The author is very grateful to Clément Deslandes for helpful discussions.†† This research was supported by the DIM Math Innov de la Région Ile-de-France.
Log-concave functions and sequences feature preeminently in many mathematical domains, including probability, statistics and combinatorics, but also more surprisingly algebraic geometry (read[Survey1] and [Survey2] for surveys of the notion).
Likewise, it is natural to consider the properties preserved by the convolution of two distributions, and in particular to consider the number of modes of the resulting distribution. It has been known for a long time that the convolution of two unimodal distributions need not be unimodal, though the convolution of two symmetric unimodal distributions will be unimodal (see [DJDUnimodality], or [ProofAlternativeDiscrete] for an alternative proof of the second fact in the discrete case).
Further explorations have shown that convolutions of log-concave real distributions are log-concave (hence unimodal) (see [ConvolutionLogConcave]), and that the convolution of a unimodal (but not symmetric) distribution with itself can have any number of modes (see [ConvolutionAnyNumberModes]). In the same line of questioning, we prove the following result:
Let be greater or equal to . Then there exists a discrete log-concave distribution and a discrete bimodal distribution , both symmetric about , such that their convolution has exactly modes.
We also prove a continuous variant, which directly contradicts the main statement111The mistake can be found in the proof of Theorem 1 in [Faux]: Equations (3) and (4) do not imply Equation (5). in [Faux]:
Let be greater or equal to . Then there exists an absolutely continuous distribution on whose density function is smooth, symmetric about and log-concave (hence unimodal), and an absolutely continuous distribution on whose density function is smooth, symmetric about and bimodal, such that the convolution has at least modes.
In fact, we can ask that have exactly modes; proving it makes the demonstration much more tedious, yet not any deeper.
In this note, we will consider absolutely continuous distributions on with continuous density functions and discrete distributions on .
The modes of an absolutely continuous distribution with continuous density function are the local maxima of .
We will say that an absolutely continuous distribution with continuous density function is strictly -modal in if there exists such that is strictly increasing on and for , and strictly decreasing on and for .
We will say that a discrete distribution with probability mass function has a mode in if . We say that it is -modal if it has exactly modes.
In the discrete case, a mode is often required to be a global maximum of the mass function. This nuance in definition matters not, as all our discrete modes will be global maxima.
The density function of an absolutely continuous distribution is called log-concave if for all and all . If is strictly positive, it is equivalent the concavity of .
The mass function of a discrete distribution is called log-concave if for all , and if its support is a contiguous interval, i.e. if there exists such that , for all and all , and for all . The second condition is sometimes omitted.
It is easy to show that log-concavity implies unimodality both in the continuous and the discrete case.
As usual, the convolution of density functions (and by extension the convolution of the two associated absolutely continous distributions) is defined as
for all .
Similarly, the convolution of mass functions (and by extension the convolution of the two associated discrete distributions) is defined as
for all .
3. Discrete case
Proof of Theorem 1.1.
The case is almost trivial: consider and , where refers to the indicator function of for any .
Now for , let us define and . The function is clearly a log-concave distribution and symmetric about .
If is even, let be as such: for , we let and . We also let , and be for any . We define on symmetrically.
Let and . Then is symmetric about , and it is a bimodal distribution whose modes are in . It has a minimum in .
See Figure 1 to see the case illustrated.
The convolution is clearly symmetric about .
For any , the finite difference satisfies . As is equal to if , to if and to otherwise, we get that .
We see that is if and is strictly negative if . For , the finite difference is equal to if
is odd and toif is even.
By symmetry of and , if , we have that
From this, we get that has exactly modes, located in .
The convolution is illustrated in Figure 2 in the case (remember that ).
The case where is odd (and strictly greater than ) is very similar. Let be as such: for we let , and for we let . We also let , and be for any .
As before, we define on symmetrically, let and . Then is symmetric about , and it is a bimodal distribution whose modes are in . It has a minimum in .
The case is illustrated in Figure 3.
As above, . Thus we see that is if and is strictly negative if . For , the finite difference is equal to if is even and to if is odd.
By symmetry, if .
From this, we get that has exactly modes, located in . The proof is complete.
Note that if we wanted to be such that there is a strict global maximum in , we could replace in the proof of Theorem 1.1 by the restriction to of for large enough, and then proceed as above.
4. Continuous case
We come up with a smooth variant of the construction used in Section 3.
Given a function , we define
Note that .
Let be two functions with finite support.
Then the convolution over (where ) is a continuous piecewise affine function which is affine on each interval for . Moreover, the convolution over (where ) coincides with the restriction to of :
In particular, and have the same modes.
The proof of Lemma 4.1 is straightforward.
Denote by the uniform norm. We need the following Lemma, which is easy to prove using classical smooth approximation tricks:
For any , there exists a family of smooth symmetric about density functions such that:
As goes to , converges to in norm .
There exists such for all .
If is even (respectively, odd and strictly greater than ), is strictly increasing from to (respectively ) and strictly decreasing from (respectively ) to (and symmetrically so on ) for all . In particular, is bimodal.
If , is strictly increasing from to and strictly decreasing from to (and symmetrically so on ) for all . In particular, is bimodal.
Proof of Lemma 4.2.
We assume that - the case is similar. Given , one can for example consider the function
illustrated in Figure 4, which is smooth and approximates the Heavyside step function as .
Let also be such that if , that if is even, and if is odd. Then for any and , one can define a smooth and symmetric about function as follows: let for any or , and let
satisfies conditions (1) to (4).
We can now prove Theorem 1.2.
Proof of Theorem 1.2.
Consider the mass functions and defined in the proof of Theorem 1.1.
For any , let be defined by
This defines a family of smooth log-concave distributions that are symmetric about . Moreover, as goes to infinity, converges in norm to .
Let be as in Lemma 4.2. Using properties (1) and (2), we see that
In particular, converges uniformly to (see Lemma 4.1).
We have seen in the proof of Theorem 1.1 that if is even,
Hence for large enough, we also have
which means (using Rolle’s theorem) that has at least modes. Moreover, the number of modes must be even, since is not a mode (and the convolution is symmetric with respect to ).
The same reasoning applies if is odd and strictly greater than , in which case
for large enough, which again means that has at least modes. Moreover, as by property (3) we see that the second derivative of in is negative if is large enough by considering (which converges to , where is the Dirac distribution in ). Hence, is a mode of , which has an odd number of modes.
If , likewise, and for large enough, which implies that has at least one mode. ∎
By adding more technical conditions and being more careful than we are in Lemma 4.2 when defining the functions , it can be tediously shown that we can make it so that has exactly modes.
The idea behind it is to make sure that the convolution alternates between being strictly convex and strictly concave, so that it admits exactly one mode on each interval on which it is convex. To do so, one needs to consider the second derivative of , and use the fact that converges nicely to a normalized sum of Dirac distributions and that can be chosen so that its derivative also converges to a weighted sum of Dirac distributions. Additional conditions concerning the behavior of on certain pairs of segments must also be added.