Second Berkeley Symposium on Mathematical Statistics and Probability, Univ. Feller, “Diffusion processes in genetics,” in Proc. Li, “Construction of continuous-state branching processes in varying environments,” arXiv: 2002.09113v2. Li, “Stochastic equations, flows and measure-valued processes,” Ann. Li, “Skew convolution semigroups and affine Markov processes,” Ann. Church, “On infinite composition products of probability generating functions,” Z. Simatos, “On the scaling limits of Galton–Watson processes in varying environments,” Electron. Simatos, “Tightness for processes with fixed points of discontinuities and applications in varying environment,” Electron. Böinghoff, “Lower large deviations for supercritical branching processes in random environment,” Proc. Shurenkov, “Transitional phenomena and the convergence of Galton–Watson processes to Jiřina processes,” Theory Probab. Agresti, “On the extinction times of varying and random environment branching processes,” J. Vatutin, “Criticality for branching processes in random environment,” Ann. Vatutin, “Limit theorems for weakly subcritical branching processes in random environment,” J. We then establish a convergence theorem about the ratio P ξ ( Z n = j ) / P ξ ( Z n = 1 ), which can be applicable in each of the subcritical, critical, and supercritical cases.V. As a by-product of the proof we also remove the exchangeability condition in another result of Athreya and Karlin (1971) for the subcritical case about the decay rate of the survival probability given the environment. In this paper we first improve this basic result by removing the exchangeability condition on the environment, and by establishing a more general result about the conditional law of Z n given the non-extinction at time n + k for each fixed k ≥ 0. Athreya and Karlin (1971) proved the basic result about the concept of subcriticality and criticality, by showing that under the quenched law P ξ, the conditional distribution of Z n given the non-extinction at time n converges in law to a proper distribution on N + = in the subcritical case, and to the null distribution in the critical case, under the condition that the environment sequence is exchangeable. We consider a branching process ( Z n ) in a stationary and ergodic random environment ξ = ( ξ n ). Strategies to reduce gut dysbiosis might improve outcomes of patients with cirrhosis. In hospitalized patients with cirrhosis, dysbiosis of the intestinal microbiota on admission (particularly changes in Protebacteria constituents) associates with increased risk of extra-hepatic organ failure, ACLF, and death, independent of clinical factors. Changes in the microbiota associated with extra-hepatic organ failure, transfer to intensive care, ACLF, and death, independently on logistic regression analyses. Taxa belonging to phylum Proteobacteria (Enterobacteriaceae, Campylobacteriaceae, and Pasteurellaceae) and Firmicutes (Enterococcaceae and Streptococcaceae) were associated with development of negative outcomes, whereas other Firmicutes members (Lachnospiraceae and Clostridiales) reduced risk of negative outcomes. The CDR was lower in subjects who developed ACLF, especially among those with renal failure. Beta diversity of the intestinal microbiome was significantly different, whereas alpha diversity was similar, between subjects with vs without outcomes. Regression analyses were performed using microbial and clinical variables for each outcome.ĪCLF developed in 8% of study subjects 16% were transferred to intensive care and 21% died. Microbiota were analyzed by 16s rRNA sequencing for alpha diversity (within groups), beta diversity (between groups), cirrhosis dysbiosis ratio (CDR), and taxa that differed between groups with vs without negative outcomes (individual organ failures, transfer to intensive care, ACLF, death, or hospice). Patients were followed for 30 days for development of ACLF, extra-hepatic organ failures, and death or hospice care. Stool samples were collected from 181 patients with cirrhosis (age 56 years mean model for end-stage liver disease score, 21 43% with infections) at time of admission, from multiple hospitals in North America. We investigated whether features of the intestinal microbiota associate organ failure, transfer to intensive care, and mortality within 30 days in patients admitted to the hospital with cirrhosis. ACLF is associated with dysbiosis of the intestinal microbiota, which might serve as a prognostic factor. Inpatients with cirrhosis are prone to develop acute-on-chronic liver failure (ACLF).
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