Extremal Eigenvalues and Eigenvectors of Deformed Wigner Matrices

We consider random matrices of the form $H = W + \lambda V$, $\lambda\in\mathbb{R}^+$, where $W$ is a real symmetric or complex Hermitian Wigner matrix of size $N$ and $V$ is a real bounded diagonal random matrix of size $N$ with i.i.d.\ entries that are independent of $W$. We assume subexponential decay for the matrix entries of $W$ and we choose $\lambda \sim 1$, so that the eigenvalues of $W$ and $\lambda V$ are typically of the same order. Further, we assume that the density of the entries of $V$ is supported on a single interval and is convex near the edges of its support. In this paper we prove that there is $\lambda_+\in\mathbb{R}^+$ such that the largest eigenvalues of $H$ are in the limit of large $N$ determined by the order statistics of $V$ for $\lambda>\lambda_+$. In particular, the largest eigenvalue of $H$ has a Weibull distribution in the limit $N\to\infty$ if $\lambda>\lambda_+$. Moreover, for $N$ sufficiently large, we show that the eigenvectors associated to the largest eigenvalues are partially localized for $\lambda>\lambda_+$, while they are completely delocalized for $\lambda<\lambda_+$. Similar results hold for the lowest eigenvalues.Comment: 47 page

Shuttle relative navigation of a tethered satellite mission with current on board software

A Shuttle mission planned in 1991 will test the feasibility of tethers in space. This mission, a joint effort between Italy and the United States, will connect a satellite (built by the Italians) to the Shuttle with a 20 km long tether. This mission poses unique navigation problems. The flight software on the Shuttle was never designed to account for the low level acceleration that is generated by the gravity gradient. IMUs on the Shuttle was never designed to account for the low level acceleration that is generated by the gravity gradient. Inertial Maneuvering Units on the shuttle will sense the acceleration of the tether but it turns out that incorporating the continuous accelerometer noise also generates large error growth. Relative navigation is another important issue since the majority of the mission will be conducted while the satellite is out of the visual range of the crew. Some kind of feedback on the motion of the satellite will be desirable. Feedback of the satellite motion can be generated by using the rendezvous radar. To process the radar measurements, the flight software uses a 13 state Kalman Filter, but unfortunately with the filter currently tuned as it is, valid measurements tend to be ignored. This is due to the constraint of the tether on the satellite, which is an unmodeled force. Analysis shows that with proper tuning, relative navigation is possible

Edge Universality for Deformed Wigner Matrices

We consider $N\times N$ random matrices of the form $H = W + V$ where $W$ is a real symmetric Wigner matrix and $V$ a random or deterministic, real, diagonal matrix whose entries are independent of $W$. We assume subexponential decay for the matrix entries of $W$ and we choose $V$ so that the eigenvalues of $W$ and $V$ are typically of the same order. For a large class of diagonal matrices $V$ we show that the rescaled distribution of the extremal eigenvalues is given by the Tracy-Widom distribution $F_1$ in the limit of large $N$. Our proofs also apply to the complex Hermitian setting, i.e., when $W$ is a complex Hermitian Wigner matrix

Time-Varying Uncertainty and the Credit Channel

We extend the Carlstrom and Fuerst (1997) agency cost model of business cycles by including time varying uncertainty in the technology shocks that affect capital production. We first demonstrate that standard linearization methods can be used to solve the model yet second moments enter the economy's equilibrium policy functions. We then demonstrate that an increase in uncertainty causes, ceteris paribus, a fall in investment supply. A second key result is that time varying uncertainty results in countercyclical bankruptcy rates - a finding which is consistent with the data and opposite the result in Carlstrom and Fuerst. Third, we show that persistence of uncertainty affects both quantitatively and qualitatively the behavior of the economy. However, the shocks to uncertainty imply a quantitatively small role for uncertainty over the business cycle.agency costs, credit channel, time-varying uncertainty

Decision-Making in Hard Times: What is a Recession, Why Do We Care and When Do We Know We Are in One?

Defining a recessionary event as one which impacts adversely on individuals’ economic well-being, the paper argues that recession is a multi-faceted phenomenon whose meaning differs from person to person as it impacts on their decision-making in real time. It argues that recession is best represented through the calculation of the nowcast of recession event probabilities. A variety of such probabilities are produced using a real-time data set for the US for the period, focusing on the likelihood of various recessionary events through 1986q1-2008q4 and on prospects beyond the end of the sample.Recession, Probability Forecasts, Real Time.