Zero-Truncated Poisson Distributions

• Zero-truncated Poisson distributions are defined by conditioning on strictly positive counts, yielding exact closed-form generating functions and moment formulas.
• Degenerate and r-truncated extensions use degenerate exponential functions and special polynomials to generalize classical results and improve error bounds in Poisson approximations.
• These models are crucial for applications in reliability theory and count data analysis where zero events are unobservable or structurally impossible.

The zero-truncated Poisson distribution modifies the classical Poisson law by restricting its support to strictly positive integers, thereby conditioning on the event of a nonzero count. This construction is ubiquitous in domains where zero events are structurally impossible or unobservable, and underlies positive Poisson, conditional Poisson, and various degenerate extensions. Recent developments in truncation and degeneracy—especially those using the "degenerate exponential," falling factorials, and Lah–Bell polynomial machinery—have systematically generalized classical results, producing closed-form probabilistic, combinatorial, and approximation-theoretic properties that admit exact computation and facilitate superior error bounds in Poisson approximation.

1. Classical Zero-Truncated Poisson Distribution

Given $X \sim \mathrm{Poisson}(\alpha)$ with pmf $P\{X = k\} = \frac{\alpha^k}{k!} e^{-\alpha}$ for $k = 0, 1, 2, \dots$, the zero-truncated Poisson is denoted by $X \mid X > 0$ and supported on $\{1, 2, \dots\}$:

$$P_{\mathrm{ZT}}(k) = \frac{\alpha^k e^{-\alpha} / k!}{1 - e^{-\alpha}}, \quad k = 1, 2, \dots$$

Its probability generating function (PGF) and moment generating function (MGF) are:

$$G_X(t) = \frac{e^{\alpha t} - 1}{e^{\alpha} - 1}, \qquad M_X(t) = \frac{e^{\alpha e^{t}} - 1}{e^{\alpha} - 1}$$

The moments are given by:

$$\mathbb{E}[X] = \frac{\alpha}{1 - e^{-\alpha}}, \qquad \operatorname{Var}[X] = \frac{\alpha(1 - e^{-\alpha} + \alpha e^{-\alpha})}{(1 - e^{-\alpha})^2}$$

These identities serve as a base case for subsequent degenerate generalizations and truncations.

2. Degenerate and Zero-Truncated Degenerate Poisson Laws

The degenerate Poisson distribution employs the degenerate exponential $e_\lambda(\alpha) = (1+\lambda \alpha)^{1/\lambda}$ ($\lambda \neq 0$) and the degenerate falling factorial $$(x)_{n,\lambda} = x(x-\lambda)\dots(x-(n-1)\lambda)$$ ($n \geq 1$). The pmf for the degenerate Poisson random variable $Y$ and its zero-truncated counterpart $X$ is:

$$P\{Y = n\} = \frac{\alpha^n}{n!} \frac{(1)_{n,\lambda}}{e_\lambda(\alpha)}, \quad n=0,1,\dots$$

$$P\{X = n\} = \frac{\alpha^n (1)_{n,\lambda}/n!}{e_\lambda(\alpha) - 1}, \quad n=1,2,\dots$$

The core generating functions generalize the classical cases:

$$G_X(t) = \frac{e_\lambda(\alpha t) - 1}{e_\lambda(\alpha) - 1}$$

$$M_X(t) = \frac{e_\lambda(\alpha e^{t}) - 1}{e_\lambda(\alpha) - 1}$$

Moments are computed using derivative identities and degenerate Bell polynomials $B_{n,\lambda}(\alpha)$:

$$\begin{aligned} \mathbb{E}[X] &= \frac{\alpha e_\lambda(\alpha)}{(1+\lambda\alpha)(e_\lambda(\alpha)-1)} \ \mathbb{E}[X^2] &= \frac{e_\lambda(\alpha)}{(1+\lambda\alpha)(e_\lambda(\alpha)-1)} + \frac{\alpha(1-\lambda) e_\lambda(\alpha)}{(1+\lambda\alpha)^2 (e_\lambda(\alpha)-1)} \ \operatorname{Var}[X] &= \mathbb{E}[X^2] - (\mathbb{E}[X])^2 \end{aligned}$$

Higher moments are given by:

$$\mathbb{E}[X^m] = \frac{B_{m,\lambda}(\alpha)}{e_\lambda(\alpha) - 1}$$

As $\lambda \to 0$, the degenerate expressions reduce to the classical zero-truncated Poisson distribution.

3. r-Truncated Degenerate Poisson Distributions

The r-truncated degenerate Poisson random variable $X_{r,a}$ is supported on $\{r+1, r+2, \dots\}$ with pmf:

$$P_{r,a}(k) = \frac{a^k}{k!} (1)_{k,\lambda} \left(e_\lambda(a) - e_{\lambda,r}(a)\right)^{-1}, \quad k \geq r+1$$

where $$e_{\lambda,r}(t) = \sum_{k=0}^r (1)_{k,\lambda} t^k / k!$$

Its generating functions are:

$$G_{r,a}(t) = \frac{e_\lambda(a t) - e_{\lambda,r}(a t)}{e_\lambda(a) - e_{\lambda,r}(a)}$$

$$M_{r,a}(t) = \frac{e_\lambda(a e^t) - e_{\lambda,r}(a e^t)}{e_\lambda(a) - e_{\lambda,r}(a)}$$

Mean and variance admit closed forms:

$$\mathbb{E}[X_{r,a}] = \frac{a \, e_\lambda(a)}{(1 + \lambda a)(e_\lambda(a) - e_{\lambda,r}(a))} - \frac{\sum_{j=0}^r j \frac{a^j}{j!}(1)_{j,\lambda}}{e_\lambda(a) - e_{\lambda,r}(a)}$$

$$\operatorname{Var}[X_{r,a}] = G_{r,a}''(1) + G_{r,a}'(1) - (G_{r,a}'(1))^2$$

The recursion for the ratio of consecutive probabilities is:

$$\frac{P_{r,a}(k+1)}{P_{r,a}(k)} = \frac{a (1 - \lambda k)}{k+1}, \quad k \geq r+1$$

Setting $r = 0$ recovers the zero-truncated degenerate Poisson law.

4. Stein's Method and Truncation for Finite Support

In contexts where random variables $W$ are naturally finite supported ($\mathrm{supp}(W) \subseteq \{0,1,\dots,n\}$), it is possible to truncate the classical Poisson distribution at the endpoint $n$ for approximation. The upper-truncated Poisson is defined:

$$\mathrm{Pois}_{[0,n]}(\lambda) = \mathcal{L}\left(Z|\ 0 \leq Z \leq n\right),\quad Z \sim \mathrm{Poisson}(\lambda)$$

$$P_{[0,n]}(k) = \frac{\lambda^k / k!}{\sum_{i=0}^n \lambda^i / i!}, \quad 0 \leq k \leq n$$

$$S_n(\lambda) = \sum_{i=0}^n \frac{\lambda^i}{i!} e^{-\lambda} = P\{Z \leq n\}$$

Moments of the truncated distribution are:

$$\mathbb{E}[X] = \frac{\lambda S_{n-1}(\lambda)}{S_n(\lambda)}$$

$$\operatorname{Var}[X] = \frac{\lambda S_{n-1}(\lambda) + \lambda^2 S_{n-2}(\lambda)}{S_n(\lambda)} - (\mathbb{E}[X])^2$$

Critical advances are achieved in Poisson approximation error bounds via Stein's method. If $W$ is supported on $\{0,1,\dots,n\}$, the Stein factor for truncated Poisson is:

$$G_{[0,n]}(\lambda) := \sup_{f:\{0,\dots,n\}\to\{0,1\}} \sup_{0\leq i\leq n} |g_f(i+1) - g_f(i)| = \frac{1 - e^{-\lambda}}{\lambda S_n(\lambda)}$$

Approximation error in total variation distance is strictly smaller when using $\mathrm{Pois}_{[0,n]}(\lambda)$:

$$d_{TV}\left(\mathcal{L}(W), \mathrm{Pois}_{[0,n]}(\lambda)\right) \leq C G_{[0,n]}(\lambda) < C G_{[0,\infty]}(\lambda)$$

where the classical factor $$G_{[0,\infty]}(\lambda) = (1 - e^{-\lambda}) / \lambda$$.

This improvement is pronounced when $\lambda$ and $n$ are comparable; for $n \gg \lambda$, the gain is minimal.

5. Moment Structures via Special Polynomials

The full moment structure of the zero-truncated degenerate Poisson and its r-truncated generalization is encoded via degenerate Bell, Stirling, and Lah–Bell polynomials. For the $\mathrm{Pois}^*_{\lambda,\alpha}$ law (Kim et al., 2021):

• Raw moments: $\mathbb{E}[X^n]$ expressed by degenerate Bell $\Bel_{k,\alpha}(\lambda)$ and Stirling numbers $S_{1,\alpha}(n,k)$
• Rising factorial moments: $E[(X)_n] = B_{0,n,\alpha}(\lambda)$ (degenerate Lah–Bell polynomials)

Generating functions are:

$$G_X(t) = \frac{e_\alpha(\lambda t) - 1}{e_\alpha(\lambda) - 1}$$

$$M_X(t) = \frac{e_\alpha(\lambda e^{t}) - 1}{e_\alpha(\lambda) - 1}$$

For sums of independent zero-truncated degenerate Poisson variables with equal degeneracy and rate:

$$P\{S=n\} = \frac{1}{(e_\alpha(\lambda) - 1)^m} \sum_{j=0}^m (-1)^{m-j} \binom{m}{j} \frac{(1)_{n,\alpha}}{n!} (j\lambda)^n$$

A plausible implication is that the polynomial structure facilitates computational evaluation, symbolic summation, and combinatorial interpretation of the associated distributions.

6. Interpretative Contexts and Limiting Behavior

The degenerate zero-truncated Poisson law interpolates between the classical distribution ($\lambda \to 0$ or $\alpha \to 0$ limits) and a richer set of probability, combinatorial, and analytical structures:

• Combinatorial: Degenerate factorials and Bell/Lah polynomials count weighted partitions and decorated permutations under deformation.
• Umbral calculus: $e_\alpha(t)$ is a delta-operator generating function with basis $(x)_{n,\alpha}$.
• Special functions & ODEs: $e_\alpha(x)$ satisfies $D_x e_\alpha(x) = e_\alpha(x)$ with step-wise product rule alteration.
• Probabilistic conditioning: The construction $X \sim Y \mid (Y > 0)$ ensures formulaic derivation of moments and generating functions.

For $\lambda < 0$ sufficiently small (that $1+\lambda\alpha>0$), analytic continuation holds, broadening distributional applicability (Kim et al., 2019).

7. Applications and Error Bound Improvements

Typical applications include modeling counts where zero is either impossible or unobserved (positive Poisson). An illustrative example is the fault process in reliability theory, where truncating the Poisson approximator to a finite endpoint $n$ (maximum attainable events, e.g., in $N$ days given repair time $R$) significantly tightens the approximation error bound for the observed process (Gan, 2017). When the expected count $\lambda$ is of the same magnitude as the support endpoint $n$, improvements are marked; however, for $n \gg \lambda$, the gains diminish (truncation becomes almost negligible).

In all constructs, limiting regimes reliably recover classical zero-truncated Poisson results. The extension to degenerate and r-truncated cases provides additional flexibility for analytic, combinatorial, and probabilistic modeling.