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Consider a non standard continuous-time renewal risk model with a constant force of interest, in which the claim sizes are assumed to be conditionally linearly wide dependent (CLWD) and belong to the intersection of dominatedly varying tailed and long tailed class, and inter-arrival times are assumed to be a sequence of independent and identically distributed random variables independent of the claim sizes. Under some technical conditions, we obtain an asymptotic formula for the tail probability of discounted aggregate claims, which holds locally uniform for all time horizon within a finite interval. When the claim sizes are further restricted to be consistently varying tailed, we show that this asymptotic formula is globally uniform for all time horizon within an infinite interval. [ABSTRACT FROM AUTHOR]