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While the forms of the distribution at very long times, i.e. $t\\to\\infty$, are very well known and are related to the Gaussian Central Limit Theorem or the L\\'{e}vy stable laws, the alternative limit of large number of renewals, i.e. $N\\to\\infty$, is much less noted. We address this limit of large $N$ and find that it attains a universal form that solely depends on the analytic properties of the distribution of renewal times. 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While the forms of the distribution at very long times, i.e. $t\\to\\infty$, are very well known and are related to the Gaussian Central Limit Theorem or the L\\'{e}vy stable laws, the alternative limit of large number of renewals, i.e. $N\\to\\infty$, is much less noted. We address this limit of large $N$ and find that it attains a universal form that solely depends on the analytic properties of the distribution of renewal times. 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