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Assuming that $a$ has a density, regularly varying at $a = 1$ with exponent $-1 < \\beta < 1$, different joint limits of normalized aggregated partial sums are shown to exist when $N^{1/(1+\\beta)}/n$ tends to (i) $\\infty$, (ii) 0, (iii) $0 < \\mu < \\infty$. 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