Hipp-Type Signed Compound Poisson Measure

• Hipp-type signed compound Poisson measure is a specialized analytical construct that refines classical approximations by matching both the mean and covariance of distributions.
• It introduces higher-order signed correction terms, yielding an extra power improvement in total variation error bounds compared to the standard compound Poisson approach.
• The method supports advanced applications in multivariate Bernoulli–Poisson and Poisson point process approximations, providing explicit error estimates in lattice and structured settings.

A Hipp-type signed compound Poisson measure is a refined analytical construction that improves the approximation of convolutions of probability measures—most notably in multivariate and compound Poisson settings—by introducing higher-order signed correction terms beyond the classic (first-order) compound Poisson approximation. The key innovation is that these signed measures can match not only the first but the first two cumulants of the original distribution, yielding strictly smaller errors—often an entire extra power in $n^{-1}$—in total variation distance. The construction generalizes naturally to multidimensional and structured settings including symmetric distributions on lattices, with precise control of the associated approximation error.

1. Definition and Construction

For a symmetric probability measure $F \in \mathcal{P}(\mathbb{Z}^d)$ on the integer lattice $\mathbb{Z}^d$, let $q = F\{\mathbf{0}\} \in (0,1)$ denote the mass at the origin, and $M = F - I$ where $I=I_{\mathbf{0}}$ is the Dirac measure at $\mathbf{0}$. The Hipp-type signed compound Poisson measure $D \in \mathcal{M}(\mathbb{Z}^d)$ is defined as

$$D = \exp\Bigl\{ M - \tfrac{1}{2} M^{*2} \Bigr\} = \sum_{j=0}^\infty \frac{1}{j!} \Bigl( M - \tfrac{1}{2} M^{*2} \Bigr)^{*j}$$

where $*$ denotes convolution of (signed) measures. For any real $F \in \mathcal{P}(\mathbb{Z}^d)$0, the generalized measure $F \in \mathcal{P}(\mathbb{Z}^d)$1 is well-defined, since $F \in \mathcal{P}(\mathbb{Z}^d)$2 has total mass zero.

In characteristic function terms,

$F \in \mathcal{P}(\mathbb{Z}^d)$3

so $F \in \mathcal{P}(\mathbb{Z}^d)$4 is an infinitely divisible signed law whose Lévy exponent matches the first two cumulants of $F \in \mathcal{P}(\mathbb{Z}^d)$5 exactly (Čekanavičius et al., 23 Jan 2026).

A more general construction arises when approximating the convolution $F \in \mathcal{P}(\mathbb{Z}^d)$6 of measures $F \in \mathcal{P}(\mathbb{Z}^d)$7 with small-parameterized remainder $F \in \mathcal{P}(\mathbb{Z}^d)$8. Introducing error operators $F \in \mathcal{P}(\mathbb{Z}^d)$9, the Hipp-type approximation sums signed correction terms up to order $\mathbb{Z}^d$0, yielding

$\mathbb{Z}^d$1

with $\mathbb{Z}^d$2, $\mathbb{Z}^d$3 (Roos, 2015).

2. Second-Order Expansion and Cumulant Matching

The central motivation is the enhanced fidelity in cumulant matching. For sums $\mathbb{Z}^d$4 of i.i.d. random vectors with law $\mathbb{Z}^d$5, the classic (first-order) compound Poisson $\mathbb{Z}^d$6 matches only the mean. By contrast, the Hipp-type measure

$\mathbb{Z}^d$7

matches both the mean (first cumulant) and the covariance (second cumulant). A standard Bergström expansion for the characteristic function shows

$\mathbb{Z}^d$8

leading, at the measure level, to

$\mathbb{Z}^d$9

where the remainder $q = F\{\mathbf{0}\} \in (0,1)$0 (Čekanavičius et al., 23 Jan 2026).

3. Total Variation Error Bounds

The Hipp-type signed compound Poisson approximation offers quantifiably sharper error bounds over the plain (unsigned) compound Poisson law. For symmetric lattice distributions,

$q = F\{\mathbf{0}\} \in (0,1)$1

with explicit constant factors tied to smoothness parameters such as

$q = F\{\mathbf{0}\} \in (0,1)$2

For finite supports of size $q = F\{\mathbf{0}\} \in (0,1)$3, the total variation simplifies to

$q = F\{\mathbf{0}\} \in (0,1)$4

Compared to the $q = F\{\mathbf{0}\} \in (0,1)$5 error for first-order (plain) compound Poisson approximation, this is a substantial improvement (Čekanavičius et al., 23 Jan 2026).

Analogous bounds hold in the context of independent small-jump factors, parameterized via smoothness parameters $q = F\{\mathbf{0}\} \in (0,1)$6, $q = F\{\mathbf{0}\} \in (0,1)$7: $q = F\{\mathbf{0}\} \in (0,1)$8 These bounds are optimal in their scaling with respect to $q = F\{\mathbf{0}\} \in (0,1)$9 and problem dimension (Roos, 2015).

4. Series Expansion and Algebraic Structure

The signed measure is expressible via a generalized exponential series. For the case of $M = F - I$0, the convolution can be expanded as

$M = F - I$1

where $M = F - I$2 (Roos, 2015).

Error operators $M = F - I$3 are given by $M = F - I$4, reflecting the key role of second-order corrections. The algebra of these expansions leverages the Banach structure of finite signed measures under convolution and the orthogonality of Charlier polynomials in bounding distinguishability in total variation.

5. Multidimensional Validity and Strengths

The measure and its associated error bounds are valid without restriction in $M = F - I$5, requiring only symmetry and lattice support. There is no essential change in the two-term expansion or the $M = F - I$6 convergence rate; only constant factors, through parameters such as $M = F - I$7 or support size $M = F - I$8, reflect the increased dimension. Correlations between coordinates or non-product support structures are accommodated natively (Čekanavičius et al., 23 Jan 2026).

Improved total variation bounds for the Hipp-type measure eliminate earlier dimension- and logarithmic-factors present in classical results (Kerstan 1964; Mattner 1970; Barbour–Utev 2000). Parameters $M = F - I$9, $I=I_{\mathbf{0}}$0 reflect local jump intensities, and the refined inequalities exploit orthogonality and Cauchy–Schwarz-type estimates within the convolution algebra (Roos, 2015).

6. Applications and Numerical Illustrations

Applications include multivariate Bernoulli–Poisson approximation, compound Poisson process approximation, and multivariate (lattice) distributions. For Poisson point processes and finite Bernoulli processes, the Hipp-type bounds provide improved scaling and explicit dependence on intensity and density parameters. For example, in a case with $I=I_{\mathbf{0}}$1 and appropriate $I=I_{\mathbf{0}}$2, empirically $I=I_{\mathbf{0}}$3, confirming the rapid decay of error with order (Roos, 2015).

When $I=I_{\mathbf{0}}$4 has strictly finite support, the explicit constants in the total variation error, along with their polynomial dependence on $I=I_{\mathbf{0}}$5 and the parameters $I=I_{\mathbf{0}}$6, quantify the advantage over classical approximations.

7. Comparative Context and Significance

The Hipp-type signed compound Poisson measure represents a principled, analytically tractable advance in stochastic approximation theory. It systematically improves upon the classical compound Poisson approach for sums of independent (possibly non-identical) variables—especially in high-dimensional or sparse-jump settings—by incorporating signed correction terms that retain multidimensional validity and optimal error scaling. The use of cumulant-matching through exponent corrections and modern smoothness inequalities distinguishes it both in accuracy and in generality within the domain of Poisson and compound Poisson approximations (Roos, 2015, Čekanavičius et al., 23 Jan 2026).