only on the properties of ðsþ for investigating the exponential decay. We can readily see that if ðsþ is a rational function, then the tail probability PðX > xþ decays exponentially since ðsþ can be expanded by partial fractions and each fraction has the inverse transform which decays exponentially. In the case of a rational function, the exponential decay and the decay rate are determined by the poles with the maximum real part. The poles with the maximum real part lie on the axis of convergence of ðsþ, and other poles do not contribute to the exponential decay. From these facts, we can expect that even when ðsþ is not rational the poles on the axis of convergence determine the exponential decay of PðX > xþ. The author discussed in [4] the exponential decay of the tail probability PðX > xþ of a discrete type random variable X. We considered the probability generating function ðzþ ¼ X n¼ PðX ¼ nþz n of X and investigated the exponential decay of PðX > xþ by noticing complex analytic properties of ðzþ. In particular, we elucidated the relation between the exponential decay of PðX > xþ and the radius of convergence and the number of poles of ðzþ on the circle of convergence. Denote by r the radius of convergence of ðzþ, then Cauchy Hadamard s formula [3] implies sup n! In addition, if < r <, then log PðX ¼ nþ ¼ log r: n sup n! log PðX > nþ ¼ log r: n ðþ

3 Tail probability of random variable and Laplace transform 5 In [4], we discussed conditions that guarantee the existence of the it (instead of the sup) of the left hand side of (), and showed the following: () If < r < and the singularities of ðzþ on the circle of convergence jzj ¼r are only a finite number of poles, then n! n log PðX > nþ exists and is equal to log r. () We gave an example of a discrete type random variable X with sup n! n log PðX > nþ 6¼ inf n! n log PðX > nþ. (3) We gave an example of M/G/ type Markov chain whose stationary distribution does not have an exponentially decaying tail. In this article, we will extend the results in [4] to continuous type random variables. While we considered the probability generating function in the case of a discrete type random variable, we focus on the Laplace Stieltjes transform of the probability distribution function FðxÞ if X is of continuous type. Let ðsþ be the Laplace Stieltjes transform of FðxÞ and be the abscissa of convergence of ðsþ. The vertical line <s ¼ is called the axis of convergence of ðsþ, where <s represents the real part of s. The main theorem of this paper is that if < < and the singularities of ðsþ on the axis of convergence are only a finite number of poles, then x! x log PðX > xþ ¼ : ð3þ The biggest difference between the case of discrete type random variables and that of continuous ones is the topological property of the domain of convergence of power series and Laplace Stieltjes transform. In the power series the circle of convergence jzj ¼r is a compact set, while the axis of convergence <s ¼ is not compact. Due to this difference, although the theorems for both types of random variables formally look similar, the proof for the continuous type random variables is much more complicated than that for discrete ones. This article is organized as follows. In section, the definition of the Laplace Stieltjes transform and its abscissa of convergence are given and some lemmas are presented. In section 3, we show an example of a random variable X whose tail probability PðX > xþ does not decay exponentially. In section 4, Tauberian theorems are introduced for the purpose of proving our main theorem. In section 5, some preinary lemmas are given for the proof of the main theorem. Ikehara s Tauberian theorem is extended to the case that the Laplace transfrom has a finite number of poles on the axis of convergence. In section 6, the main theorem is proved by using the lemmas proved in section 5. Further, we show that for the example F ðxþ in section 3 all the points on the axis of convergence are singularities of the Laplace Stieltjes transform of F ðxþ. In section 7, we conclude with the results.. Laplace Stieltjes transform of probability distribution function and its abscissa of convergence Let X be a non-negative continuous type random variable with probability distribution function FðxÞ ¼PðX xþ:

4 5 K. Nakagawa Define the Laplace Stieltjes transform of FðxÞ as ðsþ ¼ e sx dfðxþ, s ¼ þ i: ð4þ The abscissa of convergence of ðsþ is defined as the number such that the integral (4) converges for <s > and diverges for <s <. The vertical line <s ¼ is called the axis of convergence of ðsþ. Since ðþ ¼ <, wehave. Now, we consider the tail probability ~FðxÞ ¼PðX > xþ of the random variable X. We have FðxÞþ ~FðxÞ ¼. Let denote the abscissa of convergence of (4). We can easily obtain LEMMA of ~FðxÞ is equal to. Next, we have The abscissa of convergence of the Laplace Stieltjes transform e sx d ~ FðxÞ, s ¼ þ i LEMMA If <, then for s with <s >, e sx dfðxþ ¼ s e sx ~FðxÞdx: Proof See Widder [5], p. 4, Theorem.3b. g From Lemma, we have LEMMA 3 If <, then the abscissa of convergence of the Laplace transform ðsþ ¼ e sx ~ FðxÞdx of ~FðxÞ is equal to. From Lemma and Widder [5], p. 44, Theorem.4e (see Appendix), we have the next lemma. LEMMA 4 If <, then for the tail probability ~FðxÞ, we have sup x! x log ~FðxÞ ¼ : Proof We give a proof different from one in [5]. By Markov s inequality, for any with <<, we have ð5þ ~ FðxÞ e x ðþ: ð6þ

