AC^0 o MOD_2 Lower Bounds for the Boolean Inner Product.

AC0∘MOD2 lower bounds for the Boolean Inner Product

We prove large deviation results for the random sum , , where are non-negative integer-valued random variables and are i.i.d. non-negative random variables with common distribution function F, independent of . Special attention is paid to the compound Poisson process and its ramifications. The right tail of the distribution function F is supposed to be of Pareto type (regularly or extended regularly varying). The large deviation results are applied to certain problems in insurance and finance which are related to large claims.

Let be an ordered sample of n independent observations, X1,X2,…,Xn of a random variable X with distribution function F(x) and density f(x) continuous on its support set . As a nonparametric histogram estimator of the density function f(x), consider an estimator fn (x) of the form: where {An (x)} is a suitably chosen sequence of non-negative integer-valued indexing random variables; and {kn} is also an appropriately defined sequence of positive integers which depends only on the sample size n . J. Van Ryzin (1973) has given conditions under which the above estimators are pointwise consistent. In this paper we establish conditions under which such a histogram density estimator is uniformly consistent almost surely. When the density has a unique mode, the results are used to obtain a strongly consistent estimator of the mode similar to that of Venter (1967).

This paper introduces gemact, a Python package for actuarial modeling based on the collective risk model. The library supports applications to risk costing and risk transfer, loss aggregation, and loss reserving. We add new probability distributions to those available in scipy, including the (a, b, 0) and (a, b, 1) discrete distributions, copulas of the Archimedean family, the Gaussian, the Student t and the Fundamental copulas. We provide an implementation of the AEP algorithm for calculating the cumulative distribution function of the sum of dependent, nonnegative random variables, given their dependency structure specified with a copula. The theoretical framework is introduced at the beginning of each section to give the reader with a sufficient understanding of the underlying actuarial models.

Let $ACC \circ THR$ be the class of constant-depth circuits comprised of AND, OR, and MOD$m$ gates (for some constant $m > 1$), with a bottom layer of gates computing arbitrary linear threshold functions. This class of circuits can be seen as a midpoint between $ACC$ (where we know nontrivial lower bounds) and depth-two linear threshold circuits (where nontrivial lower bounds remain open). We give an algorithm for evaluating an arbitrary symmetric function of $2^{n^{o(1)}}$ $ACC \circ THR$ circuits of size $2^{n^{o(1)}}$, on all possible inputs, in $2^n \cdot poly(n)$ time. Several consequences are derived: $\bullet$ The number of satisfying assignments to an $ACC \circ THR$ circuit of subexponential size can be computed in $2^{n-n^{\varepsilon}}$ time (where $\varepsilon > 0$ depends on the depth and modulus of the circuit). $\bullet$ $NEXP$ does not have quasi-polynomial size $ACC \circ THR$ circuits, nor does $NEXP$ have quasi-polynomial size $ACC \circ SYM$ circuits. Nontrivial size lower bounds were not known even for $AND \circ OR \circ THR$ circuits. $\bullet$ Every 0-1 integer linear program with $n$ Boolean variables and $s$ linear constraints is solvable in $2^{n-\Omega(n/((\log M)(\log s)^{5}))}\cdot poly(s,n,M)$ time with high probability, where $M$ upper bounds the bit complexity of the coefficients. (For example, 0-1 integer programs with weights in $[-2^{poly(n)},2^{poly(n)}]$ and $poly(n)$ constraints can be solved in $2^{n-\Omega(n/\log^6 n)}$ time.) We also present an algorithm for evaluating depth-two linear threshold circuits (a.k.a., $THR \circ THR$) with exponential weights and $2^{n/24}$ size on all $2^n$ input assignments, running in $2^n \cdot poly(n)$ time. This is evidence that non-uniform lower bounds for $THR \circ THR$ are within reach.

A finite function f is a mapping of {0,1}n into {0,1}m∪{#}, where “#” is a symbol to be thought of as “undefined.” A family of finite functions is said to be one-way (in a circuit complexity sense) if it can be computed with polynomial size circuits, but no family of inverses of these functions can be computed with polynomial size circuits. In this paper we show that (provided functions that are not one-to-one are allowed) one-way functions exist if and only if the satisfiability problem SAT does not have polynomial size circuits.A family of functions f i (x) can be checked if some family of polynomial size circuits with inputs x and y can determine if f i (x)=y. A family of functions f i (x) can be evaluated if some family of polynomial size circuits with input x can compute f i (x). Can all families of total functions that can be checked also be evaluated? We show that this is true if and only if the nonuniform versions of the complexity classes P and UP∩co-UP are equal.A family of functions f i is one-way for constant depth circuits if f i can be computed with unbounded fan-in circuits of polynomial size and constant depth, but every family of inverses f −1i cannot. In this paper we give two provably one-way functions (in fact permutations) for constant depth circuits. The second example has the stronger property that no bit of its inverse can be computed in polynomial size and constant depth.

