Almost sure central limit theory for products of sums of partial sums

Consider a sequence of i.i.d.positive random variables.An universal result in almost sure limit theorem for products of sums of partial sums is established.We will show that the almost sure limit theorem holds under a fairly general condition on the weight dk= k-1 exp（lnβk）,0≤β〈1.And in a sense,our results have reached the optimal form.

Nullors, and Nullor Circuits;There Applications in Symbolic Circuit Analysis and Design

The objective in this presentation is to introduce some of the unique properties and applications of nullors in active circuit analysis and designs. The emphasis is to discuss the role nullors can play in symbolic representation of transfer functions. To show this we adopt the topological platform for the circuit analysis and use a recently developed Admittance Method (AM) to achieve the Sum of Tree Products (STP), replacing the determinant and cofactors of the Nodal Admittance Matrix (NAM) of the circuit. To construct a transfer function, we start with a given active circuit and convert all its controlled sources and I/O-ports to nullors. Now, with a solid nullor circuit (passive elements and nullors) we first eliminate the passive elements through AM operations. This produces the STPs. Second, the all-nullor circuit is then used to find the signs or the STPs. Finally, the transfer function (in symbolic, if chosen) is obtained from the ratio between the STPs.

A General Law of Precise Asymptotics for Products of Sums under Dependence

Let {Xn,n ≥ 1} be a strictly stationary LNQD （LPQD） sequence of positive random variables with EX1 = μ 〉 0, and VarX1 = σ^2 〈 ∞. Denote by Sn = ∑i=1^n Xi and γ = σ/μ the coefficients of variation. In this paper, under some suitable conditions, we show that a general law of precise asymptotics for products of sums holds. It can describe the relations among the boundary function, weighted function, convergence rate and limit value in the study of complete convergence.

Approximate Processing Element Design and Analysis for the Implementation of CNN Accelerators

As a primary computation unit,a processing element(PE)is key to the energy efficiency of a convolutional neural network(CNN)accelerator.Taking advantage of the inherent error tolerance of CNNs,approximate computing with high hardware efficiency has been considered for implementing the computation units of CNN accelerators.However,individual approximate designs such as multipliers and adders can only achieve limited accuracy and hardware improvements.In this paper,an approximate PE is dedicatedly devised for CNN accelerators by synergistically considering the data representation,multiplication and accumulation.An approximate data format is defined for the weights using stochastic rounding.This data format enables a simple implementation of multiplication by using small lookup tables,an adder and a shifter.Two approximate accumulators are further proposed for the product accumulation in the PE.Compared with the exact 8-bit fixed-point design,the proposed PE saves more than 29%and 20%in power-delay product for 3×3 and 5×5 sum of products,respectively.Also,compared with the PEs consisting of state-of-the-art approximate multipliers,the proposed design shows significantly smaller error bias with lower hardware overhead.Moreover,the application of the approximate PEs in CNN accelerators is analyzed by implementing a multi-task CNN for face detection and alignment.We conclude that 1)an approximate PE is more effective for face detection than for alignment,2)an approximate PE with high statistically-measured accuracy does not necessarily result in good quality in face detection,and 3)properly increasing the number of PEs in a CNN accelerator can improve its power and energy efficiency.

A Note on the ASCLT for Triangular Arrays of Random Variables with an Extension to U-statistics

In this paper, we obtain an almost sure central limit theorem for products of independent sums of positive random variables. An extension of the result gives an ASCLT for the U-statistics.

A New Improved Method for Symbolic Reliability Analysis

This paper describes a method for finding the terminal pair reliability expression of communication networks.This method incorporates bit manipulation and Boolean operations for generating the sum of disjointed products(SDP) expression in which each term has 1 to 1 correspondence with respective minimal path(or cut).This method is convenient and efficient. An example illustrates the method.

Exact Reliability Formulas for Linear Consecutive-k-out-of-n: F Systems

A consecutive k out of n: F System, with n linearly arranged i. i. d. components, is examined. This paper gives an exact formula for computing the system's reliability directly. Complexity of this algorithm is an O(n k). Further more, the system's reliability can be computed more quickly and more effectively from this formula than those previously given.