5 Tail probability of random variable and Laplace transform 53 From (6) x log FðxÞ ~ þ log ðþ; x ð7þ hence, by taking sup of both sides of (7), we have sup x! x log FðxÞ ~ ; ð8þ since is arbitrary in <<. Suppose that the equality in (8) does not hold, then there should exist with < such that ~ FðxÞ satisfies ~FðxÞ e x for sufficiently large x. But by Lemma 3 this contradicts being the abscissa of convergence. g Lemma 4 can be regarded as an analogy of Cauchy Hadamard s formula on the radius of convergence of power series. In general, the sup in (5) is different from inf. We will show such an example in the next section. 3. Example of X whose tail probability does not decay exponentially In [4], an example of a discrete type random variable X is given such that the sup in () does not coincide with inf, i.e., the it does not exist. In this section, we will show an example of a continuous type random variable such that the sup in (5) is not equal to inf by modifying the example in [4]. For any h >, we define a sequence fc n g n¼ by the following recursive formula: c ¼, c n ¼ c n þ e hc ð9þ n, n ¼,,...: A function ðxþ, x is defined by ðxþ ¼e hc n, for c n x < c nþ, n ¼,,...: ðþ ðxþ has following properties, which are easily verified. ðxþ is right continuous, piecewise constant, and non-increasing: ðþ Z cnþ c n ðxþdx ¼ e hc n ðc nþ c n Þ¼, n ¼,,...: ðþ

6 54 K. Nakagawa ðxþdx ¼: ð3þ For any >, e x ðxþdx < : ð4þ Now, for <, putting F ðxþ ¼ e x ðxþ, x, we find from () that F ðxþ is a right continuous and non-decreasing function with F ðþ ¼; F ðþ ¼, i.e., F ðxþ is a distribution function. Let us define X as a random variable whose probability distribution function is F ðxþ. Denoting by ~F ðxþ the tail probability of X,wehave ~F ðxþ ¼PðX > xþ ¼e x ðxþ, x : ð5þ Hence, from (3), we have e x ~F ðxþdx ¼ ðxþdx ¼, ð6þ and for any >, from (4), e x ~ F ðxþdx ¼ e ð Þx ðxþdx < : ð7þ Thus from (6), (7) the abscissa of convergence of the Laplace transform of ~F ðxþ is. Therefore, from Lemmas, 3, 4, we have sup x! x log F~ ðxþ ¼ : ð8þ On the other hand, for x ¼ c n, ~F ðc n Þ¼e c n ðc n Þ ¼ e ð hþ c n, and hence inf x! x log ~F ðxþ inf n! ¼ h c n log ~ F ðc n Þ < : ð9þ Thus, from (8), (9) we can conclude that the it does not exist.

7 Tail probability of random variable and Laplace transform 55 The example (5) with sup x log ~F ðxþ 6¼ inf x log ~ F ðxþ is in a sense pathological, hence there is a possibility that we can guarantee the existence of the it by posing appropriate weak conditions. In fact, we have the following theorem, that is the main theorem of this article. THEOREM For a non-negative random variable X, let FðxÞ PðX xþ be the probability distribution function of X and ~FðxÞ PðX > xþ be the tail probability of X. Denote by the abscissa of convergence of the Laplace Stieltjes transform ðsþ of FðxÞ. If < < and the singularities of ðsþ on the axis of convergence <s ¼ are only a finite number of poles, then we have x log FðxÞ ~ ¼ : x! ðþ Remark The assumption of Theorem implies that there exists an open neighborhood U of <s ¼ and ðsþ is analytic on U except for the finite number of poles on <s ¼. Remark The real point s ¼ of the axis of convergence is always a singularity of ðsþ by Widder [5], p. 58, Theorem 5b (see Appendix). Hence, under the assumption in Theorem, s ¼ is a pole of ðsþ. In the case of discrete type random variable X, the following theorem is shown in [4], which corresponds to Theorem. THEOREM (Nakagawa [4]) Let X be a discrete type random variable taking non-negative integral values. The probability function of X is denoted by n ¼ PðX ¼ nþ; n ¼ ; ;...; and the tail probability by n PðX > nþ ¼ P j>n j. Consider the probability generating function ðzþ of f n g. If the radius of convergence r of ðzþ satisfies < r < and the singularities of ðzþ on the circle of convergence are only a finite number of poles, then we have n log n ¼ log r: n! The proof of this theorem depends mainly on the following fact for power series: if a power series f ðzþ ¼ P n¼ a nz n converges in jzj < R and is continued analytically to a function analytic in jzj < R with R > R, then the power series P n¼ a nz n converges in jzj < R. However, for Laplace transforms, analogous to this property does not hold. In fact, even if the integral ðsþ ¼ e sx ðxþdx converges in <s > and ðsþ is continued analytically to a function analytic in <s > with <, the integral () does not necessarily converge in <s >. For instance, the abscissa of convergence of the integral ðsþ ¼ e sx e x sinðe x Þdx ðþ

