Probabilities Of Large Deviations Research Articles

Stochastic diffusion within expanding space–time

The paper examines stochastic diffusion within an expanding space–time framework motivated by cosmological applications. Contrary to other results in the literature, for the considered general stochastic model, the expansion of space–time leads to a class of stochastic equations with non-constant coefficients that evolve with the expansion factor. The Cauchy problem with random initial conditions is posed and investigated. The exact solution to a stochastic diffusion equation on the expanding sphere is derived. Various probabilistic properties of the solution are studied, including its dependence structure, evolution of the angular power spectrum and local properties of the solution and its approximations by finite truncations. The paper also characterizes the extremal behaviour of the random solution by establishing upper bounds on the probabilities of large deviations. Numerical studies are carried out to illustrate the obtained theoretical results.

A large-deviation principle for birth–death processes with a linear rate of downward jumps

Abstract Birth–death processes form a natural class where ideas and results on large deviations can be tested. We derive a large-deviation principle under an assumption that the rate of jump down (death) grows asymptotically linearly with the population size, while the rate of jump up (birth) grows sublinearly. We establish a large-deviation principle under various forms of scaling of the underlying process and the corresponding normalization of the logarithm of the large-deviation probabilities. The results show interesting features of dependence of the rate functional upon the parameters of the process and the forms of scaling and normalization.

Large deviation probabilities for the range of a d-dimensional supercritical branching random walk

Let {Zn}n≥0 be a d-dimensional supercritical branching random walk started from the origin. Write Zn(S) for the number of particles located in a set S⊂Rd at time n. Denote by Rn:=inf⁡{ρ≥0:Zi({|x|≥ρ})=0,∀0≤i≤n} the radius of the minimal ball (centered at the origin) containing the range of {Zi}i≥0 up to time n. In this work, we show that under some mild conditions Rn/n converges in probability to some positive constant x⁎ as n→∞. Furthermore, we study its corresponding lower and upper deviation probabilities, i.e. the decay rates ofP(Rn≤xn)forx∈(0,x⁎);P(Rn≥xn)forx∈(x⁎,∞) as n→∞.

On the Probabilities of Large Deviations of Combinatorial Sums of Independent Random Variables That Satisfy the Linnik Condition

On the Probabilities of Large Deviations of Combinatorial Sums of Independent Random Variables That Satisfy the Linnik Condition

Asymptotic local probabilities of large deviations for branching process in random environment with geometric distribution of descendants

Abstract We consider the branching process Z n = X n , 1 + ⋯ + X n Z n − 1 $ Z_{n} =X_{n, 1} + \dotsb +X_{nZ_{n-1}} $ , in random environments η , where η is a sequence of independent identically distributedvariables, for fixed η the random variables X i, j areindependent, have the geometric distribution. We suppose that the associated random walk S n = ξ 1 + ⋯ + ξ n $ S_n = \xi_1 + \dotsb + \xi_n $ has positive meanμ,0 < h<h +satisfies the right-hand Cramer’s condition E exp(h ξ i ) < ∞ for, some h +. Under theseassumptions, we find the asymptotic representation for local probabilities P (Z n =⌊exp(θ n)⌋) for θ ∈ [θ 1, θ 2]⊂</given−names><x> </x><surname>(μ;μ +) and someμ +.

Large deviations for subcomplex counts and Betti numbers in multiparameter simplicial complexes

We consider the multiparameter random simplicial complex as a higher dimensional extension of the classical Erdős–Rényi graph. We investigate appearance of “unusual” topological structures in the complex from the point of view of large deviations. We first study upper tail large deviation probabilities for subcomplex counts, deriving the order of magnitude of such probabilities at the logarithmic scale precision. The obtained results are then applied to analyze large deviations for the number of simplices of the multiparameter simplicial complexes. Finally, these results are also used to deduce large deviation estimates for Betti numbers of the complex in the critical dimension.

