Subexponential Random Variables Research Articles

Stochastic trajectory optimization for 6-DOF spacecraft autonomous rendezvous and docking with nonlinear chance constraints

This paper addresses the problem of stochastic trajectory planning for 6-DOF spacecraft close proximity in the presence of external disturbances and initial state uncertainties. Nonlinear chance constraints on collision avoidance, docking corridor, sensor field-of-view, and control magnitude are considered in the corresponding optimal control problem. Nominal trajectory along with affine feedback controller that restricts the dispersion caused by disturbances are optimized simultaneously. The nonlinear stochastic dynamics are transformed into equality constraints on the first two statistical moments of the random states by linearization and discretization, and further, the non-convex covariance update equation is relaxed to a semidefinite cone constraint. The nonlinear path chance constraints are first approximated by affine chance formations, and then are convexified by introducing auxiliary variables instead of direct linearization of the covariance matrix. Combined with the tail bound of sub-exponential random variables, the constraints on control input are replaced with conservative deterministic inequalities. The sub-optimal solution to the original problem is obtained by iteratively solving a series of semidefinite cone programs. The effectiveness of the proposed framework is verified using numerical simulations.

A Kesten-type inequality for randomly weighted sums of dependent subexponential random variables with applications to risk theory*

A Kesten-type inequality for randomly weighted sums of dependent subexponential random variables with applications to risk theory*

Concentration inequality of sums of dependent subexponential random variables and application to bounds for value-at-risk

Concentration inequalities are widely used tools in many fields such as high-dimensional statistics, machine learning, optimization, signal processing, time series analysis, and finance. Therefore, various types of concentration inequalities have been derived so far. In this study, we derived new concentration inequalities for the sum of subexponential random variables. First one is the concentration inequalities for the sum of subexponential random variables with partial dependence structure. Second one is the concentration inequalities with Pearson’s Φ . By applying obtained concentration inequalities to the problem of portfolio risk management, we obtained upper bound for the value-at-risk of financial portfolio.

Improved concentration bounds for sums of independent sub-exponential random variables

Known Bernstein-type upper bounds on the tail probabilities for sums of independent zero-mean sub-exponential random variables are improved in several ways at once. The new upper bounds have a certain optimality property.

On tail behavior of randomly weighted sums of dependent subexponential random variables

In this paper, we revisit the tail behavior of randomly weighted sums and their maxima of dependent subexponential random variables, in which the primary random variables X 1 , … , X n are real-valued and dependent following two general dependence structures, respectively, and the random weights θ 1 , … , θ n are another n positive and arbitrarily dependent random variables, but independent of X 1 , … , X n . . Under some technical conditions, we derive some asymptotic formulas for the tail probability of the randomly weighted sums and their maxima, which coincide with some existing ones in the literature. The merit of our results is that unbounded supports for the random weights are allowed and the distributions of primary random variables can be different.

Sub‐Gaussian Matrices on Sets: Optimal Tail Dependence and Applications

AbstractRandom linear mappings are widely used in modern signal processing, compressed sensing, and machine learning. These mappings may be used to embed the data into a significantly lower dimension while at the same time preserving useful information. This is done by approximately preserving the distances between data points, which are assumed to belong to . Thus, the performance of these mappings is usually captured by how close they are to an isometry on the data. Gaussian linear mappings have been the object of much study, while the sub‐Gaussian settings is not yet fully understood. In the latter case, the performance depends on the sub‐Gaussian norm of the rows. In many applications, e.g., compressed sensing, this norm may be large, or even growing with dimension, and thus it is important to characterize this dependence.We study when a sub‐Gaussian matrix can become a near isometry on a set, show that previous best‐known dependence on the sub‐Gaussian norm was suboptimal, and present the optimal dependence. Our result not only answers a remaining question posed by Liaw, Mehrabian, Plan, and Vershynin in 2017, but also generalizes their work. We also develop a new Bernstein‐type inequality for subexponential random variables, and a new Hanson‐Wright inequality for quadratic forms of sub‐Gaussian random variables, in both cases improving the bounds in the sub‐Gaussian regime under moment constraints. Finally, we illustrate popular applications such as Johnson‐Lindenstrauss embeddings, null space property for 0‐1 matrices, randomized sketches, and blind demodulation, whose theoretical guarantees can be improved by our results (in the sub‐Gaussian case). © 2021 Wiley Periodicals LLC.

