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How to calculate the following multivariate probability mass function:

$P(X_1-X = n, X_2-X = n, ..., X_{N-1}-X = n)$

Where $n$ and $N$ are positive integers, and $X_i$ and $X$ are iid random variables with the following discrete probability distribution:

$P(i)=\frac{C_{i-1}}{2^{2i-1}}$; $C_{i}$ are Catalan numbers.

Looking at: Integrating pdf times cdf squared and walking backwards, from multivariate to integral representation I got:

$ \sum _{i=0} ^{+ \infty} {P(i)P(i+n)^{N-1}}$

However, it appears that the last formula gives me correct results only for N=2. Am I doing something wrong?

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    $\begingroup$ What have you tried so far? If these are independent random variables, you can work out the answer by conditioning on the value of $X$. $\endgroup$ Sep 25, 2023 at 13:05
  • $\begingroup$ They are iids, I have amended the question. I have also added something I tried but didn't work. $\endgroup$
    – Francesco
    Sep 25, 2023 at 15:50

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The start of your solution looks correct to me. We have $$ \begin{align} p & = P(X_1-X = n, \ldots , X_{N-1}-X = n ) \\ & = \sum_{i=1}^\infty P(X_1-X = n, \ldots , X_{N-1}-X = n | X = i) P(X = i) \\ & = \sum_{i=1}^\infty P(X_1 = n+i, \ldots , X_{N-1} = n+i) P(X = i) \\ & = \sum_{i=1}^\infty f_X(n+i)^{N-1} f_X(i) \end{align} $$ by independence, where $f_X$ is the PMF. Substituting in $f_X(i)$ gives $$ p = \sum_{i=1}^\infty \left[\frac{C_{n+i-1}}{2^{2(n+i)-1}} \right]^{N-1} \frac{C_{i-1}}{2^{2i-1}} $$

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