ON CONVERGENCE OF STOCHASTIC PROCESSES

By J. Lamperti. 1962

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In Pure Mathematics

Citations in the corpus

A Theory of Multiple-Source Adaptation with Limited Target Labeled Data (2020, background): “They can be easily extended to bounds based on Rademacher complexity (Mohri et al., 2018) or pseudo-dimension (Pollard, 2012) for general losses.”
Smoothness, Low Noise and Fast Rates (2010, background): “More formally, these are hypothesis classes with ﬁnite VC-subgraph dimension [24] (aka pseudo-dimension).”
GenDICE: Generalized Offline Estimation of Stationary Values (2020, methodology): “Lemma 4 (Pollard (2012)) Let G be a permissible class of Z → [−M,M ] functions and {Zi}Ni=1 are i.”
A Model of Inductive Bias Learning (2000, methodology): “ological space by equipping it with the topology of weak convergence. B ( P ) is then the -algebra generated by this topology. The following two deﬁnitions are taken (with minor modiﬁcation s) from Pollard (1984). Deﬁnition 8. A set H of [0 ; 1] -valued functions on Z is indexed by the set T if the”
Learning in a Large Function Space: Privacy-Preserving Mechanisms for SVM Learning (2009, background): “Consider loss function `(y, ŷ) that is di erentiable, convex and L-Lipschitz in ŷ, and an RKHS H induced by nite F -dimensional feature mapping φ with bounded norm k(x,x) ≤ κ2 for all x ∈ Rd.”
On the mathematical foundations of learning (2001, background): “(ii) Bernstein's inequality without the absolute value provides a bound without the first 2, i.e. e − mε 2 2 ( σ 2 + 1 3 Mε ) (see [32]).”
Kernel dimension reduction in regression (2009, background): “The chaining lemma [25], which plays a crucial role in the uniform central limit theorem, is readily extendable to a random process in a Banach space.”
Local Rademacher complexities (2005, background): “Also,m ostpreviousresults,such asthose ofPollard [28],van de G eer[32] or Haussler [13],give com plexity term s that involve \global" m easures of com plexity oftheclass,such ascovering num bers.”
Contextual Decision Processes with low Bellman rank are PAC-Learnable (2016, methodology): “Lemma 20 (Uniform deviation bound using covering number [Pollard, 1984]; also see Devroye et al.”
Analysis of Classification-based Policy Iteration Algorithms (2016, methodology): “By Pollard's inequality (Pollard, 1984), for the bounded space F k , we have P D k sup ℓπ k ∈F k 1 N N i=1 ℓ π k (x i ) − ℓ π k (x)ρ(dx) > ǫ ≤ 8E N 1 ǫ 8 , F k , X N 1 exp − N ǫ 2 128(2Q max ) 2 .”
Weak convergence of the tail empirical process for dependent sequences (2009, background): “Let D(I ) be the space of real functions on the (finite or infinite) interval I which are right continuous and have left limits at each point, given the Lindvall–Stone extension of the Skorokhod J1-topology; see [33].”
Learning Adversary Behavior in Security Games: A PAC Model Perspective (2015, methodology): “For the parametric case, we derive sharp sample complexity bounds based on covering numbers using our techniques rather than bounding it using the standard technique of pseudo-dimension [28] or fat shattering dimension [2].”
CONSISTENCY OF CROSS VALIDATION FOR COMPARING REGRESSION PROCEDURES (2007, background): “Thus conditional on Z1, on Hn1, by Bernstein’s inequality (see, e.g., Pollard [18], page 193), for each x > 0, we have PZ1 ( n∑ i=n1+1 ( f (Xi) − f̂n1,1(Xi) )2 − n2‖f − f̂n1,1‖22 ≥ x ) ≤ exp ( −1 2 x2 n2‖f − f̂n1,1‖44 + (2(An1, )2x/3) ) .”
Local nonlinear least squares: Using parametric information in nonparametric regression (2000, background): “…; the outer probability for A is deened as P ( A ) = inffP ( B ) : A B ; B 2 Gg : The reason for needing outer probability is due to the fact that the random covering numbers need not be measurable with respect to P , even if the class of functions is permissible in the sense of Pollard ( 1984 ) .”
Size-free generalization bounds for convolutional neural networks (2019, methodology): “We will prove Theorem 3.1 by using || · ||N to witness the fact that `Fβ,ν is ( χλβeνL+β ,W ) -Lipschitz parameterized.”
Nonparametric comparison of regression curves: An empirical process approach (2003, methodology): “Let P n denote the symmetrization of Pn [see Pollard (1984), p.”