8 56 K. Nakagawa is ¼, nevertheless ðsþ is continued analytically to a function analytic in the whole finite plane jsj < ([5], p.58). This is a large difference between power series and Laplace transform. In general, in Laplace transform, we cannot discuss the convergence of the integral in the left half-plane of the axis of convergence. However, we may be able to discuss the convergence on <s ¼ with the knowledge of convergence in <s >. For this purpose, we will apply Tauberian theorems. 4. Application of Tauberian theorems 4.. Abelian theorems First, we introduce a general form of Abelian theorem for Laplace transform. A function ðxþ of bounded variation in [a, b] is said to be normalized if it satisfies ðaþ ¼, ðxþ ¼ ðxþþ þ ðx Þ, a < x < b: ABELIAN THEOREM (Widder [5], p. 83, Cor. c.) For a normalized function ðxþ of bounded variation in [, R] for every R >, if the integral ðsþ ¼ exists for s > ; then we have e sx dðxþ inf x! inf x!þ ðxþ inf s!þ ðxþ inf s! ðsþ sup s!þ ðsþ sup s! ðsþ sup ðxþ, x! ðsþ sup ðxþ: x!þ ðþ ð3þ 4.. Tauberian theorems Contrary to Abelian theorems, Tauberian theorems give sufficient conditions for the existence of x! ðxþ. The following is one of the well-known Tauberian theorems. TAUBERIAN THEOREM (Widder [5], p. 87, Theorem 3b) For a normalized function ðxþ of bounded variation in ½; RŠ for every R > ; let the integral ðsþ ¼ exist for s > and let s!þ ðsþ ¼A. Then e sx dðxþ ðxþ ¼A x!

9 holds if and only if Tail probability of random variable and Laplace transform 57 Z x tdðtþ ¼oðxÞ, x! If we consider only a special type of function ðxþ, then a weaker sufficient condition may be enough to describe the asymptotic behavior of ðxþ. The following Tauberian theorem by Ikehara is for a non-negative and non-decreasing function. This Ikehara s Tauberian theorem could be used for the proof of the prime-number theorem [5]. THEOREM (Ikehara [], see also Widder [5], p. 33, Theorem 7) Let ðxþ be a nonnegative and non-decreasing function defined in x <, and let the integral ðsþ ¼ e sx ðxþdx, s ¼ þ i converge for <s >. If for some constant and some function gðþ ðsþ ¼ gðþ!þ s uniformly in every finite interval of, then we have x! e x ðxþ ¼: Remark 3 Suppose s ¼ is a pole of ðsþ of order and the residue of ðsþ at s ¼. Further, if ðsþ has no other singularities than s ¼ on the axis of convergence <s ¼, then the condition (4) holds. More generally, suppose that the singularities of ðsþ on the axis of convergence <s ¼ are only a finite number of poles s j, j ¼,..., m. Denoting by j ðsþ the principal part of ðsþ at the pole s j, then similarly to (4),!!þ uniformly in every finite interval of. ðsþ Xm j¼ jðsþ ¼ gðþ ð4þ ð5þ 5. Preinary for proof of theorem In this section, we will prepare some lemmas for the proof of Theorem. First of all, we define and for any >, ðxþ sin x= pffiffiffiffiffi, < x <, x= k ðxþ ðxþ ¼ sin x pffiffiffi, < x < : x

