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Let $X_i$ be iid random discrete variables with pmf $f$ .

We may restrict ourselve to pmfs with finite even support: $f \in G_N$ ($ f[k] > 0 \implies |k| \le N$) or perhaps $f \in G^{+}_N$ ($ f[k] > 0 \iff |k| \le N$) .

I wonder if there is some characterization on $f$ (perhaps sufficient or necesary conditions) for this property to hold:

$$P(X_1 + \cdots X_{n+1} = 0) < P(X_1 + \cdots X_n = 0) \, \forall n\ge 1$$

Alternatively, letting $f^{(n)}$ denote the $n-$self convolution of $f$, the above is equivalent to $f^{(n+1)}[0] < f^{(n)}[0] $

In by this question it's shown that $f \in G^{+}_1$ is not enough, but $g=f^{(2)}$ is.

leonbloy
  • 66,202
  • One sufficient condition is that the support of $f$ is non-negative. In this case, the sum of the random variables being equal to zero is equivalent to them all being zero. Then as long as $f$ does not only put positive probability on zero, the probability $P\left(X_1 + \cdots + X_{n} = 0\right) = f\left(0\right)^n$ is decreasing in $n$. – Shiva Oct 06 '21 at 13:02
  • What does "but $g=f^{(2)}$ is" mean? – Bart Michels Feb 07 '22 at 17:29

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