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The second rate function and the asymptotic problems of renewal and hitting the boundary for multidimensional random walks. (English. Russian original) Zbl 0878.60023
Sib. Math. J. 37, No. 4, 647-682 (1996); translation from Sib. Mat. Zh. 37, No. 4, 745-782 (1996).
Let $$\xi$$ be a nondegenerate random vector in the $$d$$-dimensional Euclidean space $$R^d$$, $$d\geq 1$$, with distribution $$F$$. The Laplace transform of $$\xi$$ is defined by $$\varphi(\lambda)=\int e^{\langle\lambda, v\rangle}F(dv)$$, $$\lambda\in R^d$$, where $$\langle\cdot,\cdot\rangle$$ is the inner product in $$R^d$$. The deviation function (in the context of large deviations) for $$\xi$$, or its distribution $$F$$, is defined to be the Legendre transform $$\Lambda(\alpha)$$ of the function $$A(\lambda)=\ln\varphi(\lambda)$$: $\Lambda(\alpha)=\sup_\lambda\{\langle\lambda,\alpha\rangle- \ln\varphi(\lambda)\},\quad \alpha\in R^d.$ The deviation function plays a crucial role in the investigation of large deviation probabilities for sums $$S_n=\xi_1+\xi_2+\cdots+\xi_n$$, where $$\xi_i$$ are independent and distributed as $$\xi$$. The second deviation function (second rate function) is defined by $D(\alpha)=\inf_{s>0}(\Lambda(s\alpha)/s),\quad \alpha\in R^d.$ This function is important in the study of the renewal function $$H(V)=\sum_{n=1}^\infty {\mathbf P}(S_n\in V)$$. The large deviation principle for $$H(V)$$ is established and asymptotic expansions are found for $$H(tV)$$ as $$t\to\infty$$. Another goal of the article is to study the properties of the second deviation function $$D(\alpha)$$ necessary for the analysis of $$H(V)$$ and the related boundary-hitting functions.

##### MSC:
 60F10 Large deviations 60K05 Renewal theory
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##### References:
  R. T. Rockafellar, Convex Analysis [Russian translation], Mir, Moscow (1967). · Zbl 0158.38601  A. A. Borovkov and A. A. Mogul’skiî, Large Deviations and Testing of Statistical Hypotheses [in Russian], Nauka, Novosibirsk (1992).  W. Feller, An Introduction to Probability Theory and Its Applications. Vol. 2 [Russian translation], Mir, Moscow (1967). · Zbl 0158.34902  A. A. Borovkov, ”On the Cramér transform, large deviations in boundary problems, and the conditional invariance principle,” Sibirsk. Mat. Zh.,36, No. 3, 493–509 (1995). · Zbl 0915.60043  A. A. Borovkov, ”On the limit conditional distributions connected with large deviations,” Sibirsk. Mat. Zh.,37, No. 4, 732–744 (1995). · Zbl 0891.60030  A. A. Borovkov and A. A. Mogul’skiî, ”The integro-local multidimensional central limit theorem involving large deviations and asymptotic expansions,” Teor. Veroyatnost. i Primenen. (to appear).  A. A. Borovkov, ”Boundary value problems for random walks and large deviations in function spaces,” Teor. Veroyatnost. i Primenen.,12, No. 4, 635–654 (1967). · Zbl 0178.20004  A. A. Borovkov and D. A. Korshunov, ”Large deviations probabilities of one-dimensional Markov chains. I. Stationary distributions,” Teor. Veroyatnost. i Primenen.,41, No. 1, 3–30 (1996). · Zbl 0888.60025  D. A. Korshunov, ”Asymptotic behavior of the distribution on the maximum of partial sums” (to appear). · Zbl 1117.60048  A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, Jones and Burtlett Publishers Inc., Boston (1993). · Zbl 0793.60030  A. A. Borovkov and A. A. Mogulskiî, ”Large deviations for stationary Markov chains in a quarter plane” (to appear).  