10 58 K. Nakagawa For integers D and N ðdþþ=, we see k N ðx tþtd dt <, < 8x <, where k N ðx tþ is the Nth power of k ðx tþ. Next, for any >, define ( K ðxþ x, jxj,, jxj > : It follows that K is the Fourier transform of k, i.e., K ¼Fðk Þ or K ðxþ ¼ p ffiffiffiffiffi e ixt k ðtþ dt: ð6þ Writing K N ¼ K K (N-fold convolution), we have K N of K N is denoted by ½ ; Š, where ¼ N. ¼Fðk N Þ. The support 5.. On a slowly increasing function A function f ðxþ defined in x < is slowly increasing if it satisfies sup f ðx þ Þ fðxþ : x!;!þ LEMMA 5 Let f ðxþ be a slowly increasing function. Given > and an increasing sequence fu n g n¼ with u n!, we assume f ðu n Þ > ðresp. f ðu n Þ < Þ, n ¼,,... Then, there exist > and infinitely many non-overlapping intervals fj n ¼ðx n, x n þ Þg with f ðxþ > resp: f ðxþ <, x J n, n ¼,,...: Proof: First, consider the case f ðu n Þ >. For u n, let us define v n ¼ supfv j v < u n, f ðvþ <=g. Suppose the claim were not true, then for any > and sufficiently large all n, By the definition of v n, we have u n v n < : ð7þ f ðu n Þ f ðv n Þ > : ð8þ From (7), (8), we have sup ff ðx þ Þ fðxþg supff ðu n Þ fðv n Þg x!,!þ n!,

11 Tail probability of random variable and Laplace transform 59 which contradicts f ðxþ being slowly increasing. The case f ðu n Þ < can be proved in a similar way. g We have the following lemma. LEMMA 6 Let f ðxþ be a bounded and slowly increasing function. Let D and N be integers with D and N ðdþþ=. If x! k N ðx tþtd f ðtþ dt ¼ ð9þ holds for any >, then we have f ðxþ ¼: ð3þ x! Proof Suppose (3) were not true. Then there should exist > and an increasing sequence fu n g with u n! such that f ðu n Þ > or f ðu n Þ <. Assume f ðu n Þ >. The other case can be handled in the same way. From f ðu n Þ > and Lemma 5, there exists a sequence of intervals fj n ¼ðx n, x n þ Þg with f ðxþ >, x J n, n ¼,,...: Hence, we have k N ðx n tþt D f ðtþdt ¼ Z xn þ Z xn þ x n Z xn Z xn þ x n þ k N ðx n tþt D f ðtþdt x n þ k N ðx n tþt D dt sup þ x n þ x< j f ðxþj k N ðx n tþt D dt : ð3þ Each term in (3) can be evaluated as follows: Z xn þ x n Z k N ðx n tþt D dt ¼ k N ðtþðx n tþ D dt Z ¼ðÞ N N ðtþ x n t D dt Z ¼ðÞ N xn D N ðtþdt þ ðx n, Þ, ð3þ where ðx n ;Þ!asx n ;!, by the binomial expansion of x n t= D.

12 5 K. Nakagawa Similarly, we have Z xn k N ðx n tþt D dt ¼ðÞ N x D n ðx n, Þ, ð33þ and x n þ k N ðx n tþt D dt ¼ðÞ N x D n 3 ðx n ;Þ, ð34þ where ðx n, Þ!, 3 ðx n, Þ!asx n,!. Write C R N ðtþdt < and M sup x< j f ðxþj <, then there exists ~C > such that for sufficiently small, > and large, from (3) (34), we have n k N ðx n tþt D f ðtþdt o ðc Þ M ðþ N xn D ~C, but this contradicts the assumption of the lemma (9). 5.. Laplace transform with finite number of poles on its axis of convergence Let ðxþ be a non-negative and non-increasing function in x, and ðxþ L ½, RŠ for any R >. Consider the Laplace transform ðsþ ¼ e sx ðxþ dx: Let be the abscissa of convergence of ðsþ with < <. Assume the singularities of ðsþ on the axis of convergence <s ¼ are only a finite number of poles s j ¼ þ i j, j ¼,..., m. Write j ðsþ as the principal part of ðsþ at the pole s j, and A j ðxþ as the inverse Laplace transform of j ðsþ, j ¼,..., m. The order of the pole s j is denoted by d j and let D ¼ max jm d j. Define functions bðxþ and BðxÞ for x as follows: bðxþ ¼e x x Dþ ðxþ, x, ð35þ BðxÞ ¼e x Xm Dþ x j¼ A j ðxþ, x : ð36þ We will investigate the properties of bðxþ and BðxÞ. We have the following lemma.

