Sample Path Large Deviation Research Articles

Sample-Path Large Deviations for Unbounded Additive Functionals of the Reflected Random Walk

We prove a sample-path large deviation principle (LDP) with sublinear speed for unbounded functionals of certain Markov chains induced by the Lindley recursion. The LDP holds in the Skorokhod space [Formula: see text] equipped with the [Formula: see text] topology. Our technique hinges on a suitable decomposition of the Markov chain in terms of regeneration cycles. Each regeneration cycle denotes the area accumulated during the busy period of the reflected random walk. We prove a large deviation principle for the area under the busy period of the Markov random walk, and we show that it exhibits a heavy-tailed behavior. Funding: The research of B. Zwart and M. Bazhba is supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek [Grant 639.033.413]. The research of J. Blanchet is supported by the National Science Foundation (NSF) [Grants 1915967, 1820942, and 1838576] as well as the Defense Advanced Research Projects Agency [Grant N660011824028]. The research of C.-H. Rhee is supported by the NSF [Grant CMMI-2146530].

A sample-path large deviation principle for dynamic Erdős–Rényi random graphs

A sample-path large deviation principle for dynamic Erdős–Rényi random graphs

The Cramér-Lundberg model with a fluctuating number of clients

This paper considers the Cramér-Lundberg model, with the additional feature that the number of clients can fluctuate over time. Clients arrive according to a Poisson process, where the times they spend in the system form a sequence of independent and identically distributed non-negative random variables. While in the system, every client generates claims and pays premiums. In order to describe the model's rare-event behaviour, we establish a sample-path large-deviation principle. This describes the joint rare-event behaviour of the reserve-level process and the client-population size process. The large-deviation principle can be used to determine the decay rate of the time-dependent ruin probability as well as the most likely path to ruin. Our results allow us to determine whether the chance of ruin is greater with more or with fewer clients and, more generally, to determine to what extent a large deviation in the reserve-level process can be attributed to an unusual outcome of the client-population size process.

Sample-path large deviations for stochastic evolutions driven by the square of a Gaussian process.

Recently, a number of physical models have emerged described by a random process with increments given by a quadratic form of a fast Gaussian process. We find that the rate function which describes sample-path large deviations for such a process can be computed from the large domain size asymptotic of a certain Fredholm determinant. The latter can be evaluated analytically using a theorem of Widom which generalizes the celebrated Szegő-Kac formula to the multidimensional case. This provides a large class of random dynamical systems with timescale separation for which an explicit sample-path large-deviation functional can be found. Inspired by problems in hydrodynamics and atmosphere dynamics, we construct a simple example with a single slow degree of freedom driven by the square of a fast multivariate Gaussian process and analyze its large-deviation functional using our general results. Even though the noiseless limit of this example has a single fixed point, the corresponding large-deviation effective potential has multiple fixed points. In other words, it is the addition of noise that leads to metastability. We use the explicit answers for the rate function to construct instanton trajectories connecting the metastable states.

Gaussian stochastic volatility models: Scaling regimes, large deviations, and moment explosions

In this paper, we establish sample path large and moderate deviation principles for log-price processes in Gaussian stochastic volatility models, and study the asymptotic behavior of exit probabilities, call pricing functions, and the implied volatility. In addition, we prove that if the volatility function in an uncorrelated Gaussian model grows faster than linearly, then, for the asset price process, all the moments of order greater than one are infinite. Similar moment explosion results are obtained for correlated models.

Gaussian Stochastic Volatility Models: Scaling Regimes, Large Deviations, and Moment Explosions

In this paper, we establish sample path large and moderate deviation principles for log-price processes in Gaussian stochastic volatility models, and study the asymptotic behavior of exit probabilities, call pricing functions, and the implied volatility. In addition, we prove that if the volatility function in an uncorrelated Gaussian model grows faster than linearly, then, for the asset price process, all the moments of order greater than one are infinite. Similar moment explosion results are obtained for correlated models.

Sample Path Large Deviations for Stochastic Evolutionary Game Dynamics

We study a model of stochastic evolutionary game dynamics in which the probabilities that agents choose suboptimal actions are dependent on payoff consequences. We prove a sample path large deviation principle, characterizing the rate of decay of the probability that the sample path of the evolutionary process lies in a prespecified set as the population size approaches infinity. We use these results to describe excursion rates and stationary distribution asymptotics in settings where the mean dynamic admits a globally attracting state, and we compute these rates explicitly for the case of logit choice in potential games.

