## Bernoulli

• Bernoulli
• Volume 7, Number 2 (2001), 211-221.

### Sufficient conditions for finite dimensionality of filters in discrete time: a Laplace transform-based approach

#### Abstract

The discrete-time filtering problem can be seen as a dynamic generalization of the classical Bayesian inference problem. For practical applications it is important to identify filtering models that, analogously to the linear Gaussian model (Kalman filter), admit a finite-dimensional filter or, equivalently, a finite-dimensional family of filter-conjugate distributions. Our main purpose here is to give sufficient conditions for the existence of finite-dimensional filters. We use a method, based on the Laplace transform, which is also constructive.

#### Article information

Source
Bernoulli, Volume 7, Number 2 (2001), 211-221.

Dates
First available in Project Euclid: 25 March 2004

Permanent link to this document
https://projecteuclid.org/euclid.bj/1080222084

Mathematical Reviews number (MathSciNet)
MR1828503

Zentralblatt MATH identifier
0981.62077

#### Citation

Runggaldier, Wolfgang J.; Spizzichino, Fabio. Sufficient conditions for finite dimensionality of filters in discrete time: a Laplace transform-based approach. Bernoulli 7 (2001), no. 2, 211--221. https://projecteuclid.org/euclid.bj/1080222084

#### References

• [1] Bar-Lev, S., Enis, P. and Letac, G. (1994) Sampling models which admit a given general exponential family as a conjugate family of priors. Ann. Statist., 22, 1555-1586.
• [2] Barndorff-Nielsen, O. (1978) Information and Exponential Families in Statistical Theory. Chichester: Wiley.
• [3] Barndorff-Nielsen, O. and Pedersen, K. (1968) Sufficient data reduction and exponential families. Math. Scand., 22, 197-202.
• [4] Bather, J.A. (1965) Invariant conditional distributions. Ann. Math. Statist., 36, 829-846.
• [5] D'Ambrosio, E., Runggaldier, W.J. and Spizzichino, F. (1998) Construction of discrete time models admitting a finite dimensional filter: an approach based on the inverse Laplace transform. Report 32/98, Dipartimento di Matematica Guido Castelnuovo', Universitá di Roma La Sapienza',September. A vailable at http://www.math.unipd.it/ probab/home/wolfgang/wolfgangpubl.html.
• [6] Diaconis, P. and Ylvisaker, D. (1979) Conjugate priors for exponential families. Ann. Statist., 7, 269- 281.
• [7] Feller, W. (1970) An Introduction to Probability Theory and Its Applications, Vol. II. New York: Wiley.
• [8] Ferrante, M. (1993) On the existence of finite-dimensional filters in discrete time. Stochastics Stochastics Rep., 40, 169-179. Abstract can also be found in the ISI/STMA publication
• [9] Ferrante, M. and Runggaldier, W.J. (1990) On necessary conditions for the existence of finitedimensional filters in discrete time. Systems Control Lett., 14, 63-69.
• [10] Ferrante, M. and Vidoni, P. (1998) Finite dimensional filters for nonlinear stochastic difference equations with multiplicative noises. Stochastic Process. Appl., 77, 69-81. Abstract can also be found in the ISI/STMA publication
• [11] Liptser, R.S. and Shiryaev, A.N. (1977) Statistics of Random Processes I. New York: Springer-Verlag.
• [12] Sawitzki, G. (1981) Finite-dimensional filter systems in discrete time. Stochastics, 5, 107-114.
• [13] Spizzichino, F. (1990) Finite dimensional stochastic filtering in discrete time: the role of convolution semigroups, In A. Bensoussan and P.L. Lions (eds), Analysis and Optimization of Systems, Lecture Notes in Control and Inform. Sci. 144, pp. 238-248. Berlin: Springer-Verlag.
• [14] Van Schuppen, J.H. (1979) Stochastic filtering theory: a discussion on concepts, methods and results. In M. Kohlmann and W. Vogel (eds), Stochastic Control Theory and Stochastic Differential Systems, Lecture Notes in Control and Inform. Sci. 16, pp. 209-226. Berlin: Springer-Verlag.
• [15] West, M., Harrison, P.J. and Migon, H.S. (1985) Dynamic generalized linear models and Bayesian forecasting. J Amer. Statist. Assoc., 80, 73-83.