Convolution of a symmetric log-concave distribution and a symmetric bimodal distribution can have any number of modes
CCONVOLUTION OF A SYMMETRIC LOG-CONCAVEDISTRIBUTION AND A SYMMETRIC BIMODALDISTRIBUTION CAN HAVE ANY NUMBER OF MODES
CHARLES ARNAL
Abstract.
In this note, we show that the convolution of a discretesymmetric log-concave distribution and a discrete symmetric bimodaldistribution can have any strictly positive number of modes. A similarresult is proved for smooth distributions, which contradicts the mainstatement in [HLL ` Contents
Acknowledgement 11. Introduction 12. Definitions 23. Discrete case 34. Continuous case 6References 9
Acknowledgement
The author is very grateful to Cl´ement Deslandes for helpful discussions.1.
Introduction
Log-concave functions and sequences feature preeminently in many math-ematical domains, including probability, statistics and combinatorics, butalso more surprisingly algebraic geometry (read [Bak18] and [SW14] for sur-veys of the notion).Likewise, it is natural to consider the properties preserved by the con-volution of two distributions, and in particular to consider the number ofmodes of the resulting distribution. It has been known for a long time thatthe convolution of two unimodal distributions need not be unimodal, thoughthe convolution of two symmetric unimodal distributions will be unimodal(see [DJD88], or [AK07] for an alternative proof of the second fact in thediscrete case).
This research was supported by the DIM Math Innov de la R´egion Ile-de-France. a r X i v : . [ m a t h . S T ] F e b CHARLES ARNAL
Further explorations have shown that convolutions of log-concave realdistributions are log-concave (hence unimodal) (see [Mer98]), and that theconvolution of a unimodal (but not symmetric) distribution with itself canhave any number of modes (see [Sat93]). In the same line of questioning, weprove the following result:
Theorem 1.1.
Let n P N be greater or equal to . Then there exists adiscrete log-concave distribution p n and a discrete bimodal distribution q n ,both symmetric about , such that their convolution p n ˚ q n has exactly n modes. We also prove a continuous variant, which directly contradicts the mainstatement in [HLL ` Theorem 1.2.
Let n P N be greater or equal to . Then there exists an ab-solutely continuous distribution on R whose density function f n is smooth,symmetric about and log-concave (hence unimodal), and an absolutely con-tinuous distribution on R whose density function g n is smooth, symmetricabout and bimodal, such that the convolution f n ˚ g n has at least n modes. Remark 1.3.
In fact, we can ask that f n ˚ g n have exactly n modes; provingit makes the demonstration much more tedious, yet not any deeper. In what follows, the reader is reminded of the main definitions in Section2. Theorem 1.1 is proved in Section 3, and Theorem 1.2 is proved in Section4. 2.
Definitions
In this note, we will consider absolutely continuous distributions on R with continuous density functions and discrete distributions on Z .The modes of an absolutely continuous distribution with continuous den-sity function f are the local maxima of f .We will say that an absolutely continuous distribution with continuousdensity function f is strictly n -modal in t a , . . . , a n u if there exists a ă b ă a ă . . . ă b n ´ ă a n P R such that f is strictly increasing on r´8 , a s and r b i , a i ` s for i “ , . . . , n ´
1, and strictly decreasing on r a n , and r a i , b i s for i “ , . . . , n ´ p has a mode in t m, m ` , . . . , m ` k u Ă Z if p p m ´ q ă p p m q “ p p m ` q “ . . . “ p p m ` k q ą p p m ` k ` q . We say that it is n -modal if it has exactly n modes.In the discrete case, a mode is often required to be a global maximum ofthe mass function. This nuance in definition matters not, as all our discretemodes will be global maxima.The density function f of an absolutely continuous distribution is called log-concave if f p αx ` p ´ α q y q ě f p x q α f p y q ´ α for all x, y P R and all α P r , s . If f is strictly positive, it is equivalent the concavity of log ˝ f .The mass function p of a discrete distribution is called log-concave if p p m q ě p p m ´ q p p m ` q for all m P Z , and if its support is a contiguous The mistake can be found in the proof of Theorem 1 in [HLL ` interval, i.e. if there exists m , m P Z such that m ă m , p p m q “ m ď m and all m ě m , and p p m q ą m ă m ă m . The secondcondition is sometimes omitted. Remark 2.1.
It is easy to show that log-concavity implies unimodality bothin the continuous and the discrete case.
