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The cumulative return over the entire path is the sum of the returns on the individual periods: $$X = X_1 + X_2 + \ldots + X_N.$$ Two potential definitions of the volatility of this process would be $Std(X) / \sqrt{N}$ (which is exactly your "cross-section" volatility) or $Std(X_i)$ (assuming each $X_i$ has the same unconditional distribution). If the $X_i$ ...

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