Finite-time ruin probability of a perturbed risk model with dependent main and delayed claims

This paper considers a delayed claim risk model with stochastic return and Brownian perturbation in which each main claim may be accompanied with a delayed claim occurring after a stochastic period of time, and the price process of the investment portfolio is described as a geometric Lévy process. By means of the asymptotic results for randomly weighted sum of dependent subexponential random variables we obtain some asymptotics for finite-time ruin probability. A simulation study is also performed to check the accuracy of the obtained theoretical result via the crude Monte Carlo method.

Concentration inequalities for polynomials in α-sub-exponential random variables

In this work we derive multi-level concentration inequalities for polynomial functions in independent random variables with a $\alpha$-sub-exponential tail decay. A particularly interesting case is given by quadratic forms $f(X_1, \ldots, X_n) = \langle X,A X \rangle$, for which we prove Hanson-Wright-type inequalities with explicit dependence on various norms of the matrix $A$. A consequence of these inequalities is a two-level concentration inequality for quadratic forms in $\alpha$-sub-exponential random variables, such as quadratic Poisson chaos. We provide various applications of these inequalities. Among these are generalizations the results given by Rudelson-Vershynin from sub-Gaussian to $\alpha$-sub-exponential random variables, i. e. concentration of the Euclidean norm of the linear image of a random vector, small ball probability estimates and concentration inequalities for the distance between a random vector and a fixed subspace. Moreover, we obtain concentration inequalities for the excess loss in a fixed design linear regression and the norm of a randomly projected random vector.

Robust concentration inequalities in maximal exponential models

Robust concentration inequalities of Bernstein type for sub-exponential random variables are proved with respect to probability measures whose densities are connected by open exponential arcs.

Lasso guarantees for $\beta$-mixing heavy-tailed time series

Many theoretical results for lasso require the samples to be i.i.d. Recent work has provided guarantees for lasso assuming that the time series is generated by a sparse Vector Autoregressive (VAR) model with Gaussian innovations. Proofs of these results rely critically on the fact that the true data generating mechanism (DGM) is a finite-order Gaussian VAR. This assumption is quite brittle: linear transformations, including selecting a subset of variables, can lead to the violation of this assumption. In order to break free from such assumptions, we derive nonasymptotic inequalities for estimation error and prediction error of lasso estimate of the best linear predictor without assuming any special parametric form of the DGM. Instead, we rely only on (strict) stationarity and geometrically decaying $\beta$-mixing coefficients to establish error bounds for lasso for sub-Weibull random vectors. The class of sub-Weibull random variables that we introduce includes sub-Gaussian and subexponential random variables but also includes random variables with tails heavier than an exponential. We also show that, for Gaussian processes, the $\beta$-mixing condition can be relaxed to summability of the $\alpha$-mixing coefficients. Our work provides an alternative proof of the consistency of lasso for sparse Gaussian VAR models. But the applicability of our results extends to non-Gaussian and nonlinear times series models as the examples we provide demonstrate.

A Kesten-type bound for sums of randomly weighted subexponential random variables

Sums of randomly weighted subexponential random variables have become an important research topic, but most works on the topic consider randomly weighted sums of finitely many terms. To extend the study to the case of infinitely many terms, we establish a Kesten-type upper bound for the tail probabilities of sums of randomly weighted subexponential random variables. As an application, we derive a precise asymptotic formula for the tail probability of the aggregate present value of subexponential claims, where the present value factor is determined according to the zero-coupon bond price.

Tail probability of randomly weighted sums of dependent subexponential random variables with applications to risk theory

Following the work of Cheng and Cheng (2018) [6], we reexamine the tail probability of randomly weighted sums of dependent subexponential random variables. Precisely speaking, let {Xn,n≥1} be real-valued and commonly distributed random variables satisfying a general dependence structure proposed in Ko and Tang (2008) [14], and random weights {θn,n≥1} be positive, bounded above and arbitrarily dependent random variables, but independent of {Xn,n≥1}. Under some mild conditions, we achieve the asymptotic behavior of tail probability for both randomly weighted finite and infinite sums. Finally, an application of the obtained results to a nonstandard continuous-time renewal risk model is proposed.

