Free Probability

The norm of products of free random variables

Probability Theory and Related Fields, 2007, 139, 397-413

This paper investigates how the norm of the product of N free identically-distributed random variables Xi grows as N approaches infinity. It is proved that if the expectation of Xi’*Xi is 1, then the growth in the norm of the product is at most linear.

If this expectation does not equal 1, it is proved that the growth in the norm of the product is exponential and the rate equals one half of the logarithm of the expectation of Xi’*Xi.

These results generalize results for random matrices derived by Kesten and Furstenberg.

A proof of a non-commutative central limit theorem by the Lindeberg method

Electronic Communications in Probability, 2007, 12, 36-50

A Central Limit Theorem for non-commutative random variables is proved using the Lindeberg method. The theorem is a generalization of the Central Limit Theorem for free random variables proved by Voiculescu. The Central Limit Theorem in this paper relies on an assumption which is weaker than freeness.

Berry-Esseen for Free Random Variables

Journal of Theoretical Probability 2007, 20, 381-395

An analogue of the Berry-Esseen inequality is proved for the speed of convergence of free additive convolutions of bounded probability measures. The obtained rate of convergence is of the order n^{-1/2}, the same as in the classical case. An example with binomial measures shows that this estimate cannot be improved without imposing further restrictions on convolved measures.

On Superconvergence of Convolutions of Free Random Variables

Annals of Probability 2007, 35, 1931-1949

This paper derives the sufficient conditions for supeconvergence of convolutions of bounded free random variables. It also provides estimates on the rate of superconvergence.

Probability and Statistics

A Bernstein-Type Inequality for Vector Functions on Finite Markov Chains

Annals of Applied Probability 2007, 17, 1202-1221

An analogue of the Bernstein inequality is derived for partial sums of a vector-valued function on a finite reversible Markov chain. The inequality gives an upper bound for the probability of a large deviation of the partial sum. The bound depends on the chain's spectral gap, the dimension of the space where the function takes value, and the upper bound on the size and the variance of the function.

Curve Forecasting by Functional Autoregression (Joint with A. Onatski)

To appear in the Journal of Multivariate Analysis

This paper deals with prediction of curve-valued random processes. It develops a novel technique, the predictive factor decomposition, for estimation of the autoregression operator. The technique is based on finding a reduced-rank approximation to the autoregression operator that minimizes the expected norm of the prediction error.

Implementing this idea, we relate the operator approximation to an eigenvalue problem for an operator pencil of the cross-covariance and covariance operators. We develop an estimation method based on regularization of the empirical counterpart of this eigenvalue problem, prove consistency and evaluate the convergence rates.

The new method is illustrated by an analysis of the dynamics of the term structure of Eurodollar futures rates. In the subsample of normal economic growth the predictive factor technique outperforms the principal components and performs on par with the best available problem specific methods.

On the Chernoff bound for efficiency of quantum hypothesis testing

Annals of Statistics 2005, 33(2), 959-976

The paper estimates the Chernoff rate for the efficiency of quantum hypothesis testing.

For both joint and separate measurements, approximate bounds for the rate are given if both states are mixed, and exact expressions are derived if at least one of the states is pure. The efficiencies of tests with separate and joint measurements are compared. The results are illustrated by a test of quantum entanglement.

Game Theory and Economics

Coordination Games with Quantum Information

To appear in the International Journal of Game Theory

A necessary condition is derived that helps to determine whether an entangled quantum system is useful for improving coordination in a game with incomplete information.

Uncertainty of the Shapley Value

International Game Theory Review 2005, 7(4), 517-529

This paper defines a measure of bargaining uncertainty that quantifies Roth's concept of strategic risk. It shows how this measure can be used for checking reliability of the Shapley value in cost allocation problems and in the theory of competitive equilibrium. Salient properties of the new measure are investigated and illustrated by examples of majority voting and market games and by a cost allocation problem from epidemiology.

Prevention of Herding by Experts
Economics Letters, 2003, 78(3), 401-407
I consider a client-expert model with many experts, who can communicate among themselves. Two questions are addressed: (1) How to prevent herding by experts without stopping communication? (2) What would be the cost of this prevention? I show that it is possible to induce every expert to do research and report truthfully to the client as an equilibrium if and only if payments to the expert depend on other experts' reports. If the experts are risk-averse then the prevention of herding is costly.

Mathematical and non-Mathematical Finance

Lattice Option Pricing By Multidimensional Interpolation

Mathematical Finance, 2005, 15(4), 635-647
This note proposes a method for pricing high-dimensional American options based on modern methods of multidimensional interpolation. The method allows using sparse grids and thus mitigates the curse of dimensionality.

A framework of the pricing algorithm and the corresponding interpolation methods are discussed, and a theorem is demonstrated that suggests that the pricing method is less vulnerable to the curse of dimensionality. The method is illustrated by an application to rainbow options and compared to Least Squares Monte Carlo and other benchmarks.

Optimal Asset Allocation with Asymptotic Criteria
International Journal of Theoretical and Applied Finance, 2003, 6(6), 593-604
Assume (1) asset returns follow a stochastic multi-factor process with time-varying conditional expectations; (2) investments are linear functions of factors. This paper calculates asymptotic joint moments of the logarithm of investor's wealth and the factors. These formulas enable fast computation of a wide range of investment criteria. The results are illustrated by a numerical example that shows that the optimal portfolio rules are sensitive to the specification of the investment criterion.

Value Investing in Emerging Markets: Risks and Benefits
Emerging Markets Review, 2002, 3(3), 233-244 (#7 in most requested papers 2002)
This paper identifies a subset of emerging markets that have higher than average expected returns and studies risk properties of this subset by investment simulations. It is found that: (1) the portfolio of "value" emerging markets generates superior returns, and (2) statistical measures of its risk are close to the corresponding measures for the portfolio of all emerging markets. The statistical significance of these results has been checked by a bootstrap procedure. The results imply that the optimal share of emerging markets increases from 0% for an equally weighted portfolio to about 25% for the portfolio of undervalued emerging markets.