Summary
Based on the conjugate kernel studied in Iscoe et al. (1985) we derive saddlepoint expansions for either the density or distribution function of a sumf(X 1)+...+f(X n ), where theX i 's constitute a Markov chain. The chain is assumed to satisfy a strong recurrence condition which makes the results here very similar to the classical results for i.i.d. variables. In particular we establish also conditions under which the expansions hold uniformly over the range of the saddlepoint. Expansions are also derived for sums of the formf(X 1,X 0)+f(X 2,X 1)+...+f(X n ,X n−1) although the uniformity result just mentioned does not generalize.
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Bhattacharva, R.N., Ranga, Rao, R.: Normal approximation and asymptotic expansion. New York: Wiley 1976
Cox, D.R., Miller, H.D.: The theory of stochastic processes. London: Chapman and Hall 1965
Daniels, H.E.: Saddlepoint approximations in statistics. Ann. Math. Stat.25, 631–650 (1954)
Götze, F., Hipp, C.: Asymptotic expansions for sums of weakly dependent random vectors. Z. Wahrsheinlichkeitstheor. Verw. Geb.64, 211–240 (1983)
Höglund, T.: Central limit theorems and statistical inference for finite Markov chains. Z. Wahrscheinlichkeitstheor. Verw. Geb.29, 123–151 (1974)
Iscoe, I., Ney, P., Nummelin, E.: Large deviations of uniformly recurrent Markov additive processes. Adv. Appl. Math.6, 373–412 (1985)
Jensen, J.L.: A note on asymptotic expansions for Markov chains using operator theory. Adv. Appl. Math.8, 377–392 (1987)
Jensen, J.L.: Uniform saddlepoint approximations. Adv. Appl. Probab.20, 622–634 (1988)
Jensen, J.L.: Asymptotic expansions for strongly mixing Harris recurrent Markov chains. Scand. J. Stat.15, 47–63 (1989)
Jensen, J.L.: Uniform saddlepoint approximations and log-concave densities. J. R. Stat. Soc.B 53, 157–172 (1991)
Kim, G.-H., David, H.T.: Large deviations of functions of markovian transitions and mathematical programming duality. Ann. Probab.7, 874–881 (1979)
Nagaev, S.V.: Some limit theorems for stationary Markov chains. Theory Probab. Appl.2, 378–406 (1957)
Ney, P., Nummelin, E.: Markov additive processes I. Eigenvalue properties and limit theorems. Ann. Probab.15, 561–592 (1987)
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Jensen, J.L. Saddlepoint expansions for sums of Markov dependent variables on a continuous state space. Probab. Th. Rel. Fields 89, 181–199 (1991). https://doi.org/10.1007/BF01366905
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DOI: https://doi.org/10.1007/BF01366905