13 Tail probability of random variable and Laplace transform 5 LEMMA 7 Let N be an integer with N ðd þ Þ=. If x D k N ðx tþtd bðtþ BðtÞ dt is bounded in x for some >; then bðxþ BðxÞ is bounded. Proof Since the principal part j ðsþ at the pole s j is represented as jðsþ ¼ Xd j l¼ jl ðs s j Þ l, jl C, jdj 6¼, we have A j ðxþ ¼ Xd j l¼ jl ðl Þ! xl e s jx : Thus, we have from (36) BðxÞ ¼ Xm j¼ X d j l¼ jl ðl Þ! xl D e i jx : ð37þ Since l D, BðxÞ is bounded in x. Next, we will show that bðxþ is bounded. We have k N ðx tþtd BðtÞdt ¼ x ¼ XD n¼ k N ðtþðx þ tþd Bðx þ tþ dt Z D x D n k N n ðtþtn Bðx þ tþ dt: x ð38þ Since BðxÞ is bounded and x k N ðtþtd dt <, we see from (38) that there exist E > and x > with x D k N ðx tþtd BðtÞdt < E, x > x : ð39þ

14 5 K. Nakagawa Noting bðxþ, we have from the assumption of the lemma and (39), x D k N ðx tþtd bðtþ dt k N ðx tþtd bðtþ BðtÞ dt þ x D x D k N ðx tþtd BðtÞdt <E, x> x : ð4þ On the other hand, we can write x D k N ðx tþtd bðtþ dt ¼ x D hence it follows from (4) that ðþ N x D x ¼ ðþn x D N ðtþ x þ t x k N ðtþðx þ tþd bðx þ tþ dt x N ðtþ x þ t D b t xþ dt, D b t xþ dt <E, x> x : p The integrand in (4) is non-negative in ½ x, Þ and ½ ffiffiffi pffiffiffi, Š½ x, Þ, hence we have and thus ðþ N Z x D p ffiffi ffiffi p N ðtþ x þ t D b t xþ dt <E, x> x, ð4þ ðþ N Z x D p ffiffi p ffiffi N ðtþ x p ffiffiffi þ t D b x p ffiffiffi þ t p Since ðxþ is non-increasing, we have for ffiffiffi p ffiffiffi t, dt <E, x> x 3 : ð4þ x p ffiffiffi þ t D b x p ffiffiffi þ t ffiffiffiffi ¼ e x ð= p ð Þþðt=ÞÞ x p ffiffiffi e p ffiffiffiffi ð x ð= Þþðt=ÞÞðxÞ, þ t and from (4) Z e x x Dþ ðxþ ðþn pffiffiffiffiffi p ffiffi ffiffi p ffiffi e ðð= p Þ ðt=þþ N ðtþ dt <E, x> x 3 :

15 Therefore, we finally have Tail probability of random variable and Laplace transform 53 ðþ N Z bðxþ < E pffiffiffiffiffi p ffiffi ffiffi p ffiffi e ðð= p Þ ðt=þþ, N ðtþ dt x > x3, which shows that bðxþ is bounded. g LEMMA 8 Under the assumption of Lemma 7, bðxþ BðxÞ is slowly increasing. Proof It is readily seen that the sum of two slowly increasing functions is slowly increasing. We show that both bðxþ and BðxÞ are slowly increasing. Since BðxÞ has bounded derivative, BðxÞ is slowly increasing. Next, we prove that bðxþ is slowly increasing. For x > and >, we have bðx þ Þ bðxþ ¼e ðxþþ ðx þ Þ Dþ ðx þ Þ e x x Dþ ðxþ bðxþ e x D D ðx þ Þ because ðxþ is non-increasing. Since bðxþ is bounded by Lemma 7, we have sup fbðx þ Þ bðxþg sup bðxþ e x!,!þ x!,!þ ¼, which shows that bðxþ is slowly increasing. x D D ðx þ Þ 5.3. Extension of Ikehara s theorem The following theorem is an extension of Ikehara s theorem to the case that ðxþ is a non-negative and non-increasing function and its Laplace transform has a finite number of poles on the axis of convergence. THEOREM Let ðxþ be a non-negative and non-increasing function in x ; and ðxþ L ½, RŠ for any R >. Let be the abscissa of convergence of the Laplace transform ðsþ ¼ e sx ðxþdx ð43þ with < <. Assume the singularities of ðsþ on the axis of convergence <s ¼ are only a finite number of poles s j ¼ þ i j, j ¼,..., m. Write j ðsþ as the principal part of ðsþ at s j and A j ðxþ as the inverse Laplace transform of j ðsþ; j ¼ ;..., m. The order of the pole s j is denoted by d j and let D ¼ max jm d j. Then we have ( x! e x x Dþ ðxþ Xm A j ðxþ ) ¼ : j¼ ð44þ

