Suppose that (Xnj) is a triangular array of random variables taking values in a Banach space E and that (Bn) is the corresponding sequence of random paths in E. Conditions are considered under which the distributions ...
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Suppose that (Xnj) is a triangular array of random variables taking values in a Banach space E and that (Bn) is the corresponding sequence of random paths in E. Conditions are considered under which the distributions of Bn converge to a Gaussian measure on C([0,1];E). Under stronger conditions on the array it is shown that if E is of type 2 the paths enjoy certain regularity properties, which are reflected in the convergence. The technique here is to factorise the integration procedure by which one passes from the array to the sequence of paths, using fractional integrals.
We consider a Brownian semimartingale X (the sum of a stochastic integral w.r.t. a Brownian motion and an integral w.r.t. Lebesgue measure), and for each n an increasing sequence T(n, i) of stopping times and a sequen...
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We consider a Brownian semimartingale X (the sum of a stochastic integral w.r.t. a Brownian motion and an integral w.r.t. Lebesgue measure), and for each n an increasing sequence T(n, i) of stopping times and a sequence of positive F-T (n,F-i)-measurable variables Delta(n, i) such that S(n, i) : T(n, i) + Delta(n, i) <= T(n, i + 1). We are interested in the limiting behavior of processes of the form U-t(n)(g) = root delta(n)Sigma(i:S(n,i) <= t) [g(T(n, i), xi(n)(i)) - alpha(n)(i)(g)], where delta(n) is a normalizing sequence tending to 0 and xi(n)(i) - Delta(n, i)(-1/2)(X-S(n,X- i) - X-T(n,X- i)) and alpha(n)(i)(g) are suitable centering terms and g is some predictable function of (omega, t, x). Under rather weak assumptions on the sequences T(n, i) as n goes to infinity, we prove that these processes converge (stably) in law to the stochastic integral of g w.r.t. a random measure B which is, conditionally on the path of X, a Gaussian random measure. We give some applications to rates of convergence in discrete approximations for the p-variation processes and local times.
We study the weak convergence for the row sums of a general triangular array of empirical processes indexed by a manageable class of functions converging to an arbitrary limit. As particular cases, we consider random ...
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This paper will discuss the analogues between Leibniz's Harmonic Triangle and Pascal's Arithmetic Triangle by utilizing mathematical proving techniques like partial sums, committees, telescoping, mathematical ...
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This paper will discuss the analogues between Leibniz's Harmonic Triangle and Pascal's Arithmetic Triangle by utilizing mathematical proving techniques like partial sums, committees, telescoping, mathematical induction and applying George Polya's perspective. The topics presented in this paper will show that Pascal's triangle and Leibniz's triangle both have hockey stick type patterns, patterns of sums within shapes, and have the natural numbers, triangular numbers, tetrahedral numbers, and pentatope numbers hidden within. In addition, this paper will show how Pascal's Arithmetic Triangle can be used to construct Leibniz's Harmonic Triangle and show how both triangles relate to combinatorics and arithmetic through the coefficients of the binomial expansion. Furthermore, combinatorics plays an increasingly important role in mathematics, starting when students enter high school and continuing on into their college years. Students become familiar with the traditional arguments based on classical arithmetic and algebra, however, methods of combinatorics can vary greatly. In combinatorics, perhaps the most important concept revolves around the coefficients of the binomial expansion, studying and proving their properties, and conveying a greater insight into mathematics.
This article studies the infinite-width limit of deep feedforward neural networks whose weights are dependent, and modelled via a mixture of Gaussian distributions. Each hidden node of the network is assigned a nonneg...
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This article studies the infinite-width limit of deep feedforward neural networks whose weights are dependent, and modelled via a mixture of Gaussian distributions. Each hidden node of the network is assigned a nonnegative random variable that controls the variance of the outgoing weights of that node. We make minimal assumptions on these per-no de random variables: they are iid and their sum, in each layer, converges to some finite random variable in the infinite-width limit. Under this model, we show that each layer of the infinite-width neural network can be characterised by two simple quantities: a non-negative scalar parameter and a Levy measure on the positive reals. If the scalar parameters are strictly positive and the Levy measures are trivial at all hidden layers, then one recovers the classical Gaussian process (GP) limit, obtained with iid Gaussian weights. More interestingly, if the Levy measure of at least one layer is non-trivial, we obtain a mixture of Gaussian processes (MoGP) in the large-width limit. The behaviour of the neural network in this regime is very different from the GP regime. One obtains correlated outputs, with non-Gaussian distributions, possibly with heavy tails. Additionally, we show that, in this regime, the weights are compressible, and some nodes have asymptotically non-negligible contributions, therefore representing important hidden features. Many sparsity-promoting neural network models can be recast as special cases of our approach, and we discuss their infinite-width limits;we also present an asymptotic analysis of the pruning error. We illustrate some of the benefits of the MoGP regime over the GP regime in terms of representation learning and compressibility on simulated, MNIST and Fashion MNIST datasets.
As a function of applied field, we find a rich variety of ordered and partially ordered vortex lattice configurations in systems with square or triangular arrays of pinning sites. We present formulas that predict the ...
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As a function of applied field, we find a rich variety of ordered and partially ordered vortex lattice configurations in systems with square or triangular arrays of pinning sites. We present formulas that predict the matching fields at which commensurate vortex configurations occur and the vortex lattice orientation with respect to the pinning lattice. Our results are in excellent agreement with recent imaging experiments on square pinning arrays [K. Harada et al., Science 274, 1167 (1996)].
The Skorokhod representation for martingales is used to obtain a functional central limit theorem (or invariance principle) for martingales. It is clear from the method of proof that this result may in fact be extende...
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The Skorokhod representation for martingales is used to obtain a functional central limit theorem (or invariance principle) for martingales. It is clear from the method of proof that this result may in fact be extended to the case of triangular arrays in which each row is a martingale sequence and the second main result is a functional central limit theorem for such arrays. These results are then used to obtain two functional central limit theorems for processes with stationary ergodic increments following on from the work of Gordin. The first of these theorems extends a result of Billingsley for Φ-mixing sequences.
This paper presents the stronger moment conditions which are sufficient to show the finiteness of the first collision time between N-dimensional Gaussian jump particle and another N-dimensional Brownian particle in a ...
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This paper presents the stronger moment conditions which are sufficient to show the finiteness of the first collision time between N-dimensional Gaussian jump particle and another N-dimensional Brownian particle in a fractured medium. To obtain these conditions, we use the triangular arrays of the random sequence vector which gives all probable positions of the first collision at any time. This study is an extension of the analytical study which is detailed in Alzulaibani and El-Hadidy [Int. J. Mod. Phys. B 33(28) (2019) 1950334] using the uniform integrability conditions.
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