Mode of sum of two random variables (RVs) v.s. Mode of each RV

Question

I wonder is there any literature or answer of the following relationship.

What I want to know is whether the maximum peak of $\mathbf{X}+\mathbf{Y}$ is reasonably close to the maximum peak of $\mathbf{X}$. Suppose the following things.

• $\mathbf{X}$ and $\mathbf{Y}$ are two random variables on the same probability space
• $\mathbf{X}$ should be any arbitrary distribution, while $\mathbf{Y}$ is unimodal (or uniform distribution within some range).
• $\mathbf{X}$ can have multiple modes, but there is only a single mode that has a maximum probability.
• Doesn't matter whether discrete or continuous case
• for simplicity, $\mathbf{Y}$ has no shifted bias, i.e. zero-mean

Can I quantify or bound the position of the mode of $\mathbf{X}+\mathbf{Y}$ by the positions of the maximum mode of $\mathbf{X}$ and $\mathbf{Y}$? I don't want to get some simulation examples, which has been already checked by myself. What I'm interested in is an explicit inequality by each of mode. If any generic relationship doesn't exist, what will be the condition for that?

I appreciate in advance, thanks!

Answer 1

Comment. (Certainly not a definitive answer.)

I tried one example of what I believe you described: The sum $T$ (bottom) of $X$, a bimodal mixture distribution, and a normal distribution $Y$. It seems that $Y$ can control the position of the mode(s) in $T.$ It shifts the mixture distribution to the right by about 20 units and smears out its peaks a little. If the modes of $X$ were closer together, then $T$ might not be bimodal. Here you can find the PDF of $X$ and thus find the PDF of $T.$

Here is R code for this experiment, in case it is of any use to you.

 x1 = rnorm(1000, 50, 3);  x2 = rnorm(1000, 70, 5);  x = c(x1, x2)
 y = rnorm(2000, 20, 5);  t = x + y
 par(mfrow=c(3, 1))  # enables 3-panel plot
   hist(x, prob=T, col="skyblue2", xlim=c(0,100), 
        main="50:50 Mix of NORM(50,3) & NORM(70,5)")
   hist(y, prob=T, col="skyblue2", xlim=c(0,100), 
        main="NORM(20,5)")
   hist(t, prob=T, col="skyblue2", xlim=c(0,100), 
        main="SUM of Mix and NORM(20,5)")
 par(mfrow=c(1,1))  # returns to default single-panel plots

Addendum: You changed the question, so I am changing my "answer" to illustrate my latest Comment beneath your Question But one cam play with simulated examples all day long without proving anything. You need to derive the formulas of the density functions. I have shown 'kernel density estimators' to give a rough idea.

R code for plotting the first density estimator.

lines(density(x), lwd=2, col="green4")