In this paper, we study a random field constructed from the twodimensional Gaussian free field (GFF) by modifying the variance along the scales in the neighborhood of each point. The construction can be seen as a local martingale transform and is akin to the time-inhomogeneous branching random walk. In the case where the variance takes finitely many values, we compute the first order of the maximum and the log-number of high points. These quantities were obtained by Bolthausen et al. (2001) and Daviaud (2006) when the variance is constant on all scales. The proof relies on a truncated second moment method proposed by Kistler (2015), which streamlines the proof of the previous results. We also discuss possible extensions of the construction to the continuous GFF.

x aγi i is completely monotonic on (0, ∞). This result generalizes the one found by Alzer (2018) for binomial probabilities (d = 1). As a consequence of the log-convexity, we obtain some combinatorial inequalities for multinomial coefficients. We also show how the main result can be used to derive asymptotic formulas for quantities of interest in the context of statistical density estimation based on Bernstein polynomials on the d-dimensional simplex.

In this paper, we present a combinatorial proof of the Gaussian product inequality (GPI) conjecture in all dimensions when the components of the centered Gaussian vector X = (X 1 , X 2 , . . . , X d ) can be written as linear combinations, with nonnegative coefficients, of the components of a standard Gaussian vector. The proof comes down to the monotonicity of a certain ratio of gamma functions. We also show that our condition is weaker than assuming the vector of absolute values |X| := (|X 1 |, |X 2 |, . . . , |X d |) to be in the multivariate totally positive of order 2 (MTP 2 ) class on [0, ∞) d , for which the conjecture is already known to be true.

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In Siotani & Fujikoshi (1984), a precise local limit theorem for the multinomial distribution is derived by inverting the Fourier transform, where the error terms are explicit up to order N −1 . In this paper, we give an alternative (conceptually simpler) proof based on Stirling's formula and a careful handling of Taylor expansions, and we show how the result can be used to approximate multinomial probabilities on most subsets of R d . Furthermore, we discuss a recent application of the result to obtain asymptotic properties of Bernstein estimators on the simplex, we improve the main result in Carter ( 2002) on the Le Cam distance bound between multinomial and multivariate normal experiments while simultaneously simplifying the proof, and we mention another potential application related to finely tuned continuity corrections.