For non-negative integer-valued random variables, the concept of “damaged” observations was introduced, for the first time, by Rao and Rubin [Rao, C. R., Rubin, H. (1964). On a characterization of the Poisson distribution. Sankhya 26:295–298] in 1964 on a paper concerning the characterization of Poisson distribution. In 1965, Rao [Rao, C. R. (1965). On discrete distribution arising out of methods of ascertainment. Sankhya Ser. A. 27:311–324] discusses some results related with inferences for parameters of a Poisson Model when it has occurred partial destruction of observations. A random variable is said to be damaged if it is unobservable, due to a damage mechanism which randomly reduces its magnitude. In subsequent years, considerable attention has been given to characterizations of distributions of such random variables that satisfy the “Rao–Rubin” condition. This article presents some inference aspects of a damaged Poisson distribution, under reasonable assumption that, when an observation on the random variable is made, it is also possible to determine whether or not some damage has occurred. In other words, we do not know how many items are damaged, but we can identify the existence of damage. Particularly it is illustrated the situation in which it is possible to identify the occurrence of some damage although it is not possible to determine the amount of items damaged. Maximum likelihood estimators of the underlying parameters and their asymptotic covariance matrix are obtained. Convergence of the estimates of parameters to the asymptotic values are studied through Monte Carlo simulations.

Quantifying the model complexity of quantum circuits provides a guide to avoid overfitting in quantum machine learning. Previously we established a Vapnik–Chervonenkis (VC) dimension upper bound for ‘encoding-first’ quantum circuits, where the input layer is the first layer of the circuit. In this work, we prove a general VC dimension upper bound for quantum circuit learning including ‘data re-uploading’ circuits, where the input gates can be single qubit rotations anywhere in the circuit. A linear lower bound is also constructed. The properties of the bounds and approximation-estimation trade-off considerations are discussed.

Let X1, X2, · ··, Xn, · ·· be independent and identically distributed non-negative integer-valued random variables with finite mean and variance. For any positive integer n and m we consider the random vector i.e., L has the same distribution as the conditional distribution of (X1, · ··, Xm) given the condition It is easy to see that our model includes the classical urn model, the Bose–Einstein urn model and the Pólya urn model as special cases. For any non-negative integer s define G(s) = the number of Li′s such that Li = s, and U = the number of Li′s such that Li is an even number; in this paper we study the asymptotic behaviour of the random variables considered above. Some central limit theorems and a multinormal local limit theorem are proved.

Let $A_0(x), x \in Z$ be independent nonnegative integer-valued random variables with $\mu_j(x) = E(A_0(x)(A_0(x) - 1)\cdots(A_0(x) - j + 1))$. Assume that $\{\mu j(x)\}_x$ has limits for $j = 1,2$ and that it is bounded for $3 \leqq j \leqq 6$. Suppose at time zero there are $A_0(x)$ particles at $x \in Z$ and subsequently the particles move independently according to the transition function $P(x, y)$ of a recurrent random walk. For a finite nonempty subset $B$ of $Z$ denote by $A_n(B)$ the number of particles in $B$ at time $n$. Then $S_n(B) = \sum^n_{k=1} A_k(B)$ is the total occupation time of $B$ by time $n$ of all particles. Assuming that the $n$ step transition function $P_n(x, y)$ is such that there is an $\alpha$, with $1 < \alpha \leqq 2$, so that $P_n(0, x) \sim cn^{-1/\alpha}$ for all $x$, it is proved that the strong law of large numbers and the central limit theorem hold for the sequence $\{S_n(B)\}$.

We consider in detail the phenomenon of transition of distributions of sums of independent identically distributed non-negative integer-valued random variables related to the generalised allocation scheme from one lattice to another.

In this paper we study the limiting distribution of the maximum term of non-negative integer-valued moving average sequences of the form X n = Σ∞ i= - ∞β i o Z n - i where {Z n } is an iid sequence of non-negative integer-valued random variables with exponential type tails of the form 1 - F(n) ∼ Kn ζ (1 + λ) -n when n → ∞, ζ ∈ R, K, λ > 0, and o denotes binomial thinning. Several models are considered allowing different dependence structures of the thinning operations. For these models results are established which present similarities with those obtained for the classic linear Fmoving average: {X n } behaves as if it was iid regarding the limiting distribution of the maximum term. The paper concludes with some examples which apply the results to a particular model, the INAR(1), and with a simulation study to illustrate the results.

In this paper we study the limiting distribution of the maximum term of non-negative integer-valued moving average sequences of the form X n = ∑ = −∞ ∞ βi ○ Z n − i where { Z n} is an iid sequence of non-negative integer-valued random variables with regularly varying tails, β i○Z n−i denotes binomial thinning. Several models are considered allowing different dependence structures on the thinning operations. For these models we manage to establish results which are similar to the ones obtained for the classic linear moving average: { X n} > has an extremal index θ = \maxi{ β α } / (∑β α ), where α is the index of regular variation, irrespectively of the dependence structure considered. The paper is concluded with a simulation study to illustrate the results.

In this note we introduce the notion of factorial moment distance for nonnegative integer-valued random variables, and we compare it with the total variation distance. Furthermore, we study the rat...

Accurate and efiective system-level modeling has become necessary to address electromagnetic compatibility (EMC) issues in modern circuit and system design. Model order reduction (MOR) techniques provide a feasibility to approximate complex circuit models with compact reduced-order models. In this paper, an efiective MOR technique entitled multi-point moment matching (MMM) is implemented for the partial element equivalent circuit (PEEC) modeling. Moment information at multiple frequency points is used in this method in order to accurately estimate a given system over an entire frequency range of interest, and for each frequency an enhanced asymptotic waveform evaluation (AWE) is applied to obtain a reduced-order model by constructing a pole-residue representation of the original transfer function. The improvements of conventional AWE in aspects of both moment computation and moment matching can avoid ill-conditioned moment matrices and unstable dominant poles. The complex frequency hopping (CFH) technique is employed to select the multiple expansion points by using a newly developed upward- search algorithm. Numerical simulations of coupled microstrip lines in both frequency and time domain indicate the efiectiveness of the proposed method.