Amalgamated Free Lévy Processes as Limits of Sample Covariance Matrices

We prove the existence of joint limiting spectral distributions for families of random sample covariance matrices modeled on fluctuations of discretized Lévy processes. These models were first considered in applications of random matrix theory to financial data, where datasets exhibit both strong multicollinearity and non-normality. When the underlying Lévy process is non-Gaussian, we show that the limiting spectral distributions are distinct from Marčenko–Pastur. In the context of operator-valued free probability, it is shown that the algebras generated by these families are asymptotically free with amalgamation over the diagonal subalgebra. This framework is used to construct operator-valued $$^*$$ ∗ -probability spaces, where the limits of sample covariance matrices play the role of non-commutative Lévy processes whose increments are free with amalgamation.

Nomogram to predict the outcomes of patients with microsatellite instability-high metastatic colorectal cancer receiving immune checkpoint inhibitors

BackgroundThe efficacy of immune checkpoint inhibitors (ICIs) in patients with microsatellite instability (MSI)-high metastatic colorectal cancer (mCRC) is unprecedented. A relevant proportion of subjects achieving durable disease control may be considered potentially ‘cured’, as opposed to patients experiencing primary ICI refractoriness or short-term clinical benefit. We developed and externally validated a nomogram to estimate the progression-free survival (PFS) and the time-independent event-free probability (EFP) in patients with MSI-high mCRC receiving ICIs.MethodsThe PFS and EFP were estimated using a cure model fitted on a developing set of 163 patients and validated on a set of 146 patients with MSI-high mCRC receiving anti-programmed death (ligand)1 (PD-(L)1) ± anticytotoxic T-lymphocyte antigen 4 (CTLA-4) agents. A total of 23 putative prognostic factors were chosen and then selected using a random survival forest (RSF). The model performance in estimating PFS probability was evaluated by assessing calibration (internally—developing set and externally—validating set) and quantifying the discriminative ability (Harrell C index).ResultsRFS selected five variables: ICI type (anti-PD-(L)1 monotherapy vs anti-CTLA-4 combo), ECOG PS (0 vs >0), neutrophil-to-lymphocyte ratio (≤3 vs >3), platelet count, and prior treatment lines. As both in the developing and validation series most PFS events occurred within 12 months, this was chosen as cut-point for PFS prediction. The combination of the selected variables allowed estimation of the 12-month PFS (focused on patients with low chance of being cured) and the EFP (focused on patients likely to be event-free at a certain point of their follow-up). ICI type was significantly associated with disease control, as patients receiving the anti-CTLA-4-combination experienced the best outcomes. The calibration of PFS predictions was good both in the developing and validating sets. The median value of the EFP (46%) allowed segregation of two prognostic groups in both the developing (PFS HR=3.73, 95% CI 2.25 to 6.18; p<0.0001) and validating (PFS HR=1.86, 95% CI 1.07 to 3.23; p=0.0269) sets.ConclusionsA nomogram based on five easily assessable variables including ICI treatment was built to estimate the outcomes of patients with MSI-high mCRC, with the potential to assist clinicians in their clinical practice. The web-based system ‘MSI mCRC Cure’ was released.

Salvage Radical Prostatectomy for Radio-Recurrent Prostate Cancer: An Updated Systematic Review of Oncologic, Histopathologic and Functional Outcomes and Predictors of Good Response

A valid treatment option for recurrence after definite radiotherapy (RT) for localized prostate cancer (PC) is salvage radical prostatectomy (SRP). However, data on SRP are scarce, possibly resulting in an underutilization. A systematic review was performed using MEDLINE (Pubmed), Embase, and Web of Science databases including studies published between January 1980 and April 2020. Overall, 23 English language articles including a total number of 2323 patients were selected according to PRISMA criteria. The overall median follow-up was 37.5 months (IQR 35.5–52.5). Biochemical-recurrence (BCR)-free probability ranged from 34% to 83% at five years, respectively, and from 31% to 37% at 10 years. Cancer specific survival (CSS) and overall survival (OS) ranged from 88.7% to 98% and 64% to 95% at five years and from 72% to 83% and 65% to 72% at 10 years, respectively. Positive surgical margins ranged from 14% to 45.8% and pathologic organ-confined disease was reported from 20% to 57%. The rate of pathologic > T2-disease ranged from 37% to 80% and pN1 disease differed between 0% to 78.4%. Pre-SRP PSA, pre-SRP Gleason Score (GS), pathologic stage after SRP, and pathologic lymph node involvement seemed to be the strongest prognostic factors for good outcomes. SRP provides accurate histopathological and functional outcomes, as well as durable cancer control. Careful patient counseling in a shared decision-making process is recommended.

Additivity violation of quantum channels via strong convergence to semi-circular and circular elements

Additivity violation of minimum output entropy, which shows non-classical properties in quantum communication, had been proved in most cases for random quantum channels defined by Haar-distributed unitary matrices. In this paper, we investigate random completely positive maps made of Gaussian Unitary Ensembles and Ginibre Ensembles regarding this matter. Using semi-circular systems and circular systems of free probability, we not only show the multiplicativity violation of maximum output norms in the asymptotic regimes but also prove the additivity violation via Haagerup inequality for a new class of random quantum channels constructed by rectifying the above completely positive maps based on strong convergence.

Cumulant–Cumulant Relations in Free Probability Theory from Magnus’ Expansion

Relations between moments and cumulants play a central role in both classical and non-commutative probability theory. The latter allows for several distinct families of cumulants corresponding to different types of independences: free, Boolean and monotone. Relations among those cumulants have been studied recently. In this work, we focus on the problem of expressing with a closed formula multivariate monotone cumulants in terms of free and Boolean cumulants. In the process, we introduce various constructions and statistics on non-crossing partitions. Our approach is based on a pre-Lie algebra structure on cumulant functionals. Relations among cumulants are described in terms of the pre-Lie Magnus expansion combined with results on the continuous Baker–Campbell–Hausdorff formula due to A. Murua.