16 54 K. Nakagawa Remark 4 From Korevaar [], p. 495, Theorem 9. (see Appendix), if e x x Dþ ðxþ Xm j¼ A j ðxþ is differentiable and its derivative is bounded from below, then we see that (44) holds. Proof Consider the functions bðxþ and BðxÞ defined in (35), (36). For any > and an integer N with N ðdþþ=, define Since K N I ðxþ p ffiffiffiffiffi ¼Fðk N Þ, by Fubini s theorem, we have I ðxþ ¼ ¼ ¼ Z Z k N ðx tþtd fbðtþ BðtÞge t dt: Z t D fbðtþ BðtÞge t dt K N ðþe ix d K N ðþe iðx tþ d t D fbðtþ BðtÞge ð iþt dt ( ) jð þ iþ : ð45þ K N ðþe ix d ð þ iþ Xm j¼ By the assumption of theorem and Remark 3, there exists a function gðþ and!þ ( ) ð þ iþ Xm jð þ iþ ¼ gðþ j¼ ð46þ uniformly in the finite interval ½ ; Š, which is the support of K N (see the definition after (6)). Therefore, we can change the order of the it and the integration in (45) to obtain Next, consider From (37), we have I ðxþ ¼ Z K N!þ ðþe ix gðþ d: ð47þ I, B ðxþ k N ðx tþtd BðtÞe t dt: I, B ðxþ ¼ X m X d j jl ðl Þ! j¼ l¼ k N ðx tþtl e i jt e t dt: ð48þ

17 Tail probability of random variable and Laplace transform 55 Since N ðd þ Þ=, the infinite integral k N ðx tþtl e i jt dt converges, thus we have by () in the Abelian theorem!þ k N ðx tþtl e i jt e t dt ¼ k N ðx tþtl e i jt dt: Hence we have by (48) I, BðxÞ ¼!þ k N ðx tþtd BðtÞ dt: ð49þ Moreover, consider I, b ðxþ k N ðx tþtd bðtþe t dt: By (47), (49), we have I, bðxþ ¼ I ðxþþi, B ðxþ < :!þ!þ ð5þ Since bðtþ, we have but if the integral is, then from (), I, bðxþ inf!þ R! ¼ which contradicts (5). Hence we have By (), we have k N ðx tþtd bðtþdt, Z R k N ðx tþtd bðtþdt k N ðx tþtd bðtþ dt < : I ðxþ ¼ I, bðxþ I, BðxÞ!þ!þ!þ ¼ k N ðx tþtd bðtþ BðtÞ dt: ð5þ

18 56 K. Nakagawa Thus by (47), (5), it follows that Z K N ðþe ix gðþd ¼ and hence by the Riemann Lebesgue theorem k N ðx tþtd bðtþ BðtÞ dt, x! k N ðx tþtd bðtþ BðtÞ dt ¼ ð5þ holds for any >. So, the assumption of Lemma 7 is satisfied. Since bðxþ BðxÞ is bounded and slowly increasing by Lemmas 7, 8, we have by (5) and Lemma 6, x! fbðxþ BðxÞg ¼ or (!) x! e x x Dþ ðxþ Xm A j ðxþ ¼ j¼ g 5.4. Estimation of B(x) from below Our goal is to examine the exponential decay of the tail probability FðxÞ. ~ Replace ðxþ in Theorem by FðxÞ. ~ The basic approach to this problem is to approximate ~FðxÞ by the inverse Laplace transform P m j¼ A jðxþ of the rational function P m j¼ jðsþ. To this end, we first estimate BðxÞ from below. LEMMA 9 Let j, j ¼,..., m, be distinct real numbers and d j, j ¼,..., m, be integers with d j. Let jl, l ¼,..., d j, j ¼,..., m, be complex numbers with jdj 6¼. Write D ¼ max jm d j. For x, define BðxÞ ¼ Xm X d j j¼ l¼ ~ jl x l D e i jx, ~ jl jl ðl Þ!, x : ð53þ Then there exists T >, ¼ ðtþ >, and x ¼ x ðtþ, such that for any x > x at least one of jbðxþj, jbðx þ TÞj,..., jbðx þðm ÞTÞj is greater than. Proof Since j 6¼ j, j 6¼ j, there exists T > with j T 6 j T mod, j 6¼ j. For this T, we will show that there exists > such that at least one of jbðxþj,..., jbðx þðm ÞTÞj is greater than for any x > x. If not, then for any > there exists a sufficiently large x with jbðx þ ntþj, n ¼,,..., m : ð54þ Now, define j ðxþ ¼ Xd j l¼ ~ jl x l D, j ¼,..., m: ð55þ

