Confidence interval with specified precision for the mean value in a sequence of Gaussian variables

Consider the class C of real-valued stochastic processes of the form Xt=m+mt+ξt, with discrete t=1,2,⋯, such that mt→0 and the ξt are random variables with normal distributions N(0,σt), where σt→0. Required is a sequence of estimators m^t for the parameter m, determined by X1,⋯,Xt and a stopping rule τ, such that for every ε>0 and 0