As usual, the convolution of density functions f, g : R ÝÑ R (and byextension the convolution of the two associated absolutely continous distri-butions) is defined as f ˚ g p x q “ ż R f p t q g p x ´ t q dt for all x P R .Similarly, the convolution of mass functions p, q : Z ÝÑ r , s (and byextension the convolution of the two associated discrete distributions) isdefined as p ˚ q p m q “ ÿ k P Z p p k q q p m ´ k q for all m P Z . 3. Discrete case
Proof of Theorem 1.1.
The case n “ p : “ t´ , , u and q : “ t´ , u , where 1 A : Z ÝÑ t , u refers to the indicatorfunction of A for any A Ă Z .Now for n ě
2, let us define ˜ p n “ t´ n ` ,..., ,...,n ´ u and p n “ ˜ p n ¨ n ´ .The function p n is clearly a log-concave distribution and symmetric about0. If n is even, let ˜ q n be as such: for k “ , . . . , n ´
1, we let ˜ q n p k ´ q “ ˜ q n p k q “ k ` q n p n ´ ´ k q “ ˜ q n p n ´ ´ k q “ k . We also let˜ q n p n ´ q “ n , ˜ q n p q “ q n p k q be 0 for any k ą n ´
3. We define ˜ q n on Z ď symmetrically.Let C n : “ ř m P Z ˜ q n p m q and q n : “ ˜ q n C n . Then q n is symmetric about 0, andit is a bimodal distribution whose modes are in t´ n ` , n ´ u . It has aminimum in 0.See Figure 1 to see the case n “ p n ˚ q n is clearly symmetric about 0.For any m P Z , the finite difference satisfies D p p n ˚ q n qp m q “ p p n ˚ q n qp m q´p p n ˚ q n qp m ´ q “ p D p p n q˚ q n qp m q . As D p p n qp l q is equal to n ´ if l “ ´ n ` ´ n ´ if l “ n ´ D p p n ˚ q n qp m q “ q n p m ` n ´ q´ q n p m ´ n ` q n ´ .We see that D p p n ˚ q n qp m q is 0 if m ą p n ´ q ` p n ´ q “ n ´ n ´ ě m ą n ´
1. For m “ , . . . , n ´
1, the finitedifference D p p n ˚ q n qp m q is equal to p n ´ q C n if m is odd and to ´ p n ´ q C n if m is even. CHARLES ARNAL
Figure 1.
Case n “
6. The ‚ correspond to ˜ q , the ˆ to˜ p and the ` to where they take the same value.By symmetry of p n and q n , if m ď
0, we have that D p p n ˚ q n qp m q “ q n p m ` n ´ q ´ q n p m ´ n ` q n ´ “´ q n p´ m ` n ´ q ´ q n p´ m ´ n ` q n ´ “´ q n p´p m ´ q ` n ´ q ´ q n p´p m ´ q ´ n ` q n ´ “ ´ D p p n ˚ q n qp´p m ´ qq . From this, we get that p n ˚ q n has exactly n modes, located in ´ n ` , . . . , ´ , ´ , , , . . . , n ´ p n ˚ ˜ q n is illustrated in Figure 2 in the case n “ p n ˚ q n “ ˜ p n ˚ ˜ q n p n ´ q C n ).The case where n is odd (and strictly greater than 1) is very similar. Let˜ q n be as such: for k “ , . . . , n ´ ´ q n p k ´ q “ ˜ q n p k q “ k ` k “ , . . . , n ´ we let ˜ q n p n ´ ´ k q “ ˜ q n p n ´ ´ k q “ k . Wealso let ˜ q n p n ´ q “ n , ˜ q n p q “ q n p k q be 0 for any k ą n ´ q n on Z ď symmetrically, let C n : “ ř m P Z ˜ q n p m q and q n : “ ˜ q n C n . Then q n is symmetric about 0, and it is a bimodal distributionwhose modes are in t´ n ` , n ´ u . It has a minimum in 0.The case n “ D p p n ˚ q n qp m q “ q n p m ` n ´ q´ q n p m ´ n ` q n ´ . Thus we see that D p p n ˚ q n qp m q is 0 if m ą p n ´ q ` p n ´ q “ n ´ n ´ ě m ą n ´
1. For m “ , . . . , n ´
1, the finite difference D p p n ˚ q n qp m q is equal to p n ´ q C n if m is even and to ´ p n ´ q C n if m is odd. Figure 2.
The convolution product ˜ p n ˚ ˜ q n in the case n “ Figure 3.