Large deviations theory for Markov jump models of chemical reaction networks

We prove a sample path Large Deviation Principle (LDP) for a class of jump processes whose rates are not uniformly Lipschitz continuous in phase space. Building on it, we further establish the corresponding Wentzell–Freidlin (W-F) (infinite time horizon) asymptotic theory. These results apply to jump Markov processes that model the dynamics of chemical reaction networks under mass action kinetics, on a microscopic scale. We provide natural sufficient topological conditions for the applicability of our LDP and W-F results. This then justifies the computation of nonequilibrium potential and exponential transition time estimates between different attractors in the large volume limit, for systems that are beyond the reach of standard chemical reaction network theory.

Exit time asymptotics for small noise stochastic delay differential equations

Dynamical system models with delayed dynamics and small noise arise in a variety of applications in science and engineering. In many applications, stable equilibrium or periodic behavior is critical to a well functioning system. Sufficient conditions for the stability of equilibrium points or periodic orbits of certain deterministic dynamical systems with delayed dynamics are known and it is of interest to understand the sample path behavior of such systems under the addition of small noise. We consider a small noise stochastic delay differential equation (SDDE). We obtain asymptotic estimates, as the noise vanishes, on the time it takes a solution of the stochastic equation to exit a bounded domain that is attracted to a stable equilibrium point or periodic orbit of the corresponding deterministic equation. To obtain these asymptotics, we prove a sample path large deviation principle (LDP) for the SDDE that is uniform over initial conditions in bounded sets. The proof of the uniform sample path LDP uses a variational representation for exponential functionals of strong solutions of the SDDE. We anticipate that the overall approach may be useful in proving uniform sample path LDPs for other infinite-dimensional small noise stochastic equations.

Large Deviations for Brownian Motion on Scale Irregular Sierpinski Gaskets

We study sample path large deviation principles for Brownian motion on scale irregular Sierpinski gaskets which are spatially homogeneous but do not have any exact self-similarity. One notable point of our study is that the rate function depends on a large deviation parameter and as such, we can only obtain an example of large deviations in an incomplete form. Instead of showing the large deviations principle we would expect to hold true, we show Varadhan’s integral lemma and exponential tightness by using an incomplete version of such large deviations.

Large deviations for product of sums of random variables

Abstract In this paper, we obtain sample path and scalar large deviation principles for the product of sums of positive random variables. We study the case when the positive random variables are independent and identically distributed and bounded away from zero or the left tail decays to zero sufficiently fast. The explicit formula for the rate function of a scalar large deviation principle is given in the case when random variables are exponentially distributed.

Large deviations for affine diffusion processes on [formula omitted

This paper proves the large deviation principle for affine diffusion processes with initial values in the interior of the state space R+m×Rn. We approach this problem in two different ways. In the first approach, we first prove the large deviation principle for finite dimensional distributions, and then use it to establish the sample path large deviation principle. For this approach, a more careful examination of the affine transform formula is required. The second approach exploits the exponential martingale method of Donati-Martin et al. for the squares of Ornstein–Uhlenbeck processes. We provide an application to importance sampling of affine diffusion models.

On the Queue-Overflow Probability of Wireless Systems: A New Approach Combining Large Deviations With Lyapunov Functions

In this paper, we study the queue-overflow probability of wireless scheduling algorithms. In wireless networks operated under queue-length-based scheduling algorithms, there often exists a tight coupling between the service-rate process, the system backlog process, the arrival process, and the stochastic process governing channel variations. Although one can use sample-path large-deviation techniques to form an estimate of the queue-overflow probability, the formulation leads to a difficult multidimensional calculus-of-variations problem. In this paper, we present a new technique to address this complexity issue. Using ideas from the Lyapunov function approach in control theory, this technique maps the complex multidimensional calculus-of-variations problem to a 1-D calculus-of-variations problem, and the latter is often much easier to solve. Further, under appropriate conditions, we show that when a scheduling algorithm minimizes the drift of a Lyapunov function at each point of every fluid sample path, the algorithm will be optimal in the sense that it maximizes the asymptotic decay rate of the probability that the Lyapunov function value exceeds a given threshold. We believe that these results can potentially be used to study the queue-overflow probability of a large class of wireless scheduling algorithms and to design new scheduling algorithms with optimal overflow probabilities.

Large Deviation Delay Analysis of Queue-Aware Multi-User MIMO Systems With Two-Timescale Mobile-Driven Feedback

Multi-user multi-input-multi-output (MU-MIMO) systems transmit data to multiple users simultaneously using the spatial degrees of freedom with user feedback channel state information (CSI). Most of the existing literatures on the reduced feedback user scheduling focus on the throughput performance, while the user queueing delay is usually ignored. As the delay is very important for real-time applications, a low feedback queue-aware user scheduling algorithm is desired for the MU-MIMO system. This paper proposes a two-stage queue-aware user scheduling algorithm, which consists of a queue-aware mobile-driven feedback filtering stage and a user scheduling stage, where the feedback filtering policy is obtained from optimizations. We evaluate the queueing performance of the proposed scheduling algorithm by using the sample path large deviation analysis. We show that the large deviation decay rate for the proposed algorithm is much larger than that of the CSI-only user scheduling algorithm. The numerical results also demonstrate that the proposed algorithm performs much better than the CSI-only algorithm, while requiring only a small amount of feedback.