Moderate deviation probabilities for empirical distribution of the branching random walk

Given a super-critical branching random walk {Zn}n≥0 on R, let Zn(A) be the number of particles located in some Borel set A⊂R at generation n. Under some mild conditions, it's well-known (e.g., [4]) that Zn(nA)/Zn(R) converges almost surely to ν(A) as n→∞, where ν(⋅) is the standard Gaussian measure on R. Large deviation probabilities of Zn(nA)/Zn(R) have been well studied (see [10] and [27]). In this work, we investigate its moderate deviation probabilities, i.e. the convergence rate ofP(Zn(nA)/Zn(R)≥ν(A)+Δn) as n→∞, where {Δn}n≥0 is some positive sequence tending to zero.

On probabilities of large deviations of combinatorial sums of independent random variables satisfying Linnik’s condition

We derive new results on asymptotic behaviour for probabilities of large deviations of combinatorial sums of independent random variables satisfying Linnik’s condition. We find zones in which these probabilities are equivalent to the tail of the standard normal law. The author earlier obtained such results under Bernstein’s condition. The truncations method is applied in proofs of results.

Large deviations for values of $L$-functions attached to cusp forms in the level aspect

We study the distribution of extreme values of automorphic $L$-functions in a family of holomorphic cusp forms with prime levels. First, we prove an asymptotic formula for a certain density function closely related to this value-distribution. Then we apply it to estimate the probabilities of large deviations for values of automorphic $L$-functions.

The information geometry of two-field functional integrals

Two-field functional integrals (2FFI) are an important class of solution methods for generating functions of dissipative processes, including discrete-state stochastic processes, dissipative dynamical systems, and decohering quantum densities. The stationary trajectories of these integrals describe a conserved current by Liouville’s theorem, despite the absence of a conserved kinematic phase space current in the underlying stochastic process. We develop the information geometry of generating functions for discrete-state classical stochastic processes in the Doi-Peliti 2FFI form, and exhibit two quantities conserved along stationary trajectories. One is a Wigner function, familiar as a semiclassical density from quantum-mechanical time-dependent density-matrix methods. The second is an overlap function, between directions of variation in an underlying distribution and those in the directions of relative large-deviation probability that can be used to interrogate the distribution, and expressed as an inner product of vector fields in the Fisher information metric. To give an interpretation to the time invertibility implied by current conservation, we use generating functions to represent importance sampling protocols, and show that the conserved Fisher information is the differential of a sample volume under deformations of the nominal distribution and the likelihood ratio. We derive a pair of dual affine connections particular to Doi-Peliti theory for the way they separate the roles of the nominal distribution and likelihood ratio, distinguishing them from the standard dually-flat connection of Nagaoka and Amari defined on the importance distribution, and show that dual flatness in the affine coordinates of the coherent-state basis captures the special role played by coherent states in Doi-Peliti theory.

Large deviations of Poisson Telecom processes

AbstractWe study large-deviation probabilities of Telecom processes appearing as limits in a critical regime of the infinite-source Poisson model elaborated by I. Kaj and M. Taqqu. We examine three different regimes of large deviations (LD) depending on the deviation level. A Telecom process $(Y_t)_{t \ge 0}$ scales as $t^{1/\gamma}$ , where t denotes time and $\gamma\in(1,2)$ is the key parameter of Y. We must distinguish moderate LD ${\mathbb P}(Y_t\ge y_t)$ with $t^{1/\gamma} \ll y_t \ll t$ , intermediate LD with $ y_t \approx t$ , and ultralarge LD with $ y_t \gg t$ . The results we obtain essentially depend on another parameter of Y, namely the resource distribution. We solve completely the cases of moderate and intermediate LD (the latter being the most technical one), whereas the ultralarge deviation asymptotics is found for the case of regularly varying distribution tails. In all the cases considered, the large-deviation level is essentially reached by the minimal necessary number of ‘service processes’.

