The method of cumulants for the normal approximation

Probabilistic limit theorems induced by the zeros of polynomials

“…We present the result in a more general form, although we had 𝛾 = 0 in all our applications. Statement (i) of the above lemma corresponds to Theorem 2.4 in [15] and to Corollary 2.1 in [36]. The latter gives the precise value…”

“…In order to keep this paper self-contained, we briefly summarize the probabilistic consequences which can be drawn from the Statulevičius condition. This approach is known as the method of cumulants and we refer to the monograph [36] as well as the recent survey [15] for a detailed account of this method. Recall that for a sequence of random variables (𝑋 𝑁 ) 𝑁≥1 , we denote the cumulant of order 𝑚 of 𝑋 𝑁 by 𝜅 𝑚,𝑁 ∶= 𝜅 𝑚 (𝑋 𝑁 ) and the respective cumulant of the normalization 𝑋 * 𝑁 ∶= (𝑋 𝑁 − 𝔼𝑋 𝑁 )∕ √ var(𝑋 𝑁 ) by 𝜅 * 𝑚,𝑁 ∶= 𝜅 𝑚 (𝑋 * 𝑁 ).…”

Normal approximation of subgraph counts in the random-connection model

“…In addition, by Theorem 1.1 of [DE13], N G satisfies a moderate deviation principle with speed a 2 λ = o(λ 1/(2r−3) ) and rate function x 2 /2, see Lemma A.1-iii) in appendix. The cumulant bounds (6.7), (6.9), (6.11) show that the centered and normalized subgraph count N G satisfies the Statulevičius condition (A.1) below, see [RSS78,DJS22], with γ := r − 2.…”

“…A number of probabilistic conclusions can be derived from the behavior of cumulants of random variables using the Statulevičius condition, including convergence rates in the Kolmogorov distance and moderate deviation principles, see [SS91], [DE13], [DJS22]. In [GT18a,GT18b], this method has been used to derive concentration inequalities, normal approximation with error bounds, and moderate deviation principles for random polytopes.…”

Beyond the delta method

“…This might be combined easily with our approach to obtain higher moments. We expect to compute cumulants [18] as well in a wide variety of cases, and to identify some corrections to enforce the quality of the estimation [42].…”