## 1 Introduction  
State space models represent an extremely well-established framework for the predictive modelling of time series, which has found countless applications in statistics, engineering, econometrics, neuroscience, and other fields [1; 2; 3; 4; 5; 6]. If both the dynamical processes within the system to be studied and the observation process can be assumed to be linear, the optimal algorithm for recursively estimating the current state of the system is given by the widely used Kalman filter [7]; for estimating past states, smoothing algorithms are available, many of which themselves are based on the results of Kalman filtering. A well-known example of a recursive smoothing algorithm is given by the Rauch-Tung-Striebel (RTS) smoother [8]. There are numerous ways to derive the equations of the Kalman filter and the RTS smoother. This paper starts with an approach that was introduced by [PERSON] and [PERSON] in 1972 [9]. Their approach is based on expressing the problem of state estimation as a weighted least-squares problem. The first purpose of the present paper is to generalise the weighted least-squares approach to the case of correlations being present between the two noise terms arising in state space models, i.e., the dynamical noise term and the observation noise term; in the paper by [PERSON] and [PERSON], it was assumed that these noise terms are uncorrelated. [PERSON] and [PERSON]'s approach to state estimation corresponds with estimating the states for all time points simultaneously by inverting a single matrix, usually of large dimensions; Kalman filtering and RTS smoothing can be interpreted as a recursive approach to inverting this matrix. However, non-recursive approaches would also be possible, and some authors actually advocate for replacing Kalman filtering and RTS smoothing with other inversion algorithms. As an example, [PERSON] and [PERSON] [10] are mentioned, who have \"rediscovered\" the weighted least-squares approach to state estimation, based on earlier work by [PERSON] and [PERSON] (1997), apparently without noticing the much earlier paper by [PERSON] and [PERSON]. [PERSON] (1997, 1998) has reformulated and extended the original work of [PERSON] and [PERSON] in various directions, but excluding the case of correlated noise terms. He approaches the aforementioned task of inverting a large matrix through a blockwise LDL decomposition of that matrix; [PERSON] and [PERSON] (2000), in contrast, employ Cholesky decomposition. [PERSON] has shown that by applying a recursive inversion algorithm for one-block-banded matrices, originally formulated by [PERSON] (1998), to the LDL decomposition, the Kalman filter (in its \"information filter\" variant) and the RTS smoother can be retrieved. The derivation of square-root information filters via the Duncan-Horn approach has been discussed by [PERSON] and [PERSON] (1998) and by [PERSON] (1999). The second purpose of this paper is to discuss the fact that for the task of state estimation in the case of correlated noise terms, two different generalised Kalman filter algorithms exist that are not equivalent. Most available textbooks on state space modelling and Kalman filtering present either one or the other of these algorithms; [PERSON] (1998) is a rare example of an author presenting both algorithms. As also mentioned by [PERSON], these two algorithms correspond to two slightly different forms of defining the underlying state space model. The difference between these two forms is given solely by the time index of the dynamical noise term, relative to the time index of the state vector; [PERSON] labels these two forms the \"contemporaneous form\" and the \"future form\" ([PERSON], 1998). While in control engineering, the \"future form\" is used almost exclusively, in statistics, the \"contemporaneous form\" is also frequently used due to its relationship with autoregressive moving-average (ARMA) modelling, which represents an important tool for time series analysis ([PERSON], 1998).