In the problem of some sequential estimation, the stopping times may be written in the form $N(c) = inf{n geq n_0; n geq c^2 S^2_n/delta^2 (ar{X}_n)}$ where ${s^2_n}$ and ${ar{X}_n}$ are the sequences of sample variance and sample mean of the independently and identically distributed (i.i.d.) random variables with distribution $F_{ heta}(x), heta in Theta$, respectively, and $delta$ is either constant or any given positive real valued function. We obtain some asymptotic normality and asymptotic expectation of the N(c) in various limiting situations. Specially, uniform asymptotic normality and uniform asymptotic expectation of the N(c) are given.