On asymptotic finite-time ruin probability of a renewal risk model with subexponential main claims and delayed claims

This paper considers a non-standard renewal risk model with constant force of interest, where each main claim may derive a delayed claim occurring after a random period of time. By means of recent asymptotic results for randomly weighted sums of subexponential random variables, we obtain some precise asymptotic expansions for the finite-time ruin probability when the main claims have a common subexponential tail.

An IHT Algorithm for Sparse Recovery From Subexponential Measurements

A matrix whose entries are independent subexponential random variables is not likely to satisfy the classical restricted isometry property in the optimal regime of parameters. However, it is known that uniform sparse recovery is still possible with high probability in the optimal regime if ones uses $\ell _1$ -minimization as a recovery algorithm. We show in this letter that such a statement remains valid if one uses a new variation of iterative hard thresholding as a recovery algorithm. The argument is based on a modified restricted isometry property featuring the $\ell _1$ -norm as the inner norm.

Randomly weighted sums of dependent subexponential random variables with applications to risk theory

For any fixed integer , let be real-valued random variables with a common subexponential distribution, and let be positive random variables which are bounded above and independent of . Under some rather loose conditional dependence assumptions on the primary random variables , this paper proves that the asymptotic relationshold as , where are arbitrarily dependent. In particular, it is shown that the above results hold true for with certain Samarnov distributions. The obtained results on randomly weighted sums are applied to estimating the finite-time ruin probability in a discrete-time risk model with both insurance and financial risks.

Uniform approximation of the tail probability of weighted sums of subexponential random variables

ABSTRACTFor subexponential random variables Xi, 1 ⩽ i ⩽ n, the uniform asymptotic resultis established for 0 < an(x) ⩽ ci ⩽ bn(x), 1 ⩽ i ⩽ n, where an(x) decreases to 0 and b(x) goes to infinity as x tends to infinity. It extends the current result when the subexponential random variables Xi’s are independent and gives the first uniform asymptotic result when Xi’s are dependent.

Approximations of value-at-risk as an extreme quantile of a random sum of heavy-tailed random variables

This paper studies the approximation of extreme quantiles of random sums of heavy-tailed random variables, or, more specifically, subexponential random variables. A key application of this approximation is the calculation of operational value-at-risk (VaR) for financial institutions in order to determine operational risk capital requirements. This paper follows work by Böcker, Klüppelberg and Sprittulla and makes several advances. These include two new approximations of VaR and an extension to multiple loss types where the VaR relates to a sum of random sums, each of which is defined by different distributions. The proposed approximations are assessed via a simulation study.

An Improved Hazard Rate Twisting Approach for the Statistic of the Sum of Subexponential Variates

In this letter, we present an improved hazard rate twisting technique for the estimation of the probability that a sum of independent but not necessarily identically distributed subexponential Random Variables (RVs) exceeds a given threshold. Instead of twisting all the components in the summation, we propose to twist only the RVs which have the biggest impact on the right-tail of the sum distribution and keep the other RVs unchanged. A minmax approach is performed to determine the optimal twisting parameter which leads to an asymptotic optimality criterion. Moreover, we show through some selected simulation results that our proposed approach results in a variance reduction compared to the technique where all the components are twisted.

Randomly weighted sums of subexponential random variables with application to capital allocation

We are interested in the tail behavior of the randomly weighted sum $ \sum _{i=1}^{n}\theta _{i}X_{i}$ , in which the primary random variables X 1, …, X n are real valued, independent and subexponentially distributed, while the random weights 𝜃 1, …, 𝜃 n are nonnegative and arbitrarily dependent, but independent of X 1, …, X n . For various important cases, we prove that the tail probability of $\sum _{i=1}^{n}\theta _{i}X_{i}$ is asymptotically equivalent to the sum of the tail probabilities of 𝜃 1 X 1, …, 𝜃 n X n , which complies with the principle of a single big jump. An application to capital allocation is proposed.

Tail probability of randomly weighted sums of subexponential random variables under a dependence structure

This paper deals with the asymptotic behavior for the tail probability of randomly weighted sums of subexponential random variables under a dependence structure, where the random weights and the corresponding summands are dependent.