Cramér large deviation expansions for martingales under Bernstein’s condition

An expansion of large deviation probabilities for martingales is given, which extends the classical result due to Cramér to the case of martingale differences satisfying the conditional Bernstein condition. The upper bound of the range of validity and the remainder of our expansion is the same as in the Cramér result and therefore are optimal. Our result implies a moderate deviation principle for martingales.     

White noise driven SPDEs with reflection: Existence,uniqueness and large deviation principles

In the first part of this paper, we prove the uniqueness of the solutions of SPDEs with reflection, which was left open in the paper [C. Donati-Martin, E. Pardoux, White noise driven SPDEs with reflection, Probab. Theory Related Fields 95 (1993) 1–24]. We also obtain the existence of the solution for more general coefficients depending on the past with a much shorter proof. In the second part of the paper, we establish a large deviation principle for SPDEs with reflection. The weak convergence approach is proven to be very efficient on this occasion.     

Long strange segments,ruin probabilities and the effect of memory on moving average processes

We obtain the rate of growth of long strange segments and the rate of decay of infinite horizon ruin probabilities for a class of infinite moving average processes with exponentially light tails. The rates are computed explicitly. We show that the rates are very similar to those of an i.i.d. process as long as the moving average coefficients decay fast enough. If they do not, then the rates are significantly different. This demonstrates the change in the length of memory in a moving average process associated with certain changes in the rate of decay of the coefficients.     

The gambler’s ruin problem with delays

A recent paper by  Pozdnyakov and Steele (2010) is devoted to the so-called binary-plus-passive design. Two problems that the authors do not consider can be identified with the classical gambler’s ruin problem in which delays are allowed.     

Renewal theory with a trend

We prove some analogs of results from renewal theory for random walks in the case when there is a drift, more precisely when the mean of the kth summand equals k^γμ, k≥1, for some μ>0 and 0<γ≤1.     

The effect of memory on functional large deviations of infinite moving average processes

The large deviations of an infinite moving average process with exponentially light tails are very similar to those of an i.i.d. sequence as long as the coefficients decay fast enough. If they do not, the large deviations change dramatically. We study this phenomenon in the context of functional large, moderate and huge deviation principles.     

An asymptotic theory for sample covariances of Bernoulli shifts

Covariances play a fundamental role in the theory of stationary processes and they can naturally be estimated by sample covariances. There is a well-developed asymptotic theory for sample covariances of linear processes. For nonlinear processes, however, many important problems on their asymptotic behaviors are still unanswered. The paper presents a systematic asymptotic theory for sample covariances of nonlinear time series. Our results are applied to the test of correlations.     

Bootstrap of deviation probabilities with applications

We show that under different moment bounds on the underlying variables, bootstrap approximation to the large deviation probabilities of standardized sample sum, based on independent random variables, is valid for a wider zone of n, the sample size, compared to the classical normal tail probability approximation. As an application, different notions of efficiency for statistical tests are considered from Bayesian point of view. In particular, efficiency due to Pitman (1938) [11], Chernoff (1952) [1], and Bayes risk efficiency due to Rubin and Sethuraman (1965) [12] turn out to be special cases with the choice of the weight function; i.e., prior density times loss.     

Lower deviation probabilities for supercritical Galton-Watson processes

There is a well-known sequence of constants cn describing the growth of supercritical Galton-Watson processes Zn. By lower deviation probabilities we refer to P(Zn=kn) with kn=o(cn) as n increases. We give a detailed picture of the asymptotic behavior of such lower deviation probabilities. This complements and corrects results known from the literature concerning special cases. Knowledge on lower deviation probabilities is needed to describe large deviations of the ratio Zn+1/Zn. The latter are important in statistical inference to estimate the offspring mean. For our proofs, we adapt the well-known Cramér method for proving large deviations of sums of independent variables to our needs.     

An extension of a logarithmic form of Cramér’s ruin theorem to some FARIMA and related processes

Cramér’s theorem provides an estimate for the tail probability of the maximum of a random walk with negative drift and increments having a moment generating function finite in a neighborhood of the origin. The class of (g,F)$(g,F)$-processes generalizes in a natural way random walks and fractional ARIMA models used in time series analysis. For those (g,F)$(g,F)$-processes with negative drift, we obtain a logarithmic estimate of the tail probability of their maximum, under conditions comparable to Cramér’s. Furthermore, we exhibit the most likely paths as well as the most likely behavior of the innovations leading to a large maximum.     

