The call function plays a crucial role in pricing the collateralized dept obligation (CDO) and we will generalize the refined Lindeberg principle developed in Chen, Shao, and Xu (2023) to study the normal approximatio...
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The call function plays a crucial role in pricing the collateralized dept obligation (CDO) and we will generalize the refined Lindeberg principle developed in Chen, Shao, and Xu (2023) to study the normal approximation for the call function. In this article, we will give the uniform and non uniform bounds on normal approximation for the call function, under the assumptions that the third and (3+delta)-th moments of random variables exist, respectively, here delta is an element of(0,1].
This paper derives normal approximation results for subgraph counts written as multiparameter stochastic integrals in a random-connection model based on a Poisson point process. By combinatorial arguments we express t...
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This paper derives normal approximation results for subgraph counts written as multiparameter stochastic integrals in a random-connection model based on a Poisson point process. By combinatorial arguments we express the cumulants of general subgraph counts using sums over connected partition diagrams, after cancellation of terms obtained by Mobius inversion. Using the Statulevicius condition, we deduce convergence rates in the Kolmogorov distance by studying the growth of subgraph count cumulants as the intensity of the underlying Poisson point process tends to infinity. Our analysis covers general subgraphs in the dilute and full random graph regimes, and tree-like subgraphs in the sparse random graph regime.
We prove normal approximation bounds for statistics of randomly weighted (simplicial) complexes. In particular, we consider the complete d-dimensional complex on n vertices with d-simplices equipped with i.i.d. weight...
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We prove normal approximation bounds for statistics of randomly weighted (simplicial) complexes. In particular, we consider the complete d-dimensional complex on n vertices with d-simplices equipped with i.i.d. weights. Our normal approximation bounds are quantified in terms of stabilization of difference operators, i.e., the effect on the statistic under addition/deletion of simplices. Our proof is based on Chatterjee's normal approximation bound and is a higher-dimensional analogue of the work of Cao on sparse Erdos-Renyi random graphs but our bounds are more in the spirit of 'quantitative two-scale stabilization' bounds by Lachieze-Rey, Peccati, and Yang. As applications, we prove a CLT for nearest face-weights in randomly weighted dcomplexes and give a normal approximation bound for local statistics of random d-complexes.
We consider the normal approximation of Kabanov-Skorohod integrals on a general Poisson space. Our bounds are for the Wasserstein and the Kolmogorov distance and involve only difference operators of the integrand of t...
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We consider the normal approximation of Kabanov-Skorohod integrals on a general Poisson space. Our bounds are for the Wasserstein and the Kolmogorov distance and involve only difference operators of the integrand of the Kabanov-Skorohod integral. The proofs rely on the Malliavin-Stein method and, in particular, on multiple applications of integration by parts formulae. As examples, we study some linear statistics of point processes that can be constructed by Poisson embeddings and functionals related to Pareto optimal points of a Poisson process.
We derive quantitative bounds in the Wasserstein distance for the approximation of stochastic integrals with respect to Hawkes processes by a normally distributed random variable. In the case of deterministic and nonn...
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We derive quantitative bounds in the Wasserstein distance for the approximation of stochastic integrals with respect to Hawkes processes by a normally distributed random variable. In the case of deterministic and nonnegative integrands, our estimates involve only the third moment of the integrand in addition to a variance term using a squared norm of the integrand. As a consequence, we are able to observe a "third moment phenomenon" in which the vanishing of the first cumulant can lead to faster convergence rates. Our results are also applied to compound Hawkes processes, and improve on the current literature where estimates may not converge to zero in large time or have been obtained only for specific kernels such as the exponential or Erlang kernels.
The generalized perturbative approach is an all-purpose variant of Stein???s method used to obtain rates of normal approximation. Originally developed for functions of independent random variables, this method is here...
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The generalized perturbative approach is an all-purpose variant of Stein???s method used to obtain rates of normal approximation. Originally developed for functions of independent random variables, this method is here extended to functions of the realization of a hidden Markov model. In this dependent setting, rates of convergence are provided in some applications, leading, in each instance, to an extra log-factor vis-??-vis the rate in the independent case.
We derive normal approximation bounds in the Wasserstein distance for sums of generalized U-statistics, based on a general distance bound for functionals of independent random variables of arbitrary distributions. Tho...
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We derive normal approximation bounds in the Wasserstein distance for sums of generalized U-statistics, based on a general distance bound for functionals of independent random variables of arbitrary distributions. Those bounds are applied to normal approximation for the combined weights of subgraphs in the Erdos-Renyi random graph, extending the graph counting results of Barbour et al. (A central limit theorem for decomposable random variables with applications to random graphs, J. Combin. Theory Ser. B 47(2) (1989), pp. 125-145) to the setting of weighted graphs. Our approach relies on a general stochastic analytic framework for functionals of independent random sequences.
We derive normal approximation results for a class of stabilizing functionals of binomial or Poisson point process, that are not necessarily expressible as sums of certain score functions. Our approach is based on a f...
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We derive normal approximation results for a class of stabilizing functionals of binomial or Poisson point process, that are not necessarily expressible as sums of certain score functions. Our approach is based on a flexible notion of the add-one cost operator, which helps one to deal with the second-order cost operator via suitably appropriate first-order operators. We combine this flexible notion with the theory of strong stabilization to establish our results. We illustrate the applicability of our results by establishing normal approximation results for certain geometric and topological statistics arising frequently in practice. Several existing results also emerge as special cases of our approach.
In this paper, we use Stein's method to obtain optimal bounds, both in Kolmogorov and in Wasserstein distance, in the normal approximation for the empirical distribution of the ground state of a many-interacting-w...
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In this paper, we use Stein's method to obtain optimal bounds, both in Kolmogorov and in Wasserstein distance, in the normal approximation for the empirical distribution of the ground state of a many-interacting-worlds harmonic oscillator proposed by Hall, Deckert and Wiseman (Phys. Rev. X 4 (2014) 041013). Our bounds on the Wasserstein distance solve a conjecture of McKeague and Levin (Ann. Appl. Probab. 26 (2016) 2540-2555).
In this paper, we estimate the difference |Eh(Z(n)) - Eh(Y)| between the expectations of real finite Lipschitz function h of the sum Z(n) = (X-1 + & ctdot;+ X-n)/B-n, where B-n(2) = E(X-1 + & ctdot;+ X-n)(2) &...
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In this paper, we estimate the difference |Eh(Z(n)) - Eh(Y)| between the expectations of real finite Lipschitz function h of the sum Z(n) = (X-1 + & ctdot;+ X-n)/B-n, where B-n(2) = E(X-1 + & ctdot;+ X-n)(2) > 0, and a standard normal random variable Y, where real centered random variables X-1,X-2,& mldr;satisfy the phi-mixing condition, defined between the "past" and " future", or are m-dependent. In particular cases, under the condition & sum;(infinity)(r=1)r phi(r)
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