2
$\begingroup$

This is perhaps a well-known result and I'd appreciate a reference if that's the case. Let $X$, $Y$ be iid random variables with support $S \subset \mathbb{Z}$ and let

$$P(X = x) = f(x, \theta)$$

where $\theta$ represents a vector of a countable number of parameters.

The question is twofold. Firstly, does there exist non-trivial distributions such that

$$P(X + Y = x) = f(x, \theta_1(\theta)), \;\;\; P(XY = x) = f(x, \theta_2(\theta))$$

for some $\theta_1$, $\theta_2$? By trivial here, I mean any case that, for each $x \in S$, there is a corresponding parameter $\theta_x$ (and thus non-trivial means there isn't a correspondence between elements of the support set and parameters).

Secondly, within this class of distributions, does there exist distributions with a finite number of parameters?

Improve this question
$\endgroup$
10
  • $\begingroup$ Can't the entire class of random variables on $\mathbb{Z}$ be represented by a countable number of parameters (say, $P(0), P(1|\text{not}0), P(-1|\text{not}0, \text{not}1), \dots$)? I'm not sure I understand your bijection condition. $\endgroup$
    – user44191
    Commented Nov 14, 2019 at 6:57
  • $\begingroup$ @user44191 This is precisely the type of situation I wished to exclude. In your case, for each element in the support set, there is a corresponding parameter. I've updated the question to hopefully be more clear about this. $\endgroup$
    – bursneh
    Commented Nov 14, 2019 at 12:09
  • 2
    $\begingroup$ This isn't a full answer, but you can phrase the pdf of the product of two (independent) normal distributions in terms of Bessel Functions, which can then be phrased in terms of Meijer-G functions. The standard exponential can phrased in terms of Meijer-G functions as well (and Meijer-G functions satisfy a certain convolution theorem). By identifying a suitable parametrization of Meijer-G functions you likely can get a family of probability distributions. This doesn't yet answer how to compute the product of arbitrary ... $\endgroup$
    – Mark Schultz-Wu
    Commented Nov 14, 2019 at 14:45
  • $\begingroup$ Random variables in this hypothesized family (only the special case that they're normal), although the general case may be possible. Moreover if you wanted to gain intuition by looking at the pdf of any mentioned distribution, this may be more difficult to do for Meijer-G functions. $\endgroup$
    – Mark Schultz-Wu
    Commented Nov 14, 2019 at 14:47
  • 1
    $\begingroup$ @bursneh: Lesson learned: GGC is closed under multiplication, see Theorem 6.2.1 in Bondesson's book. Then a natural guess would be that "generalised negative binomial convolutions" share similar properties in the discrete setting. To my surprise, the name I just made up is in use, and a whole chapter of Bondesson's book is devoted precisely to this subject. Unfortunately, I did not find a variant of the multiplication theorem for GNBC there. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Nov 15, 2019 at 9:40

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .