Let B(k,mu)=∑_i k brace imu^i denote the Bell polynomial, and write B_x=B(x,1). For a Poisson random variable with mean mu, its kth raw moment is B(k,mu). Bell-number moment bounds. For all k>0 and mu≥ 1, B_k/mu^1/(k/mu) ≤ fracB(k,mu)^1/kmu ≤ B_k/mu+1^1/(k/mu+1). Furthermore, for 0<mu≤ 1, fracB(k,mu)^1/kmu ≤ B_k/mu^1/(k/mu). These inequalities would give matching upper and lower bounds for normalized Poisson moments in terms of Bell numbers, complementing the preceding lower bound and the known asymptotic and uniform estimates for large k/mu.

Bell-number bounds for normalized raw moments of the Poisson distribution

From papers

Let $B(k,\mu)=\sum_i {k\brace i}\mu^i$ denote the Bell polynomial, and write $B_x=B(x,1)$ . For a Poisson random variable with mean $\mu$ , its $k$ th raw moment is $B(k,\mu)$ . Bell-number moment bounds. For all $k>0$ and $\mu\ge 1$ ,

B_{k/\mu}^{1/(k/\mu)} \le \frac{B(k,\mu)^{1/k}}{\mu} \le B_{k/\mu+1}^{1/(k/\mu+1)}.

Furthermore, for $0<\mu\le 1$ ,

\frac{B(k,\mu)^{1/k}}{\mu} \le B_{k/\mu}^{1/(k/\mu)}.

These inequalities would give matching upper and lower bounds for normalized Poisson moments in terms of Bell numbers, complementing the preceding lower bound and the known asymptotic and uniform estimates for large $k/\mu$ .

Progress summary

Open

The conjecture remains unproved: one integer-mean lower-bound case is known, but the proposed upper bounds and the full real-parameter statement have no verified resolution.

The inequalities were formulated as Conjecture 1 in a 2021 paper on sharp bounds for raw Poisson moments. The paper records numerical evidence for the upper bound but does not prove the conjecture.

Known results

The lower bound for integer $\mu \ge 1$ is proved; its extension to all real $\mu \ge 1$ is left conjectural.
The paper also gives separate asymptotic and moment-generating-function bounds, but neither establishes the conjectured upper bounds.
The $0<\mu\le 1$ assertion remains part of the conjecture.

Current status (as of August 2026): The integer- $\mu$ lower bound and separate auxiliary estimates are settled, while the full lower bound for real $\mu \ge 1$ , both proposed upper-bound regimes, and any claimed counterexample remain without independently verified public resolution.

Sources

arxiv.org

Sources & referencesView supporting material

Primary source

Thomas D. Ahle, “Sharp and Simple Bounds for the raw Moments of the Binomial and Poisson Distributions”, arXiv:2103.17027 (2021).

1 week ago

The conjectured upper bound is false, with infinitely many positive integer counterexamples.

Let

T_k(μ)=∑_{j=0}^k S(k,j)μ^j=E[Z_μ^k], Z_μ∼Poisson(μ),

and B_j=T_j(1). Take k=3200 and μ=32, so k/μ=100. The proposed upper bound becomes

T₃₂₀₀(32)^{1/3200}/32 ≤ B₁₀₁^{1/101}.

Because every quantity is positive, this is equivalent to the entirely integer inequality

T₃₂₀₀(32)¹⁰¹ ≤ 32³²³²⁰⁰ B₁₀₁³²⁰⁰.

Compute S(n,j) using the division-free recurrence

S(0,0)=1,

S(n,j)=jS(n−1,j)+S(n−1,j−1),

with zero values outside 0≤j≤n. It gives

B₁₀₁= 1409730628836818079651989640877750147939649550972810633687810677623397460767033916688901247498361259223729545318022772.

Exact integer arithmetic then gives

bitlength(T₃₂₀₀(32)¹⁰¹)=2,861,322,

bitlength(32³²³²⁰⁰ B₁₀₁³²⁰⁰)=2,861,316.

The left-hand integer therefore strictly exceeds the right-hand integer, reversing the conjectured inequality. A complete reproducible integer certificate is:

k, mu, r = 3200, 32, 100 s = [0] * (k + 1) s[0] = 1 for n in range(1, k + 1): for j in range(n, 0, -1): s[j] = j * s[j] + s[j - 1] s[0] = 0 if n == r + 1: bell = sum(s[:n + 1]) touchard = 0 for coefficient in reversed(s): touchard = mu * touchard + coefficient left = touchard ** (r + 1) right = mu ** (k * (r + 1)) * bell ** k assert left.bit_length() == 2861322 assert right.bit_length() == 2861316 assert left > right

Moreover, this yields an infinite family without numerical extrapolation. For any integer t≥1, take independent Z₁,…,Z_t∼Poisson(32) and let W=Z₁+⋯+Z_t∼Poisson(32t). The arithmetic–geometric mean inequality gives pointwise

[W/(32t)]^{3200t} ≥ ∏_{j=1}^t [Z_j/32]^{3200}.

Taking expectations and using independence yields

∥W/(32t)∥{3200t} ≥∥Z₁/32∥{3200}

B₁₀₁^{1/101}.

Since (3200t)/(32t)+1=101, every pair

(k,μ)=(3200t,32t), t=1,2,3,…,

violates the upper bound. The separate proposed lower bound and the μ≤1 assertion are not addressed by this counterexample.

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Shivam Patel · 21 hours ago