W. L. Smith, ”Renewal theory and its ramifications,” J. Roy. Statist. Soc. Ser. B,20, No. 2, 95–150 (1961).  C. Stone, ”On characteristic function and renewal theory,” Trans. Amer. Math. Soc.,120, No. 2, 327–342 (1965). · Zbl 0133.40504 · doi:10.1090/S0002-9947-1965-0189151-0  A. A. Borovkov, Stochastic Processes in Queueing Theory [in Russian], Nauka, Moscow (1972). · Zbl 0275.60102  M. Essen, ”Banach algebra methods in renewal theory,” J. Anal. Math.,26, 303–335 (1973). · Zbl 0327.60052 · doi:10.1007/BF02790434  R. A. Doney, ”An analog of the renewal theorem in higher dimensions,” Proc. London Math. Soc.,16, 669–684 (1966). · Zbl 0147.16301 · doi:10.1112/plms/s3-16.1.669  A. J. Stam, ”Renewal theory inr-dimensions. I and II,” Composito Math., I:21, 383–399 (1969); II:23, 1–13 (1971). · Zbl 0192.54601  A. V. Nagaev, ”Renewal theory inR d ,” Teor. Veroyatnost. i Primenen.,24, No. 3, 565–573 (1979). · Zbl 0408.60088  P. J. Bickel and J. A. Yahav, ”Renewal theory in the plane,” Ann. Math. Statist.,36, 946–955 (1965). · Zbl 0138.40702 · doi:10.1214/aoms/1177700067  J. J. Hunter, ”Renewal theory in two dimensions: asymptotic results,” Adv. in Appl. Probab.,6, 546–562 (1974). · Zbl 0316.60058 · doi:10.2307/1426233  A. J. Stam, ”Local central limit theorem for first entrance of a random walk into a half space,” Composito Math.,23, 15–23 (1971). · Zbl 0208.44205  A. A. Borovkov, An Asymptotic Exit Problem for Multidimensional Markov Chains [Preprint, No. 23], Sobolev Institute of Mathematics (Novosibirsk), Novosibirsk (1995).  A. A. Borovkov, ”Sharp exponential estimators for the distribution of the sum of a random number of random variables,” Teor. Veroyatnost. i Primenen.,40, No. 2, 231–244 (1995).  V. V. Sazonov, ”On the multidimensional concentration function,” Teor. Veroyatnost. i Primenen.,11, No. 4, 683–690 (1966). · Zbl 0161.37204  V. P. Chistyakov, ”A theorem on sums of independent positive random variables and one of its applications to branching random processes,” Teor. Veroyatnost. i Primenen.,9, No. 4, 640–648 (1964). · Zbl 0203.19401  B. A. Rogozin, ”Estimation of the remainder in the limit theorems of renewal theory,” Teor. Veroyatnost. i Primenen.,18, No. 4, 703–717 (1973). · Zbl 0323.60082  B. A. Rogozin, ”Asymptotics of the renewal function,” Teor. Veroyatnost. i Primenen.,21, No. 4, 689–706 (1976). · Zbl 0367.60102  W. Feller, An Introduction to Probability Theory and Its Applications. Vol. 1 [Russian translation], Mir, Moscow (1967). · Zbl 0158.34902  A. A. Borovkov, ”New limit theorems in boundary problems for sums of independent terms,” Sibirsk. Mat. Zh.,3, No. 5, 645–694 (1962).  R. R. Bahadur and R. Ranga Rao, ”On deviations of the sample mean,” Ann. Math. Statist.,31, No. 4, 1015–1027 (1960). · Zbl 0101.12603 · doi:10.1214/aoms/1177705674  H. Cramér, Collective Risk Theory: A Survey of Theory from the Point of View of the Theory of Stochastic Processes, Scandia Insurance Company, Stockholm (1955).  N. Veraverbeke, ”Asymptotic behavior of Wiener-Hopf factors of random walk,” Stochastic Process. Appl.,5, 27–38 (1977). · Zbl 0353.60073 · doi:10.1016/0304-4149(77)90047-3  J. L. Teugels, ”The sub-exponential class of probability distributions,” Teor. Veroyatnost. i Primenen.,19, No. 4, 854–855 (1974).
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