...and encoding it as a probability distribution.
Suppose we have a sequence of non-negative integers that is periodic with period $N$:
\begin{equation*} A_{1},A_{2},...,A_{N},A_{1}... \end{equation*}
Each $A_{k}$ takes on a value no greater than some constant $B$:
\begin{equation*} 0 \leq A_{k} \leq B \end{equation*}
We then take this sequence and do a simple convolution, for some constant $L > 0$ and $1 \leq n \leq N$:
\begin{equation*} S_{L}(n) = A_{n} + A_{n+1} +...+ A_{n+L-1}. \end{equation*}
From $S_{L}(n)$ we then form a probability distribution $P(n)$ which gives the frequency of each of its values. Let $e_{j}(k) = 1$ if $j = k$ and $0$ otherwise. Then:
\begin{equation*} P(n) = (e_{n}(S_{L}(1)) + e_{n}(S_{L}(2)) +...+ e_{n}(S_{L}(N))) / N. \end{equation*}
What I would like to find out is the extent to which this process can be reversed. I have two data points:
1) I know (pretty much) everything about the probability distribution $P(n)$: the distribution itself, its mean, range, variance, skewness, kurtosis, etc.
2) I can tell you the frequency of values of $A_{k}$ in one period, so that if the sequence is 1,0,2,3,1,0, I can tell you there are two 0's, two 1's, one 2, and one 3.
To what extent am I able to reconstruct the sequence $A_{k}$ from these two data points?