Case n “
7. The ‚ correspond to ˜ q , the ˆ to˜ p and the ` to where they take the same value.By symmetry, D p p n ˚ q n qp m q “ ´ D p p n ˚ q n qp´p m ´ qq if m ď p n ˚ q n has exactly n modes, located in ´ n ` , . . . , ´ , , , . . . , n ´
1. The proof is complete. (cid:3)
Remark 3.1.
Note that if we wanted p n to be such that there is a strictglobal maximum in , we could replace ˜ p n in the proof of Theorem 1.1 bythe restriction to Z of x ÞÑ exp ˆ ´ ” xn ´ ` ı i ˙ for i P N large enough, andthen proceed as above. CHARLES ARNAL Continuous case
We come up with a smooth variant of the construction used in Section 3.Given a function p : Z ÝÑ Z , we defineΦ p p q : “ ÿ k P Z p p k q s k ´ ,k ` s : R ÝÑ R . Note that Φ p p q| Z “ p . Lemma 4.1.
Let p, q : Z ÝÑ Z be two functions with finite support.Then the convolution Φ p p q ˚ R Φ p q q over R (where Φ p p q ˚ R Φ p q qp x q “ ş R Φ p p qp t q Φ p q qp x ´ t q dt ) is a continuous piecewise affine function which isaffine on each interval r l, l ` s for l P Z . Moreover, the convolution p ˚ Z q over Z (where p ˚ Z q p m q “ ř k P Z p p k q q p m ´ k q ) coincides with the restrictionto Z of Φ p p q ˚ R Φ p q q : p Φ p p q ˚ R Φ p q qq| Z “ p ˚ Z q. In particular, p ˚ Z q and Φ p p q ˚ R Φ p q q have the same modes. The proof of Lemma 4.1 is straightforward.Denote by || ´ || : f ÞÑ || f || “ sup x P R p| f p x q|q the uniform norm. Weneed the following Lemma, which is easy to prove using classical smoothapproximation tricks: Lemma 4.2.
For any n P N , there exists a family of smooth symmetricabout density functions t g Nn u N P N such that: (1) As N goes to , g Nn converges to Φ p q n q in norm L . (2) There exists M ą such || g Nn || ď M for all N . (3) If n is even (respectively, odd and strictly greater than ), g Nn isstrictly increasing from ´8 to ´ n ` (respectively ´ n ` ) andstrictly decreasing from ´ n ` (respectively ´ n ` ) to (and sym-metrically so on R ` ) for all N . In particular, g Nn is bimodal. (4) If n “ , g N is strictly increasing from ´8 to ´ and strictly de-creasing from ´ to (and symmetrically so on R ` ) for all N . Inparticular, g N is bimodal.Proof of Lemma 4.2. We assume that n ě n “ N P N , one can for example consider the function h N : “ x ÞÑ ` exp p Nxx ´ q t´ ă x ă u ` t x ě u , illustrated in Figure 4, which is smooth and approximates the Heavysidestep function as N Ñ 8 .Let also b N : Z Ñ R be such that b N p k q “ k R t´ n ` , . . . , n ´ u ,that b N p k q “ N if k P t´ n ` , . . . , n ´ u is even, and b N p k q “ ´ N if k P t´ n ` , . . . , n ´ u is odd. Then for any n ě N P N , onecan define a smooth and symmetric about 0 function ˜ g Nn as follows: let˜ g Nn p x q “ x ă ´ n ` x ą n ´
2, and let˜ g Nn p x q “ p ˜ q n p k ` q ` b N p k ` q ´ ˜ q n p k q ´ b N p k qq¨ h N p p x ´ k ´ . qq` ˜ q n p k q` b N p k q Figure 4.
The graph of h N : x ÞÑ ` exp p Nxx ´ q t´ ă x ă u ` t x ě u for N “ Figure 5.
The graph of ˜ g Nn , which serves as a smooth ap-proximation of Φ p ˜ q n q , for n “ N “ x P r k, k ` s and k P t´ n ` , n ´ u , where ˜ q n is as in the proof ofTheorem 1.1. The case n “ N “
10 is illustrated in Figure 5. Then˜ g Nn converges to Φ p ˜ q n q in norm L as N Ñ 8 , and the family of functions g Nn : “ ˜ g Nn ş R ˜ g Nn p x q dx satisfies conditions (1) to (4). (cid:3) We can now prove Theorem 1.2.
Proof of Theorem 1.2.
