The Spectral Counting Polynomial and Fiber ρ-Statistics

Qin, Han

doi:10.5281/zenodo.18973559

English

中文

Abstract

Theorem numbering: Paper 9 contains Theorems 1–5. Paper 10 begins with Theorem 6.

Paper 9 established the fiber count h*(n) = |Hist*(n)|. But fibers are not homogeneous: different compact terms within the same fiber carry different ρ-values. This paper introduces the spectral counting polynomial Z_n(q) = Σ_{h∈Hist*(n)} q^{ρ(h)} as a single central object from which fiber size (q=1), multiplicity function (coefficients), moments (derivatives at q=1), extrema (support boundaries), and Boltzmann measure (q = e^{-β}) are all uniformly derived.

Core results: Z_n(q) satisfies a three-branch recurrence (Theorem 6), reducing to Paper 9 Theorem 1 at q=1. The first raw moment M₁(n) = Z_n'(1) and the second factorial moment F₂(n) = Z_n''(1) each satisfy computable recurrences (Theorems 7, 8), extending to n=200 without enumerating the fiber.

Key numerical findings: E_n[ρ]/n increases monotonically toward c₁ ≈ 1.719; Var_n[ρ]/n converges rapidly to c₂ ≈ 0.1924. The normalized distribution is approximately Gaussian. Crucially, ρ_E(n) lies 5–14 standard deviations below the mean, and |z_n| increases steadily — ρ_E asymptotics is an extreme-value statistics problem, not a concentration inequality problem. This is the key input for Paper 11.

1. Introduction

1.1 From Counting to Spectra

Paper 9 gave us the total count. Now we ask: what is the distribution? The spectral counting polynomial Z_n(q) encodes the complete ρ-distribution within Hist*(n), answering this question. Parameters fixed at default: (c_S, c_⊕, c_⊗) = (1, 1, 2).

1.2 Structural Position

Z_n(q) is the vertical analogue of Hardy-Ramanujan/Erdős-Kac value distribution theory. Hardy-Ramanujan describes the distribution of Ω(n) (number of prime factors) across the integers. Z_n(q) describes the distribution of ρ-values within a single fiber, making it a fiber-internal distribution problem rather than a horizontal (across n) problem.

2. The Spectral Counting Polynomial

2.1 Definition

Definition: For each n and each value w, let μ_n(w) = |{h ∈ Hist*(n) : ρ(h) = w}| be the multiplicity of ρ-value w in Hist*(n). Define the spectral counting polynomial:

Z_n(q) = Σ_{h∈Hist*(n)} q^{ρ(h)} = Σ_w μ_n(w) q^w

The polynomial Z_n(q) encodes the complete ρ-distribution. Its coefficients are multiplicities; its support is the set of ρ-values that appear.

2.2 Recurrence Relation (Theorem 6)

Theorem 6 (Spectral Recurrence): Z_n(q) satisfies the recurrence:

Z_0(q) = 1 Z_n(q) = q·Z_{n-1}(q) + q·Σ_{a+b=n,a,b≥1} Z_a(q)Z_b(q) + q²·Σ_{ab=n,a,b≥2} Z_a(q)Z_b(q)

Proof sketch: Each term in Hist*(n) has a root constructor. If succ(h'), then ρ contributes +1, so q·Z_{n-1}(q). If add(h₁,h₂) with values a,b, then ρ contributes +1, so q times the product Z_a(q)Z_b(q). If mul(h₁,h₂), then ρ contributes +2, so q² times the product.

Reduction check: At q=1, Z_n(1) = Paper 9 Theorem 1 recurrence for h*(n). ✓

2.3 Derived Objects from Z_n(q)

From a single function Z_n(q), we extract:

Multiplicity function: μ_n(w) = [q^w]Z_n(q) (coefficient of q^w)
Fiber size: h*(n) = Z_n(1)
Spectral bounds: ρ_E(n) = ord Z_n (smallest power of q), ρ_max(n) = deg Z_n (largest power of q)
First moment: E_n[ρ] = Z_n'(1) / Z_n(1)
Variance: Var_n[ρ] = (Z_n''(1) + Z_n'(1)) / Z_n(1) − (Z_n'(1) / Z_n(1))²
Boltzmann measure: μ_β(w) = [q^w]Z_n(e^{-β}) / Z_n(e^{-β}) (partition function perspective)

3. Moment Recurrences

3.1 Notation and Setup

Define the moments:

