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This paper studies the tail probability of weighted sums of the form$\sum_{i=1}^n c_i X_i$, where random variables $X_i$'s are either independentor pairwise quasi-asymptotical independent with heavy tails. Using$h$-insensitive function, the uniform asymptotic equivalence of the tailprobabilities of $\sum_{i=1}^n c_iX_i$, $\max_{1\le k\le n}\sum_{i=1}^k c_iX_i$and $\sum_{i=1}^n c_iX_i^+$ is established, where $X_i$'s are independent andfollow the long-tailed distribution, and $c_i$'s take value in a broadinterval. Some further uniform asymptotic results for the weighted sums of$X_i$'s with dominated varying tails are obtained. An application to the ruinprobability in a discrete-time insurance risk model is presented.

Uniform asymptotics for the tail probability of weighted sums with heavy tails