Measure Theory and Probability Theory

By Krishna B. Athreya, Soumendra Nath Lahiri. 2006

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Sufficient Forecasting Using Factor Models (2015, methodology): “Since E | u it | 8 ≤ M for some constant M and any t by Assumption 3.3(1), by using Davydov’s inequality (Corollary 16.2.4 in Athreya and Lahiri (2006)), there exist constants C 1 and C 2 such that for all ( i, j ), where α ( · ) is the mixing coeﬃcient, ρ ∈ (0 , 1), and α ( | t − s | ) cρ | t −…”
A penalized empirical likelihood method in high dimensions (2013, unknown): “…< · · ·< tr ≤ 1, 1≤ r <∞, (Zn(t1), . . . ,Zn(tr))→d (Z(t1), . . . ,Z(tr)),(6.7) (ii) For any ε > 0, δ > 0, there exists N =N(ε, δ) ∈N such that lim sup n→∞ P ( ∞ ∑ k=N |〈Zn, φk〉|2 > δ ) < ε.(6.8) Part (i) can be proved using Theorem 11.1.6 of Athreya and Lahiri (2006); we omit the routine details.”
Large Covariance Estimation by Thresholding Principal Orthogonal Complements (2011, unknown): “Stock and Watson (1998, 2002), Bai and Ng (2002), Bai (2003) and Doz et al. (2011), among others. Our contribution is on the estimation of covariance matrices and their inverse in large factor models. The static model that is considered in this paper is to be distinguished from the dynamic factor mo”
Measure Transformed Independent Component Analysis (2013, background): “Therefore, by Proposition 4.1.2 in (Athreya and Lahiri, 2006) it is implied that PX is absolutely continuous w.r.t.”
On Bayes Risk Lower Bounds (2014, background): “Because of the inequality R ≤ R Q (proved in Lemma 7.1 in the Appendix) and Lemma 3.1, it should be clear that (14) provides an implicit lower bound for R = R Bayes ( w, L ; Θ) .”
Variable selection in functional data classification: a maxima-hunting proposal (2013, methodology): “e in statistical applications, in particular to the classication problem; see, e.g., Lindquist and McKeague (2009). Let us now recall some basic notions and results to be used throughout [see, e.g., Athreya and Lahiri (2006), ch. 4, for details]: 0 is said to be absolutely continuous with respect t”
Statistical Inference for High-Dimensional Matrix-Variate Factor Models (2020, unknown): “4 in Athreya and Lahiri (2006)), there is a constantC > 0, for all i, l ∈ [p], j,h ∈ [q], and s, t ∈ [T ], ∣∣∣∣Cov [et,ijet,lh, es,ijes,lh]∣∣∣∣ ≤ Cα(|t − s|)1−2/γ .”
On computing the distribution function for the Poisson binomial distribution (2013, methodology): “Our approach is based on the characteristic function (CF) for the Poisson binomial distribution (see, for example, Athreya and Lahiri, 2006, Chapter 10 for details on CF). Fernández and Williams (2010) provided the same closed-form expression for the cdf, which was derived by using the polynomial in”
STATISTICAL INFERENCE FOR THE MEAN OUTCOME UNDER A POSSIBLY NON-UNIQUE OPTIMAL TREATMENT STRATEGY (2016, background): “, Athreya and Lahiri (2006)] would allow us to show that the leading term on the right-hand side converges to a standard normal random variable.”
Growth of preferential attachment random graphs via continuous-time branching processes (2007, unknown): “9 [6]).”