This paper studies Graphical SLOPE for precision matrix estimation, with emphasis on its ability to recover both sparsity and clusters of edges with equal or similar strength. In a fixed-dimensional regime, we establish that the root-$n$ scaled estimation error converges to the unique minimizer of a strictly convex optimization problem defined through the directional derivative of the SLOPE penalty. We also establish convergence of the induced SLOPE pattern, thereby obtaining an asymptotic characterization of the clustering structure selected by the estimator. A comparison with GLASSO shows that the grouping property of SLOPE can substantially improve estimation accuracy when the precision matrix exhibits structured edge patterns. To assess the effect of departures from Gaussianity, we then analyze Gaussian-loss precision matrix estimation under elliptical distributions. In this setting, we derive the limiting distribution and quantify the inflation in variability induced by heavy tails relative to the Gaussian benchmark. We also study TSLOPE, based on the multivariate $t$-loss, and derive its limiting distribution. The results show that TSLOPE offers clear advantages over GSLOPE under heavy-tailed data-generating mechanisms. Simulation evidence suggests that these qualitative conclusions persist in high-dimensional settings, and an empirical application shows that SLOPE-based estimators, especially TSLOPE, can uncover economically meaningful clustered dependence structures.

Two Literal Crypto Bros Built a Real Estate Empire. In 2019, two Canadian brothers blew into Detroit with an irresistible pitch: For $50, almost anyone could become a property owner. When houses decayed and the city intervened, the blame games began. A fire broke out at 10410 Cadieux in March 2025, burning a hole in the roof. The smell hit me first: damp brick, stagnant water, mold, and bleach. I was partway down a flight of wooden stairs that led to the basement of a 1920s duplex in east Detroit, Michigan. Leading the way was Cornell Dorris, a tenant in the building for nearly a decade. Dorris is in his early forties, has two daughters who visit on weekends, and makes a living smoking meat and cooking for events. As my eyes adjusted, I made out rodent droppings and a black puddle that spread across the basement floor. "Anytime it rains, the water comes down," Dorris said. The air was unnaturally heavy, and I felt a nagging urge to leave. Dorris doesn't have a typical landlord. Almost four years ago, his building was acquired by a startup called RealToken, or RealT.

Online portfolio selection is a sequential decision-making problem in which a learner repetitively selects a portfolio over a set of assets, aiming to maximize long-term return. In this paper, we study the problem with the cardinality constraint that the number of assets in a portfolio is restricted to be at most k, and consider two scenarios: (i) in the full-feedback setting, the learner can observe price relatives (rates of return to cost) for all assets, and (ii) in the bandit-feedback setting, the learner can observe price relatives only for invested assets. We propose efficient algorithms for these scenarios that achieve sublinear regrets. We also provide regret (statistical) lower bounds for both scenarios which nearly match the upper bounds when k is a constant. In addition, we give a computational lower bound which implies that no algorithm maintains both computational efficiency, as well as a small regret upper bound.

The central objective of empirical asset pricing is to identify firm-level signals that explain the cross-section of expected stock returns--whether through exposure to risk factors or persistent mispricing. The dominant paradigm, grounded in the assumption of self-predictability, asserts that a firm's own characteristics forecast its own returns (see, e.g., Cochrane (2011); Harvey et al. (2016)). Complementing this view is a growing literature on cross-predictability--the idea that the characteristics or returns of one asset can help forecast the returns of others (see, e.g., Lo and MacKinlay (1990); Hou (2007); Cohen and Frazzini (2008); Cohen and Lou (2012); Huang et al. (2021, 2022)). A key mechanism underpinning this phenomenon is the presence of lead-lag effects, whereby price movements or information from one firm precede and predict those of related firms. Such effects can stem from staggered information diffusion, peer influence within industries, supply chain linkages, or correlated trading by institutional investors that induces price pressure across related assets. Despite recent methodological advances in modeling cross-stock predictability, several foundational questions remain unresolved. Chief among them is how a mean-variance investor can analytically integrate multiple predictive signals when returns are interconnected across assets. Equally crucial is developing a framework that jointly captures both the relevance of individual signals and the structure of return spillovers--enhancing portfolio performance while preserving interpretability .

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In modern portfolio management, one may require anm-sparse (no more thanm active assets) portfolio to save managerial and financial costs.

A large and rapidly expanding literature demonstrates that machine learning (ML) methods substantially improve out-of-sample asset return prediction relative to conventional linear benchmarks, and that these statistical gains often translate into economically meaningful portfolio performance. Seminal contributions such as Gu et al. (2020) document large Sharpe ratio improvements from nonlinear learners in U.S. equities, while subsequent work extends these findings to stochastic discount factor estimation (Chen et al. 2024), international equity markets (Leippold et al. 2022), and bond return forecasting (Kelly et al. 2019, Bianchi et al. 2020). Collectively, this literature establishes ML as a powerful tool for extracting conditional expected returns in environments characterized by noisy signals, nonlinear interactions, and pervasive multicollinearity.

We investigate machine learning models for stock return prediction in non-stationary environments, revealing a fundamental nonstationarity-complexity tradeoff: complex models reduce misspecification error but require longer training windows that introduce stronger non-stationarity. We resolve this tension with a novel model selection method that jointly optimizes model class and training window size using a tournament procedure that adaptively evaluates candidates on non-stationary validation data. Our theoretical analysis demonstrates that this approach balances misspecification error, estimation variance, and non-stationarity, performing close to the best model in hindsight. Applying our method to 17 industry portfolio returns, we consistently outperform standard rolling-window benchmarks, improving out-of-sample $R^2$ by 14-23% on average. During NBER-designated recessions, improvements are substantial: our method achieves positive $R^2$ during the Gulf War recession while benchmarks are negative, and improves $R^2$ in absolute terms by at least 80bps during the 2001 recession as well as superior performance during the 2008 Financial Crisis. Economically, a trading strategy based on our selected model generates 31% higher cumulative returns averaged across the industries.

Acquisition would further expand SoftBank's investments in artificial intelligence as it tries to center itself in the boom SoftBank Group will acquire digital infrastructure investor DigitalBridge Group in a deal valued at $4bn, the companies said on Monday, as the Japanese investment firm looks to deepen its AI-related portfolio. The acquisition would expand SoftBank's exposure to digital infrastructure as the Japanese conglomerate is positioning its portfolio to focus on artificial intelligence. SoftBank's billionaire founder Masayoshi Son is seeking to capitalize on surging demand for the computing capacity that underpins artificial intelligence applications. DigitalBridge invests in digital infrastructure sectors such as datacenters, cell towers, fiber networks, small-cell systems and edge infrastructure, with a portfolio including companies such as Vantage Data Centers, Zayo, Switch and AtlasEdge. Founded in 1991 as real estate-focused Colony Capital, the firm pivoted under CEO Marc Ganzi into digital infrastructure and rebranded as DigitalBridge in 2021 after shedding most of its legacy property assets.