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Let $X$ be a discrete random variable supported on $\{−5,\dots, 6\}$ in which the outcomes have the following respective probabilities: $$\begin{array}{rc} n & p(n) \\ \hline −5 & 6/36 \\ −4 & 0 \\ −3 & 0 \\ −2 & 6/36 \\ −1 & 5/36 \\ 0 & 5/36 \\ 1 & 2/36 \\ 2 & 2/36 \\ 3 & 3/36 \\ 4 & 4/36 \\ 5 & 2/36 \\ 6 & 1/36 \end{array}$$ and let the random variable $S_n$ be the sum of $n$ IID copies of $X$. Since $X$ has expected value 0, so does $S_n$. Let $p_n$ be ${\rm Prob}[S_n > 0]$ and $q_n$ be ${\rm Prob}[S_n < 0]$, so that for instance $p_1 =14/36$ and $q_1 = 17/36 > p_1$. Is it true that $q_n > p_n$ for all $n$ besides $n=4$? (It is not hard to show that $p_4 > q_4$, and it has been checked that $q_n > p_n$ for all other $n$ up to 100.)

More broadly, one can ask what kinds of behavior the sign of $p_n-q_n$ can exhibit if we replace $X$ by an arbitrary finitely-supported integer-valued mean-zero random variable. Must the sign be eventually constant (as it appears to be in my specific example), or can it exhibit more complicated forms of behavior?

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  • $\begingroup$ Have you tried looking at the Fourier transform of the distribution? I think you can describe $p$ and $q$ by weighted sums of powers of the coefficients, which might be helpful. $\endgroup$
    – Daniel Weber
    Commented Apr 6 at 12:39
  • $\begingroup$ Have you tried graphing $p$ and $q$, by the way? Do they approach some values? $\endgroup$
    – Daniel Weber
    Commented Apr 6 at 12:49
  • $\begingroup$ Alternatively maybe you can use the Cornish–Fisher expansion? $\endgroup$
    – Daniel Weber
    Commented Apr 6 at 13:19
  • $\begingroup$ Hi @James Propp, this is not an answer. I am just suggesting a reformulation to see if that helps you. If the random variables had a density function (continuous case), they being iid, their joing deinsity would be self-convolution n times. Then what you are asking can be converted to a question of "does the RHS area of this density function exceed the LHS area, after n convolutions?" . I am thinking of approaching this as study of symetry in the smoothening property of convolution. $\endgroup$
    – Srini
    Commented Apr 8 at 14:22
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    $\begingroup$ @Iosef Pinelis: I would accept it as an answer if it showed that $q_n > p_n$ for all $n>4$, not just for all sufficiently large $n$, with full details included. $\endgroup$
    – James Propp
    Commented Jun 2 at 13:35

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The sign will be eventually constant, except perhaps for a few special distributions of $X$.

This follows from the asymptotic expansion in powers of $n^{-1/2}$ of the cdf of the sum of iid integer-valued random variables with finite moments of some order -- see e.g. Theorem 6 in Section 3 of Chapter VI in Petrov's book.

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  • $\begingroup$ How does the sign of $p_n-q_n$ connect with the asymp. expansion in powers of $n^{-1/2}$? $\endgroup$
    – MikeG
    Commented Apr 7 at 20:16
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    $\begingroup$ @MikeG : The connection of the cdf (say $G_n$) of the sum $S_n$ of iid integer-valued random variables is as follows: $p_n-q_n=1-G_n(0)-G_n(-1)$. $\endgroup$
    – Iosif Pinelis
    Commented Apr 7 at 20:48

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