Consider the mass functions p n and q n defined in theproof of Theorem 1.1.For any N P N ą , let f Nn be defined by f Nn p x q “ exp p´p xn ´ ` q N q ş R exp p´p yn ´ ` q N q dy if n ě f N p x q “ exp p´p x q N q ş R exp p´p y q N q dy CHARLES ARNAL if n “ N goes to infinity, f Nn converges in norm L to Φ p p n q .Let t g Nn u N P N be as in Lemma 4.2. Using properties (1) and (2), we seethat || f Nn ˚ g Nn ´ Φ p p n q ˚ Φ p q n q|| ď|| f Nn ˚ g Nn ´ Φ p p n q ˚ g Nn || ` || Φ p p n q ˚ g Nn ´ Φ p p n q ˚ Φ p q n q|| ď M || f Nn ´ Φ p p n q|| L ` || g Nn ´ Φ p q n q|| L N Ñ8 ÝÝÝÝÑ . In particular, p f Nn ˚ g Nn q| Z converges uniformly to p Φ p p n q ˚ Φ p q n qq| Z “ p n ˚ q n (see Lemma 4.1).We have seen in the proof of Theorem 1.1 that if n is even, p n ˚ q n p´ n q ă p n ˚ q n p´ n ` q ą p n ˚ q n p´ n ` q ă . . . ă p n ˚ q n p´ q ą p n ˚ q n p q ă p n ˚ q n p q ą . . . ą p n ˚ q n p n ´ q ă p n ˚ q n p n ´ q ą p n ˚ q n p n q . Hence for N large enough, we also have f Nn ˚ g Nn p´ n q ă f Nn ˚ g Nn p´ n ` q ą f Nn ˚ g Nn p´ n ` q ă . . . ă f Nn ˚ g Nn p´ q ą f Nn ˚ g Nn p q ă f Nn ˚ g Nn p q ą . . . ą f Nn ˚ g Nn p n ´ q ă f Nn ˚ g Nn p n ´ q ą f Nn ˚ g Nn p n q , which means (using Rolle’s theorem) that f Nn ˚ g Nn has at least n modes.Moreover, the number of modes must be even, since 0 is not a mode (andthe convolution is symmetric with respect to 0).The same reasoning applies if n is odd and strictly greater than 1, inwhich case p n ˚ q n p´ n q ă p n ˚ q n p´ n ` q ą p n ˚ q n p´ n ` q ă . . . ą p n ˚ q n p´ q ă p n ˚ q n p q ą p n ˚ q n p q ă . . . ą p n ˚ q n p n ´ q ă p n ˚ q n p n ´ q ą p n ˚ q n p n q and f Nn ˚ g Nn p´ n q ă f Nn ˚ g Nn p´ n ` q ą f Nn ˚ g Nn p´ n ` q ă . . . ą f Nn ˚ g Nn p´ q ă f Nn ˚ g Nn p q ą f Nn ˚ g Nn p q ă . . . ą f Nn ˚ g Nn p n ´ q ă f Nn ˚ g Nn p n ´ q ą f Nn ˚ g Nn p n q for N large enough, which again means that f Nn ˚ g Nn has at least n modes.Moreover, as p g Nn q p´ n ` ´ q “ ´p g Nn q p n ´ ` q ą f Nn ˚ g Nn in 0 is negative if N is largeenough by considering p f Nn q (which converges to ´ ´ δ ´ n ` ´ ` δ n ´ ` ¯ ,where δ x is the Dirac distribution in x P R ). Hence, 0 is a mode of f Nn ˚ g Nn ,which has an odd number of modes. If n “
1, likewise, p ˚ q p´ q ă p ˚ q p q ą p ˚ q p q and f N ˚ g N p´ q ă f N ˚ g N p q ą f N ˚ g N p q for N large enough, which implies that f N ˚ g N has at least one mode. (cid:3) Remark 4.3.
By adding more technical conditions and being more carefulthan we are in Lemma 4.2 when defining the functions g Nn , it can be tediouslyshown that we can make it so that f Nn ˚ g Nn has exactly n modes.The idea behind it is to make sure that the convolution alternates betweenbeing strictly convex and strictly concave, so that it admits exactly one modeon each interval on which it is convex. To do so, one needs to considerthe second derivative p f Nn q ˚ p g Nn q of f Nn ˚ g Nn , and use the fact that p f Nn q converges nicely to a normalized sum of Dirac distributions and that g Nn canbe chosen so that its derivative also converges to a weighted sum of Diracdistributions. Additional conditions concerning the behavior of g Nn on certainpairs of segments must also be added. References [AK07] Mohammed I. Ageel and Anwer Khurshid. Simple proofs of two results on con-volutions of discrete unimodal distributions.
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Charles Arnal, Univ. Paris 6, IMJ-PRG, France.
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