conditional central limit theorem
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Biometrika ◽  
2020 ◽  
Author(s):  
D C Ahfock ◽  
W J Astle ◽  
S Richardson

Summary Sketching is a probabilistic data compression technique that has been largely developed by the computer science community. Numerical operations on big datasets can be intolerably slow; sketching algorithms address this issue by generating a smaller surrogate dataset. Typically, inference proceeds on the compressed dataset. Sketching algorithms generally use random projections to compress the original dataset, and this stochastic generation process makes them amenable to statistical analysis. We argue that the sketched data can be modelled as a random sample, thus placing this family of data compression methods firmly within an inferential framework. In particular, we focus on the Gaussian, Hadamard and Clarkson–Woodruff sketches and their use in single-pass sketching algorithms for linear regression with huge samples. We explore the statistical properties of sketched regression algorithms and derive new distributional results for a large class of sketching estimators. A key result is a conditional central limit theorem for data-oblivious sketches. An important finding is that the best choice of sketching algorithm in terms of mean squared error is related to the signal-to-noise ratio in the source dataset. Finally, we demonstrate the theory and the limits of its applicability on two datasets.





2012 ◽  
Vol 15 (02) ◽  
pp. 1250011 ◽  
Author(s):  
NICOLAS DIENER ◽  
ROBERT JARROW ◽  
PHILIP PROTTER

This paper uses a conditional law of large numbers and a conditional central limit theorem to provide simplified asymptotic valuation formulas for credit derivatives on baskets, including synthetic and cash-flow CDOs. In particular, approximate pricing procedures are provided for synthetic and cash-flow CDOs. In the process, this paper also clarifies the relation between the "top-down" and "bottom-up" approaches for pricing credit derivatives.



2011 ◽  
Vol 11 (01) ◽  
pp. 71-80 ◽  
Author(s):  
DALIBOR VOLNÝ ◽  
MICHAEL WOODROOFE ◽  
OU ZHAO

The Central Limit Theorem is studied for stationary sequences that are sums of countable collections of linear processes. Two sets of sufficient conditions are obtained. One restricts only the coefficients and is shown to be best possible among such conditions. The other involves an interplay between the coefficients and the distribution functions of the innovations and is shown to be necessary for the Conditional Central Limit Theorem in the case of a causal process with independent innovations.



2003 ◽  
Vol 108 (2) ◽  
pp. 229-262 ◽  
Author(s):  
Jérôme Dedecker ◽  
Florence Merlevède






1987 ◽  
Vol 15 (2) ◽  
pp. 776-782 ◽  
Author(s):  
Dieter Landers ◽  
Lothar Rogge




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