Limit laws for rational continued fractions and value distribution of quantum modular forms

We study the limiting distributions of Birkhoff sums of a large class of cost functions (observables) evaluated along orbits, under the Gauss map, of rational numbers in (0, 1] ordered by denominators. We show convergence to a stable law in a general setting, by proving an estimate with power-saving error term for the associated characteristic function. This extends results of Baladi and Vallée on Gaussian behaviour for costs of moderate growth. We apply our result to obtain the limiting distribution of values of several key examples of quantum modular forms. We obtain the Gaussian behaviour of central values of the Esterman function ∑ n ⩾ 1 τ ( n ) e 2 π i n x / n $\sum _{n\geqslant 1} \tau (n) {\rm e}^{2\pi i n x}/\sqrt {n}$ ( x ∈ Q $x\in {\mathbb {Q}}$ ), a problem for which known approaches based on Eisenstein series have been so far ineffective. We give a new proof, based on dynamical systems, that central modular symbols associated with a holomorphic cusp form for S L ( 2 , Z ) $SL(2,{\mathbb {Z}})$ have a Gaussian distribution, and give the first proof of an estimate for their probabilities of large deviations. We also recover a result of Vardi on the convergence of Dedekind sums to a Cauchy law, using dynamical methods.

While stability lasts: A stochastic model of noncustodial stablecoins

AbstractThe “Black Thursday” crisis in cryptocurrency markets demonstrated deleveraging risks in over‐collateralized noncustodial stablecoins. We develop a stochastic model that helps explain deleveraging crises in these over‐collateralized systems. In our model, the stablecoin supply is decided by speculators who optimize the profitability of a leveraged position while incorporating the forward‐looking cost of collateral liquidations, which involves the endogenous price of the stablecoin. We formally characterize regimes that are interpreted as stable and unstable for the stablecoin. We prove bounds on quadratic variation (QV) and the probability of large deviations in the stable domain and we demonstrate distinctly greater price variance in the unstable domain. We identify a deflationary deleveraging spiral by means of a submartingale. These deleveraging spirals, which resemble short squeezes, lead to faster collateral drawdown (and potential shortfalls) and are accompanied by higher price variance, as experienced on Black Thursday. We conclude by discussing noncustodial ways in which the issues raised in this paper can be mitigated.

Supremum of the Euclidean Norms of the Multidimensional Wiener Process and Brownian Bridge: Sharp Asymptotics of Probabilities of Large Deviations

For T > 0, we prove theorems concerning sharp asymptotics of the probabilities$$ \mathbf{P}\left\{\underset{t\in \left[0,T\right]}{\sup}\sum \limits_{j=1}^n{w}_j^2(t)>{u}^2\right\},\kern1em \mathbf{P}\left\{\underset{t\in \left[0,T\right]}{\sup}\sum \limits_{j=1}^n{w}_{j0,T}^2(t)>{u}^2\right\}, $$as u → ∞, where wj(t), j = 1, . . . , n, are independent Wiener processes and wj0,T (t), j = 1, . . . , n, are independent Brownian bridges on the segment [0, T]. Our research method is the double sum method for the Gaussian processes and fields. We also give an application of the obtained results to the statistical tests for the homogeneity hypothesis of k one-dimensional samples.

Asymptotic Behavior of Large Deviation Probabilities for a Simple Oscillating Random Walk

This paper considers simple oscillating random walks with \( {\overset{\sim }{S}}_n=\sum_{i=1}^n{\overset{\sim }{X}}_t \), under the assumption that \( \mathbf{P}\left({\overset{\sim }{X}}_{n+1}=1\left|{\overset{\sim }{S}}_n\right.>0\right)=p>1/2 \). We show that the asymptotic behavior of probability to reach high level for the oscillating random walk and a standard random walk are similar up to a constant multiplier. The asymptotics for the maximum of a random walk and for the moment of the first exit beyond the high level are obtained.

