The "diversification return" of a portfolio is the difference between the geometric average (compound) return of a rebalanced portfolio and the weighted sum of the geometric average returns of the assets. My understanding is this diversification return is always nonnegative in a portfolio with positive asset weights (only long positions)
I have not found a rigorous proof that diversification return is never negative for the most general case involving $N$ assets. I have found references that show diversification return is approximately the difference between the weighted average of asset variances and the portfolio variance which must be nonnegative. But I would like to see a proof that does not rely on any approximation for an arbitrary number of assets.
To formalize the problem, suppose we have a portfolio with N assets that is always rebalanced to maintain constant asset weights $w_1, w_2, \ldots, w_N$. Let $r_{ij}$ denote the return of asset $i$ in holding period $j$ where $i=1,2,\ldots,N$ and $j = 1,2,\ldots, T$. The return of the portfolio in period $j$ is $r_{Pj} = w_1r_{1j} + \ldots w_N r_{Nj}$ and so the geometric average return of the portfolio over the $T$ periods is
$$g_P = [(1+r_{P1})(1+r_{P2})\cdots (1+r_{PT})]^{1/T} -1$$
The geometric average return of asset $i$ is $$g_i = [(1+r_{i1})(1+r_{i2})\cdots (1+r_{iT})]^{1/T} -1$$
Can it be proved rigorously that $g_P \geq w_1g_1 + w_2 g_2 + \ldots + w_N g_N$?