19 Tail probability of random variable and Laplace transform 57 Denote by m the number of indices j that attains D ¼ max jm d j. Without loss of generality, we can assume d ¼¼d m ¼ D. Then by (55) we can write j ðxþ ¼ ~ jd þ j ðxþ, j ¼,..., m j ðxþ, j ¼ m þ,..., m, where j ðxþ is a function with x! j ðxþ ¼, j ¼,..., m. Defining ðxþ ¼ P m j¼ jðxþ, we have from (53), (55) BðxÞ ¼ Xm j¼ j ðxþe i jx ¼ Xm j¼ ~ jd e i jx þ ðxþ: Therefore we have Bðx þ ntþ ¼ Xm j¼ ~ jd e i jðxþntþ þ ðx þ ntþ, n ¼,,..., m : ð56þ The matrix representation of (56) is BðxÞ e i x e i x... e i mx Bðx þ TÞ B. A ¼ e i ðxþtþ e i ðxþtþ... e i mðxþtþ B A Bðx þðm ÞTÞ e i ðxþðm ÞTÞ e i ðxþðm ÞTÞ... e i mðxþðm ÞTÞ ~ D.. ðxþ ~ m D ðx þ TÞ þ. B A : B. A ðx þðm ÞTÞ ð57þ Writing V as the matrix in the right-hand side of (57), we have... det V ¼ Ym e ijx e i T e i T... e i mt detb A j¼ e i ðm ÞT e i ðm ÞT... e i mðm ÞT ¼ Ym e ijx ð Þ Y mðm Þ= e ijt e i j T : ðvandermonde determinantþ j<j j¼

20 58 K. Nakagawa Since e i jt 6¼ e i j T by j T 6 j T mod, j 6¼ j,wehave j det Vj ¼ Y j<j je i jt e i j T j > : Denoting by jn the cofactor of V, we have by (57) ~ jd ¼ Xm jn Bðx þ ntþ ðxþntþ, j ¼,..., m : det V n¼ ð58þ For any > there exists a sufficiently large x such that jðxþj < holds for any x > x. Since jn is ð Þ jþn multiplied by the sum of ðm Þ! complex numbers of absolute value, we have j jn jðm Þ!. Hence, by (54), (58) j ~ jd j m! j det Vj ð þ Þ, j ¼,..., m : Because, are arbitrary, we have ~ jd ¼, but this is a contradiction. The following lemma is an immediate consequence of Lemma 9. LEMMA There exist L >, c ¼ cðlþ >, and an increasing sequence fx g ¼ with x!which is contained in ½x ; Þ for x sufficiently large, such that g x x L, jbðx Þj c, ¼,,...: ð59þ ð6þ Proof Let L ¼ mt for m and T in the proof of Lemma 9. g 6. Proof of theorem Based on the lemmas in the preceding sections, we will prove our main theorem. Proof of Theorem By Lemma 4, hence, we only need to prove sup x! x log FðxÞ ~ ¼, inf x! x log ~FðxÞ : Consider the increasing sequence fx g with L and c which are given in Lemma.

21 Now, by Theorem, for any with <<c, taking sufficiently large x, we have Thus, it follows that ( ) e x x Dþ FðxÞ ~ Xm A j ðxþ <, x > x : ~FðxÞ > Xm j¼ Particularly for x ¼ x, by (6) we have j¼ A j ðxþ e x x D ¼ e x x D jbðxþj, x > x : ~ Fðx Þ > e x x D ðc Þ: ð6þ For sufficiently large any x, take the index with x < x x. Then by (59), (6) and the decreasing property of FðxÞ, ~ we have FðxÞ ~ Fðx ~ Þ > ðc Þe x x D ðc Þe ðxþlþ x D : ð6þ Therefore from (6), we have Tail probability of random variable and Laplace transform 59 inf x! x log ~FðxÞ : g Theorem gives a sufficient condition in which the number of poles on the axis of convergence <s ¼ is finite. But in a special case, there exists x! x log FðxÞ ~ even if the number of poles on <s ¼ is infinite. We show the following theorem. THEOREM P 3 Let ¼f n g n¼ be a discrete type probability distribution and ðzþ ¼ n¼ nz n be the probability generating function of. We assume that the radius of convergence r of ðzþ satisfies < r < and the singularities of ðzþ on the circle of convergence jzj ¼r are only a finite number of poles. Let X be a continuous type random variable whose probability distribution function is defined by FðxÞ ¼ X nx n, x : Then the tail probability ~FðxÞ of X satisfies x log ~FðxÞ ¼ log r: x! Proof By Theorem 3 in Nakagawa [4]. g