Compound Poisson approximation for unbounded functions on a group,with application to large deviations

Summary A second order error bound is obtained for approximating h d by h d , where is a convolution of measures andQ a compound Poisson measure on a measurable abelian group, and the functionh is not necessarily bounded. This error bound is more refined than the usual total variation bound in the sense that it contains the functionh. The method used is inspired by Stein's method and hinges on bounding Radon-Nikodym derivatives related to . The approximation theorem is then applied to obtain a large deviation result on groups, which in turn is applied to multivariate Poisson approximation.Research of the second author was supported by Schweizerischer Nationalfonds     

Hoeffding’s inequality for supermartingales

We give an extension of Hoeffding’s inequality to the case of supermartingales with differences bounded from above. Our inequality strengthens or extends the inequalities of Freedman, Bernstein, Prohorov, Bennett and Nagaev.     

On the asymptotic behaviour of Lévy processes,Part I: Subexponential and exponential processes

We study tail probabilities of the suprema of Lévy processes with subexponential or exponential marginal distributions over compact intervals. Several of the processes for which the asymptotics are studied here for the first time have recently become important to model financial time series. Hence our results should be important, for example, in the assessment of financial risk.     

On functional limits of short- and long-memory linear processes with GARCH(1,1) noises

This paper considers the short- and long-memory linear processes with GARCH (1,1) noises. The functional limit distributions of the partial sum and the sample autocovariances are derived when the tail index α$\alpha$ is in (0,2)$(0,2)$, equal to 2, and in (2,∞)$(2,\infty )$, respectively. The partial sum weakly converges to a functional of α$\alpha$-stable process when α<2$\alpha <2$ and converges to a functional of Brownian motion when α≥2$\alpha \ge 2$. When the process is of short-memory and α<4$\alpha <4$, the autocovariances converge to functionals of α/2$\alpha /2$-stable processes; and if α≥4$\alpha \ge 4$, they converge to functionals of Brownian motions. In contrast, when the process is of long-memory, depending on α$\alpha$ and β$\beta$ (the parameter that characterizes the long-memory), the autocovariances converge to either (i) functionals of α/2$\alpha /2$-stable processes; (ii) Rosenblatt processes (indexed by β$\beta$, 1/2<β<3/4$1/2<\beta <3/4$); or (iii) functionals of Brownian motions. The rates of convergence in these limits depend on both the tail index α$\alpha$ and whether or not the linear process is short- or long-memory. Our weak convergence is established on the space of càdlàg functions on [0,1]$[0,1]$ with either (i) the _J1$J_1$ or the _M1$M_1$ topology (Skorokhod, 1956); or (ii) the weaker form S$S$ topology (Jakubowski, 1997). Some statistical applications are also discussed.     

On the ruin probability in a dependent discrete time risk model with insurance and financial risks

This paper considers the discrete-time risk model with insurance risk and financial risk in some dependence structures. Under assumptions that the insurance risks are heavy tailed (belong to the intersection of the long-tailed class and the dominatedly varying-tailed class) and the financial risks satisfy some moment conditions, the asymptotic and uniformly asymptotic relations for the finite-time and ultimate ruin probabilities are derived.     

An invariance principle for stationary random fields under Hannan’s condition

We establish an invariance principle for a general class of stationary random fields indexed by Z^d${\mathbb{Z}}^d$, under Hannan’s condition generalized to Z^d${\mathbb{Z}}^d$. To do so we first establish a uniform integrability result for stationary orthomartingales, and second we establish a coboundary decomposition for certain stationary random fields. At last, we obtain an invariance principle by developing an orthomartingale approximation. Our invariance principle improves known results in the literature, and particularly we require only finite second moment.     

The asymptotic behavior of linear placement statistics

Orban and Wolfe (1982) and Kim (1999) provided the limiting distribution for linear placement statistics under null hypotheses only when one of the sample sizes goes to infinity. In this paper we prove the asymptotic normality and the weak convergence of the linear placement statistics of Orban and Wolfe (1982) and Kim (1999) when the sample sizes of each group go to infinity simultaneously.     

Absolute continuity under flows generated by SDE with measurable drift coefficients

We consider the Itô SDE with a non-degenerate diffusion coefficient and a measurable drift coefficient. Under the condition that the gradient of the diffusion coefficient and the divergences of the diffusion and drift coefficients are exponentially integrable with respect to the Gaussian measure, we show that the stochastic flow leaves the reference measure absolutely continuous.     

A fractional credit model with long range dependent default rate

Motivated by empirical evidence of long range dependence in macroeconomic variables like interest rates we propose a fractional Brownian motion driven model to describe the dynamics of the short and the default rate in a bond market. Aiming at results analogous to those for affine models we start with a bivariate fractional Vasicek model for short and default rate, which allows for fairly explicit calculations. We calculate the prices of corresponding defaultable zero-coupon bonds by invoking Wick calculus. Applying a Girsanov theorem we derive today’s prices of European calls and compare our results to the classical Brownian model.