Gao, Fuqing Large deviations for fields with stationary independent increments. (English) Zbl 0824.60020 Acta Math. Appl. Sin., Engl. Ser. 10, No. 3, 288-301 (1994). Consider a field \((\{x(t),t \subset R_ +^ d\}, D)\) with stationary independent increments with \(X(t) = 0\) for \(t\) on the axes and define \(Z_ n(t) = X(nt)/(n_ 1, \dots, n_ d)\) where \(n = (n_ 1, \dots, n_ d) > 0\) and \(nt = (n_ 1t_ 1, \dots, n_ dt_ d)\). The probability distribution \(P_ n\) of \(\{Z_ n(t), t \in [0,1]^ d\}\) can be considered to be a probability measure on \(D[0,1]^ d\) endowed with Skorokhod topology. Define \(J(a) : = \sup_{\theta \in R^ 1} (a \theta - \log \varphi (\theta))\) where \(\varphi (\theta) : = E e^{\theta X (1,1, \dots, 1)}\) which is assumed to be finite for \(\theta\) in the neighbourhood of 0. The author asserts that if \(\lim_{| a | \to \infty} (J(a)/ | a |) = \infty\), then \(\{P_ n\}\) satisfies the large deviation principle on \(D[0,1]^ d\). Also, if \(\{X(t), t \in R^ d_ +\}\) has no Gaussian component, then \(P_ n\) can be considered to be a probability measure on the space of finite signed Borel measures on \([0,1]^ d\) endowed with the weak convergence topology, on which \(\{P_ n\}\) satisfies the large deviation principle. Some general related results have also been proved. Reviewer: S.K.Basu (Calcutta) MSC: 60F10 Large deviations 60J99 Markov processes Keywords:independent increments; signed Borel measure; stationary independent increments; Skorokhod topology; large deviation principle; weak convergence topology PDFBibTeX XMLCite \textit{F. Gao}, Acta Math. Appl. Sin., Engl. Ser. 10, No. 3, 288--301 (1994; Zbl 0824.60020) Full Text: DOI References: [1] Adler, R.J., Monrad, D., Scissors, R.H. and Wilson, R.J. Representations, Decompositions and Sample Function Continuity of Random Fields with Independent Increments.Stoch. Proc. Appl., 1983, 15: 3–30. · Zbl 0507.60043 [2] Chernoff, H. A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the Sum of Observations.Ann. Math. Statist., 1952, 23: 493–507. · Zbl 0048.11804 [3] Chow, Y.S. and Teicher, H. Probability Theory. Springer-Verlag, 1978. [4] Ellis, R.S. Large Deviations for a General Class of Random Vectors.Ann. Probab., 1984, 12: 1–12. · Zbl 0534.60026 [5] Freidlin, M.I. and Wentzell, A.D. Random Perturbations of Dynamical Systems. Springer-Verlag, 1984. · Zbl 0522.60055 [6] Gihman, I.I. and Skorohod, A.V. The Theory of Stochastic Processes I. Springer-Verlag, 1974. · Zbl 0291.60019 [7] Lynch, J. and Setturaman, J. Large Deviations for Processes with Independent Increments.Ann. Probab., 1987, 15: 610–627. · Zbl 0624.60045 [8] Olla, S. Large Deviations for Gibbs Random Fields.Probab. Th. Rel. Fields, 1988, 77: 343–357. · Zbl 0621.60031 [9] Straf, M. Weak Convergence of Stochastic Processes with Several Parameters.Proc. 6th Berkeley Sympos. Math. Statist. Probab. Univ. Calif, 1972, I: 187–221. · Zbl 0255.60019 [10] Stroock, D.W. An Introduction to the Theory of Large Deviations. Heidelberg, Berlin, Springer-Verlag, New York, 1984. · Zbl 0552.60022 [11] Varadhan, S.R.S. Asymptotic Probabilities and Differential Equations.Comm. Pure Appl. Math., 1966, 19: 261–286. · Zbl 0147.15503 [12] Varadhan, S.R.S. Large Deviations and Applications. SIAM, Philadephia, 1984. · Zbl 0549.60023 [13] Wichura, M.J. Some Strassen-type Laws of the Iterated Logarithm for Multiparameter Stochastic Processes with Independent Increments.Ann. Probab., 1973, 1: 272–296. · Zbl 0288.60030 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.