SummaryLetX1X2,.be i.i.d. random variables and let Un= (n r)‐1S̀(n,r)h (Xi1,.,Xir,) be aU‐statistic withEUn= v,vunknown. Assume that g(X1) =E[h(X1,.,Xr) ‐v|X1]has a strictly positive variance s̀2. Further, let a be such that φ(a) ‐ φ(‐a) =α for fixed α, 0<α0, and a confidence interval forvof length 2d,of the formIn,d= [Un,‐d, Un+ d].We assume that VarUnis unknown, and hence, no fixed sample size method is available for finding a confidence interval forvof prescribed width 2d and prescribed coverage probability α Turning to a sequential procedure, letIN(d),dbe a sequence of sequential confidence intervals forv.The asymptotic consistency of this procedure, i.e. limd → 0P(v ∈ IN(d),d)=α follows from Sproule (1969). In this paper, the rate at which |P(v ∈ IN(d),d)converges to α is investigated. We obtain that |P(v ∈ IN(d),d)‐ α| = 0 (d1/2‐(1+k)/2(1+m)), d → 0, where K = max {0,4 ‐ m}, under the condition thatE|h(X1, Xr)|m2. This improves and extends recent results o