22 5 K. Nakagawa Remark 5 The Laplace Stieltjes transform ðsþ of FðxÞ in Theorem 3 satisfies ðsþ ¼ ¼ X n¼ ¼ X n¼ e sx dfðxþ e sn n n z n, z ¼ e s, so, by the change of variable z ¼ e s, we see that ðsþ has infinitely many poles on the axis of convergence <s ¼ log r. Then, how about the case of X which was defined in (5)? For X, let F ðxþ denote the probability distribution function of X, ðsþ the Laplace Stieltjes transform of F ðxþ, and ~F ðxþ the tail probability of X. Since the it x! x log F~ ðxþ does not exist, ðsþ may have different type of infinitely many singularities from those in Theorem 3. We will show that, by taking h as a special value, all the points on the axis of convergence <s ¼ are singularities of ðsþ. First, in the definition (9) of the sequence fc n g, take h such that e h is an integer. Then all the c n are non-negative integers. For fc n g, define ðxþ by (), and define F ðxþ by Write ~ F ðxþ ¼ F ðxþ and F ðxþ ¼ e x ðxþ, <, x : ðsþ ¼ e sx df ðxþ, <s > : Since d ~F ðxþ ¼ ~ F ðxþdx þ e x dðxþ, we have by Lemma, ðsþ ¼ e sx d F~ ðxþ ¼ e sx ~F ðxþ dx ¼ s ¼ s ð ðsþ Þ e ð sþx dðxþ e sx df ðxþdx e ð sþx dðxþ e ð sþx dðxþ: ð63þ Defining ðsþ ¼ e ð sþx dðxþ,

23 Tail probability of random variable and Laplace transform 5 we have by (63) ðsþ ¼ þ sðsþ s : ð64þ Since we can write ðsþ as ðsþ ¼ X n¼ e ð sþc n ðe hc n e hc n Þ, and c n is a non-negative integer, by the change of variable e s ¼ z, define ðzþ as ðzþ ðsþ ¼ X n¼ ðe hc n e hc n Þz c n : Since the abscissa of convergence ðsþ is, that of ðsþ is also by (64). Thus, the radius of convergence of ðzþ is. The sequence fc n g is a gap sequence, i.e., inf n! c nþ =c n >, hence all the points on the circle of convergence jzj ¼ are singularities of ðzþ by Hadamard s gap theorem shown in Appendix. Therefore, by the correspondence e s ¼ z, all the points on the axis of convergence <s ¼ are singularities of ðsþ, hence by (64) these are singularities of ðsþ, too. 7. Conclusion We investigated the exponential decay of the tail probability PðX > xþ of a continuous type random variable X based on the analytic properties of the Laplace Stieltjes transform ðsþ of the probability distribution function PðX xþ. We saw that the singularities of ðsþ on the axis of convergence play an essential role. For the proof of the exponential decay of PðX > xþ, Ikehara s Tauberian theorem was extended and applied. The exponential decay rate is determined by the abscissa of convergence and s ¼ is always a singularity of ðsþ. Through the proof of the main theorem, we believe the following conjecture is true. Conjecture: If < < and s ¼ is a pole of ðsþ, then the tail probability decays exponentially. References [] Ikehara, S., 93, An extension of Landau s theorem in the analytic theory of numbers. Journal of Mathematics and Physics, MIT no.,. [] Korevaar, J.,, A century of complex Tauberian theory. Bulletin of AMS, 39(4), [3] Markushevich, A.I., 985, Theory of Functions of a Complex Variable (New York: Chelsea Publishing Company). [4] Nakagawa, K., 4, On the exponential decay rate of the tail of a discrete probability distribution. Stochastic Models, (), 3 4. [5] Widder, D.V., 94, The Laplace Transform (Princeton, NJ: Princeton University Press).

24 5 K. Nakagawa A. Citation of theorems THEOREM (Widder [5], p. 44, Theorem.4e) If e sx df ðxþ has a negative abscissa of convergence, then f ðþ exists and ¼ sup x! log j f ðxþ fðþj: x THEOREM (Widder [5], p. 58, Theorem 5b) the axis of convergence of If f ðxþ is monotonic, then the real point of ðsþ ¼ e sx df ðxþ is a singularity of ðsþ. THEOREM (Widder [5], p. 4, Theorem.b) If e sx df ðxþ converges for s with <s ¼ <, then f ðþ exists and f ðxþ fðþ ¼ oðe x Þ: THEOREM (Korevaar [], p. 495, Theorem 9.) t and Let f ðxþ be bounded from below for ðsþ ¼ e sx f ðxþdx converge for s with <s >. If ðsþ is continued analytically to the closed half-plane <s, then R f ðxþdx exists and is equal to ðþ. THEOREM (Hadamard s Gap Theorem, see [3]) If a power series f ðzþ ¼ P n¼ a c n z c n satisfies c nþ inf >, n! c n then all the points on the circle of convergence are singularities of f ðzþ.