LemmaStatement: AI-generatedProof: AI-generated

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Finite sieve determines the probability of a zero count

Statement

Let $X_{n}$ be nonnegative integer-valued random variables on finite probability spaces, and define $(\binom{x}{r}) = (x)_{r} / r!$ for integers $0 \leq r \leq x$ , and zero for $r > x$ . Powers $λ^{r}$ use natural integer exponents, with $λ^{0} = 1$ also when $λ = 0$ . If, for each fixed $r \geq 0$ , $E (X_{n})_{r} \to λ^{r}$ for a fixed $λ \geq 0$ , then $P (X_{n} = 0) \to e^{- λ}$ . No assertion about other values of $X_{n}$ is needed.

Facts & Assumptions

Given: The objects and hypotheses of the Statement.

Goal: Prove the Statement.

[F1]

Expectation is a finite weighted sum, preserving pointwise inequalities (Expectation of a real random variable on a finite probability space).

[F2]

Factorials and falling factorials have their finite product definitions (The factorial n! and the falling factorial n^k̲, defined by recursion in ℕ).

[F3]

The exponential is the everywhere convergent series $\exp (t) = \sum_{r \geq 0} t^{r} / r!$ (The real exponential function and the number e by a power series).

[F4]

Convergence is interpreted along the sequence of finite probability spaces (Asymptotic quantifiers and uniform parameter ranges).

[F5]

A finite probability space $(Ω_{n}, w_{n})$ has nonnegative weights summing to one; every subset is an event whose probability is the sum of its weights. A real random variable is a function $X_{n} : Ω_{n} \to R$ (Finite probability spaces, outcome weights, events, and event probabilities, Real random variables on finite probability spaces and their finite distributions).

[F6]

For $a > 0$ and $t \in R$ , real powers satisfy $a^{t} = \exp (t \log a)$ , and the inverse logarithm satisfies $\log (\exp (t)) = t$ (Real powers for positive bases, with the zero-base positive-exponent convention, The natural logarithm as the inverse of the exponential function).

[F7]

Natural integer powers satisfy $a^{0} = 1$ and $a^{r + 1} = a^{r} a$ for every real $a$ and integer $r \geq 0$ , including $a = 0$ (Integer powers aᵐ).

Proof

technique · direct

1.1F2F7algebra

For integers $x \geq 1$ and $m \geq 0$ , Pascal's identity, obtained by separating subsets according to whether they contain a specified element, telescopes to $\sum_{r = 0}^{m} (- 1)^{r} (\binom{x}{r}) = (- 1)^{m} (\binom{x - 1}{m})$ ; the binomial coefficient is zero beyond its range. The factorial formula agrees with subset counting because each $r$ -subset admits $r!$ orders. For $x = 0$ , the sum is one. Write $1_{{x = 0}}$ for the function equal to one when $x = 0$ and zero otherwise. Thus, for every $M \geq 0$ , $\sum_{r = 0}^{2 M + 1} (- 1)^{r} (\binom{x}{r}) \leq 1_{{x = 0}} \leq \sum_{r = 0}^{2 M} (- 1)^{r} (\binom{x}{r})$ .

2.1step 1.1F1F2F5F7algebra

On $(Ω_{n}, w_{n})$ , the function $ω \mapsto 1_{{X_{n} (ω) = 0}}$ has expectation $\sum_{ω : X_{n} (ω) = 0} w_{n} (ω) = P (X_{n} = 0)$ . Put $q_{n} = P (X_{n} = 0)$ and $T_{n, k} = \sum_{r = 0}^{k} (- 1)^{r} E (X_{n})_{r} / r!$ . Taking expectations in step 1.1 and distributing the finite weighted sum over the finite sieve sums gives $T_{n, 2 M + 1} \leq q_{n} \leq T_{n, 2 M}$ for every $M \geq 0$ .

3.1step 2.1F2F3F4F7givenchoosealgebra

Put $s_{k} = \sum_{r = 0}^{k} (- 1)^{r} λ^{r} / r!$ and $L = \exp (- λ)$ . The natural-power recursion gives $(- λ)^{r} = (- 1)^{r} λ^{r}$ by induction on $r$ , starting at $r = 0$ with both sides equal to one. Thus $s_{k} \to L$ by the exponential series, including when $λ = 0$ . Fix $ε > 0$ and choose $M$ so that both $∣ s_{2 M + 1} - L ∣ < ε / 3$ and $∣ s_{2 M} - L ∣ < ε / 3$ . For each of the finitely many $r \in {0, \dots, 2 M + 1}$ , the given moment convergence supplies an $N_{r}$ such that $∣ E (X_{n})_{r} - λ^{r} ∣ < ε r! / [3 (2 M + 2)]$ for $n \geq N_{r}$ . For $n \geq N = \max_{0 \leq r \leq 2 M + 1} N_{r}$ , the triangle inequality gives $∣ T_{n, k} - s_{k} ∣ \leq \sum_{r = 0}^{k} ∣ E (X_{n})_{r} - λ^{r} ∣ / r! < ε / 3$ for both $k = 2 M$ and $k = 2 M + 1$ . The bounds from step 2.1 now imply $L - 2 ε / 3 < q_{n} < L + 2 ε / 3$ , so $∣ q_{n} - L ∣ < ε$ . This proves $q_{n} \to \exp (- λ)$ directly from the convergence definition.

4.1step 3.1F3F6algebra∎

The series at one gives $e = \exp (1) > 0$ , and the inverse logarithm gives $\log e = 1$ ; hence $e^{- λ} = \exp (- λ \log e) = \exp (- λ)$ by the real-power definition. Substituting this identity into step 3.1 yields the stated zero-count limit.

Depends on

Used by · 1

Connectivity in the logarithmic random-graph window

Dependency tree · 2 levels

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