A hidden Markov regime is a Markov process that governs the time or space dependent distributions of an observed stochastic process. Recursive algorithms can be used to estimate parameters in mixed distributions governed by a Markov regime. The authors derive a recursive algorithm for estimation of parameters in a Markov-modulated Poisson process also called a Cox point process. By this the authors mean a doubly stochastic Poisson process with a time dependent intensity that can take on a finite number of different values. The intensity switches randomly between the possible values according to a Markov process. The authors consider two different ways to observe the Markov-modulated Poisson process: in the first model the observations consist of the observed time intervals between events, and in the second model they use the total number of events in successive intervals of fixed length. They derive an algorithm for recursive estimation of the Poisson intensities and the switch intensities between the two states and illustrate the algorithm in a simulation study. The estimates of the switch intensities are based on the observed conditional switch probabilities.

A recursive algorithm is proposed for estimation of parameters in mixture models, where the observations are governed by a hidden Markov chain. The often badly conditioned information matrix is estimated, and its inverse is incorporated into the algorithm. The performance of the algorithm is studied by simulations of a symmetric normal mixture. The algorithm seems to be stable and produce approximately normally distributed estimates, provided the adaptive matrix is kept well conditioned. Some numerical examples are included.>

A Slepian model is derived for the shape of dicks in modulated FM transmission. The modulated signal may be a deterministic periodic function or a stationary stochastic process, not necessarily Gaussian. To describe the clicks one first derives the stochastic properties of noise and signal at the clicks. (For a periodic signal this amounts to giving the relative number of clicks in any given part of the period.) Then the conditional distributions of noise and signal near clicks are derived in explicit form, using the noise and signal values at the click as (random) parameters, and finally these conditional distributions are mixed into a total model for the click shapes. For increasing carrier-to-noise power ratioA^{2}/2, the Slepian model converges after normalization to a rational function with random coefficients. The amplitude of a click is shown to be of the orderA^{2}, in contrast to the unmodulated ease, where click amplitudes are of the orderA.

The stochastic behavior of FM-clicks in an unmodulated carrier plus noise is studied in terms of the conditional behavior of stationary Gaussian processes after level crossings. By a systematic use of thc Slepian model processes it is shown that the normalized shape of a positive FM-click approaches a certain rational function with simple random coefficients as the carrier-to-noise ratio tends to infinity. The outcomes of this random function illustrate the typical click shapes that have been observed experimentally. The distribution of the duration of a click is studied by means of similar model processes, and its exact asymptotic density is derived. It is seen to be very similar to a Maxwell density, although of smaller order for large durations.