Two-step empirical CDF from one-step empirical CDF

(cross-post)

Suppose I have a random variable X_i which changes by X_{i+1}=X_i+\delta_i from one timestep to the next. Suppose I do an experiment where I observe N values d_1,d_2,\ldots,d_N of $\delta_0$ and make an experimental CDF of \delta_0, by sorting d so that d_1 \leq d_2 \cdots \leq d_N, and then approximating the CDF of y as \frac{i}{N} where $d_i \leq y \leq d_{i+1}$.

Question: What is the most efficient way to compute the empirical CDF of two steps of X, assuming that the process for going from X_i to X_{i+1} follows the same empirical distribution? The brute force way that occurs to me is to create the set E={d_i+d_j: 1\leq i\leq N, 1\leq j\leq N}, sorting E=e_1,\ldots,e_{N^2} and then approximating the two-step CDF of y as \frac{i}{N^2} where e_i \leq y \leq e_{i+1}.

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