On large-deviation probabilities for the empirical distribution of branching random walks with heavy tails

AbstractGiven a branching random walk $(Z_n)_{n\geq0}$ on $\mathbb{R}$ , let $Z_n(A)$ be the number of particles located in interval A at generation n. It is well known that under some mild conditions, $Z_n(\sqrt nA)/Z_n(\mathbb{R})$ converges almost surely to $\nu(A)$ as $n\rightarrow\infty$ , where $\nu$ is the standard Gaussian measure. We investigate its large-deviation probabilities under the condition that the step size or offspring law has a heavy tail, i.e. a decay rate of $\mathbb{P}(Z_n(\sqrt nA)/Z_n(\mathbb{R})>p)$ as $n\rightarrow\infty$ , where $p\in(\nu(A),1)$ . Our results complete those in Chen and He (2019) and Louidor and Perkins (2015).

Characterization of Large Deviation Probabilities for Regenerative Sequences

Characterization of Large Deviation Probabilities for Regenerative Sequences

Large and moderate deviations for record numbers in some non–nearest neighbor random walks

The deviation principles of record numbers in random walk models have not been completely investigated, especially for the non–nearest neighbor cases. In this paper, we derive the asymptotic probabilities of large and moderate deviations for the number of “weak records” (or “ladder points”) in two kinds of one–dimensional non–nearest neighbor random walks. The proofs depend only on the direct analysis of random walks. We illustrate that the traditional method of analyzing the local time of Brownian motions, which is often adopted for the simple random walks, would lead to wrong conjectures for our cases.

Variations of flow direction in solar wind streams of different types

Studying the direction of the solar wind flow is a topical problem of space weather forecasting. As a rule, the quiet and uniform solar wind propagates radially, but significant changes in the solar wind flow direction can be observed, for example, in compression regions before the interplanetary coronal mass ejections (Sheath) and Corotating Interaction Regions (CIR) that precede high-speed streams from coronal holes. In this study, we perform a statistical analysis of the longitude (φ) and latitude (θ) flow direction angles and their variations on different time scales (30 s and 3600 s) in solar wind large-scale streams of different types, using WIND spacecraft data. We also examine the relationships of the value and standard deviations SD of the flow direction angles with various solar wind parameters, regardless of the solar wind type.
 We have established that maximum values of longitude and latitude angle modulus, as well as their variations, are observed for Sheath, CIR, and Rare, with the probability of large deviations from the radial direction (>5°) increasing. The dependence on the solar wind type is shown to decrease with scale. We have also found that the probability of large values of SD(θ) and SD(φ) increases with increasing proton temperature (Tp) in the range 5–10 eV and with increasing proton velocity (Vp) in the range 400–500 km/s.

Вариации направления потока в различного типа течениях солнечного ветра

Studying the direction of the solar wind flow is a topical problem of space weather forecasting. As a rule, the quiet and uniform solar wind propagates radially, but significant changes in the solar wind flow direction can be observed, for example, in compression regions before the interplanetary coronal mass ejections (Sheath) and Corotating Interaction Regions (CIR) that precede high-speed streams from coronal holes. In this study, we perform a statistical analysis of the longitude (φ) and latitude (θ) flow direction angles and their variations on different time scales (30 s and 3600 s) in solar wind large-scale streams of different types, using WIND spacecraft data. We also examine the relationships of the value and standard deviations SD of the flow direction angles with various solar wind parameters, regardless of the solar wind type.
 
 We have established that maximum values of longitude and latitude angle modulus, as well as their variations, are observed for Sheath, CIR, and Rare, with the probability of large deviations from the radial direction (>5°) increasing. The dependence on the solar wind type is shown to decrease with scale. We have also found that the probability of large values of SD(θ) and SD(φ) increases with increasing proton temperature (Tp) in the range 5–10 eV and with increasing proton velocity (Vp) in the range 400–500 km/s.