assumes that the estimate sequence generated by the algorithm is bounded, which is not easy to be verified in practice. For many theoretical problems, due to the complexity of the system noise and less _a priori_ information on the estimate sequence, neither PM nor the ODE method can be applied. In order to essentially relax the assumptions required by PM and the ODE method, in the past 35 years an expanding truncation technique was introduced into SAA by [PERSON] and his colleagues, and a so-called stochastic approximation algorithm with expanding truncations (SAAWET) has been developed [22; 23]. The key idea of the expanding truncation technique lies in that if the magnitude of the estimate exceeds the truncation bound, then the estimate is reset to the initial value but with a decreased stepsize, and, at the same time, an enlarged truncation bound will be used for the next recursion, while if the magnitude of estimate is smaller than the truncation bound, then SAAWET evolves as the classical SAA in the next recursion and the truncation bound remains unchanged. The expanding truncation technique introduces a way of adaptively choosing the step-sizes of the algorithm and essentially relaxes the conditions on the noises and the root-searching function. Furthermore, to guarantee the convergence of SAAWET, one only needs to verify the noise condition along the index sequence \(\{n_{k}\}_{k\geq 0}\) of any convergent subsequence \(\{x_{n_{k}}\}_{k\geq 0}\) rather than along the whole sequence \(\{x_{k}\}_{k\geq 0}\), which greatly facilitates the application of such an algorithm. The convergence analysis method of SAAWET is accordingly named the trajectory-subsequence (TS) method. In the past 35 years, SAAWET has successfully been applied to recursive identification of a multivariate ARMAX system [24], recursive identification of nonlinear systems [25; 26], adaptive regulation of a nonlinear system [27], consensus control of multiagent systems [28], convergence of a distributed randomized PageRank algorithm [29], and so forth. In this paper, we first review the probabilistic method (PM) and the ordinary differential equation (ODE) method for SAA and the trajectory-subsequence method for SAAWET. Then, we will give an application example of SAAWET to the recursive principal component analysis. All the above algorithms and methods are applied in a centralized situation. In recent years, distributed algorithms have been extensively investigated--for example, the consensus problem [30; 31], distributed estimation [32; 33], and so on. The distributed algorithms work in a network situation, where there is no central agent collecting and processing the data over the whole network but the agents cooperatively accomplish a global objective by using their local observations and information obtained from communication with their adjacent neighbours [34; 35; 36; 37; 38]. Recently, many distributed problems have been solved by SAA-based algorithms, such as distributed estimation [32; 39], and distributed convex optimization [34; 35]. As a result, the distributed algorithm for SAA has naturally drawn much attention from researchers. Distributed SAA (DSAA) was proposed in [40; 41; 42; 43] and so forth, in which agents cooperatively find roots of a function, being a sum of local functions associated with agents connected in a network. It is noticed that almost all the above DSAA algorithms require restrictive conditions to guarantee convergence; for example, in [41] it requires the global Lipschitz continuity of local functions and the observation noise as a martingale difference sequence. However, these assumptions may not hold for many problems, including the distributed principal component analysis and the distributed gradient-free optimization, and so forth. Aiming at solving the distributed root-seeking problem under weaker conditions compared with those used in [41], a distributed version of SAAWET (DSAAWET) was recently developed [44].