Exponential martingale and large deviations for a Cox risk process with Poisson shot noise intensity

We introduce an exponential martingale for a Cox risk process with Poisson shot noise intensity. As applications of the exponential martingale, we obtain the sample path large and moderate deviation principles. Finally, we give an estimate of ruin probability.

Large deviations for a feed-forward network

We consider a feed-forward network with a single-server station serving jobs with multiple levels of priority. The service discipline is preemptive in that the server always serves a job with the current highest level of priority. For this system with discontinuous dynamics, we establish the sample path large deviation principle using a weak convergence argument. In the special case where jobs have two different levels of priority, we also explicitly identify the exponential decay rate of the total population overflow probabilities by examining the geometry of the zero-level sets of the system Hamiltonians.

Large deviations for a feed-forward network

We consider a feed-forward network with a single-server station serving jobs with multiple levels of priority. The service discipline is preemptive in that the server always serves a job with the current highest level of priority. For this system with discontinuous dynamics, we establish the sample path large deviation principle using a weak convergence argument. In the special case where jobs have two different levels of priority, we also explicitly identify the exponential decay rate of the total population overflow probabilities by examining the geometry of the zero-level sets of the system Hamiltonians.

Large deviation principles for sequences of logarithmically weighted means

In this paper we consider several examples of sequences of partial sums of triangular arrays of random variables { X n : n ⩾ 1 } ; in each case X n converges weakly to an infinitely divisible distribution (a Poisson distribution or a centered Normal distribution). For each sequence we prove large deviation results for the logarithmically weighted means { 1 log n ∑ k = 1 n 1 k X k : n ⩾ 1 } with speed function v n = log n . We also prove a sample path large deviation principle for { X n : n ⩾ 1 } defined by X n ( ⋅ ) = ∑ i = 1 n U i ( σ 2 ⋅ ) n , where σ 2 ∈ ( 0 , ∞ ) and { U n : n ⩾ 1 } is a sequence of independent standard Brownian motions.

On the large deviations of a class of modulated additive processes

We prove that the large deviation principle holds for a class of processes inspired by semi-Markov additive processes. For the processes we consider, the sojourn times in the phase process need not be independent and identically distributed. Moreover the state selection process need not be independent of the sojourn times. We assume that the phase process takes values in a finite set and that the order in which elements in the set, called states, are visited is selected stochastically. The sojourn times determine how long the phase process spends in a state once it has been selected. The main tool is a representation formula for the sample paths of the empirical laws of the phase process. Then, based on assumed joint large deviation behavior of the state selection and sojourn processes, we prove that the empirical laws of the phase process satisfy a sample path large deviation principle. From this large deviation principle, the large deviations behavior of a class of modulated additive processes is deduced. As an illustration of the utility of the general results, we provide an alternate proof of results for modulated Levy processes. As a practical application of the results, we calculate the large deviation rate function for a processes that arises as the International Telecommunications Union's standardized stochastic model of two-way conversational speech.

Information ranking and power laws on trees

In this paper we consider the stochastic analysis of information ranking algorithms of large interconnected data sets, e.g. Google's PageRank algorithm for ranking pages on the World Wide Web. The stochastic formulation of the problem results in an equation of the formwhereN,Q, {Ri}i≥1, and {C,Ci}i≥1are independent nonnegative random variables, the {C,Ci}i≥1are identically distributed, and the {Ri}i≥1are independent copies ofstands for equality in distribution. We study the asymptotic properties of the distribution ofRthat, in the context of PageRank, represents the frequencies of highly ranked pages. The preceding equation is interesting in its own right since it belongs to a more general class of weighted branching processes that have been found to be useful in the analysis of many other algorithms. Our first main result shows that if ENE[Cα] = 1, α > 0, andQ,Nsatisfy additional moment conditions, thenRhas a power law distribution of index α. This result is obtained using a new approach based on an extension of Goldie's (1991) implicit renewal theorem. Furthermore, whenNis regularly varying of index α > 1, ENE[Cα] < 1, andQ,Chave higher moments than α, then the distributions ofRandNare tail equivalent. The latter result is derived via a novel sample path large deviation method for recursive random sums. Similarly, we characterize the situation when the distribution ofRis determined by the tail ofQ. The preceding approaches may be of independent interest, as they can be used for analyzing other functionals on trees. We also briefly discuss the engineering implications of our results.