By Walter Philipp

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**Additional resources for Almost sure invariance principles for partial sums of weakly dependent random variables**

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For example, one can work in groups of seven, with any permutation of HHLLLLL used, and the permutation can vary with time. Alternatively, work in groups of four, where the ﬁrst three are any permutation of HLL and the fourth is selected at random, with L being selected with probability 6/7. If one form of the algorithm is well deﬁned and convergent, all the suggested forms will be. The various alternatives can be alternated among each other, etc. Again, the convergence proofs show that it is only the “local averages” that determine the limit points.

However, during a training period many samples of the pairs (yn , φn ) were available, and a recursive linear least squares algorithm was used to sequentially get the optimal weights for the aﬃne decision function. Thus, during the training period, we used a sequence of inputs {φn } and chose θ so that the outputs vn = θ φn matches the sequence of correct decisions yn as closely as possible in the mean square sense. Neural networks serve a similar purpose, but the output vn can be a fairly general nonlinear function of the input [8, 97, 193, 205, 253].

6) shows that it is preferable to use χ+ n,i = χn,i if possible, since it eliminates the dominant 1/cn factor in the eﬀective noise. That is, ψn,i 16 1. 6) and is not inversely proportional to cn . − The use of χ+ n,i = χn,i can also be advantageous, even without diﬀerentiability. Fixing θn = θ, letting EF (θ ± cn ei , χ± n,i ) = f (θ ± cn ei ), and F (θ, χ) = F (θ, χ) − f (θ), the variance of the eﬀective noise is E F (θ + cn ei , χ+ n,i ) 2 +E F (θ − cn ei , χ− n,i ) 2 −2E F (θ + cn ei , χ+ n,i ) F (θ − cn ei , χ− n,i ) , divided by 4c2n , which suggests that the larger the correlation between χ± n,i , the smaller the noise variance will be when cn is small.