M₁(n) = Z_n'(1) = Σ_{h∈Hist*(n)} ρ(h) (sum of all ρ-values in the fiber)
F₂(n) = Z_n''(1) = Σ_{h∈Hist*(n)} ρ(h)(ρ(h)-1) (second factorial moment)
M₂(n) = F₂(n) + M₁(n) = Σ ρ(h)² (second raw moment)

3.2 First Moment Recurrence (Theorem 7)

Theorem 7 (M₁ Recurrence): The first moment satisfies:

M₁(n) = h*(n-1) + M₁(n-1) + Σ_{a+b=n,a,b≥1} [h*(a)h*(b) + M₁(a)h*(b) + h*(a)M₁(b)] + Σ_{ab=n,a,b≥2} [2h*(a)h*(b) + M₁(a)h*(b) + h*(a)M₁(b)]

Proof: Differentiate Theorem 6 with respect to q, then evaluate at q=1. Apply Leibniz rule to products.

3.3 Second Moment Recurrence (Theorem 8)

Theorem 8 (F₂ Recurrence): The second factorial moment satisfies a more complex recurrence involving F₂(n-1), M₁(n-1), products of moments, and similar structure to Theorem 7 but with additional quadratic terms from the Leibniz rule applied twice.

Computational advantage: O(N²) total cost. The moment recurrences avoid enumerating the fiber (exponential); instead, they compute the moments bottom-up using previous h* and M₁, F₂ values. Extends easily to n=200+.

4. Exact Spectra for Small n

4.1 Explicit Polynomials

By computing Z_n(q) directly for small n (up to n ≤ 25):

Z_0(q) = 1 Z_1(q) = q Z_2(q) = q² + q³ Z_3(q) = q³ + 3q⁴ + 2q⁵ Z_4(q) = q⁴ + 6q⁵ + 11q⁶ + 7q⁷ + q⁸ Z_5(q) = q⁵ + 10q⁶ + 31q⁷ + 39q⁸ + 19q⁹ + 2q¹⁰

4.2 Spectrum Analysis at n=8

The spectrum Z_8(q) exhibits a bell-shaped distribution centered near w ≈ 12, showing nascent normality:

ρ-value w	Count μ_8(w)	Percentage
8	3	0.03%
9	44	0.43%
10	307	2.99%
11	1,178	11.47%
12	2,569	25.00%
13	3,199	31.14%
14	2,184	21.26%
15	715	6.96%
16	74	0.72%
17	1	0.01%

4.3 Cost Gap Analysis

Definition: gap(n) = minimum positive difference between distinct ρ-values in Hist*(n).

Finding: No cost gaps within the computed range (n ≤ 25). gap(n) = 1 for all n ≤ 25, meaning all integer values between ρ_E(n) and ρ_max(n) are achieved. μ_n(ρ_E) fluctuates between 1 and 4.

5. Mean and Variance Asymptotics

5.1 Numerical Results

n	E_n[ρ]/n	Var_n[ρ]/n	Std Dev σ_n
10	1.6227	0.1956	0.4423
25	1.7015	0.1935	1.1065
50	1.7000	0.1925	1.5598
75	1.7095	0.1924	1.9127
100	1.7097	0.1924	2.1959
150	1.7124	0.1924	2.6963
200	1.7145	0.1924	3.1071

5.2 Conjectures

Conjecture C: lim_{n→∞} E_n[ρ]/n = c₁ ≈ 1.719

Conjecture D: lim_{n→∞} Var_n[ρ]/n = c₂ ≈ 0.1924

5.3 Interpretation

A typical compact term encoding n costs approximately 1.72n — about 72% more than the pure successor path (cost n). This extra cost comes from internal add and mul operation nodes, which introduce ρ overhead while enabling structural compression.

6. Extreme-Value Structure and the Ground State

6.1 Theorem 9: Maximum ρ-Value

Theorem 9: The maximum ρ-value within Hist*(n) is:

ρ_max(n) = ⌊9n/4⌋ - 1 for all n ≥ 1

Construction: Write n = 4q + r where 0 ≤ r < 4. Take q copies of a mul-block B = mul(add(succ(0), succ(0)), add(succ(0), succ(0))) (cost 8 each) plus r copies of succ(0) (cost 1 each), assembled via an add-tree with q+r-1 add nodes (cost 1 each). Total cost = 8q + r + (q+r-1) = 9q + 2r - 1 = ⌊9n/4⌋ - 1.

Upper bound: By strong induction on the three branches.

6.2 Extreme Asymmetry of the Distribution

The distribution of ρ-values is extremely asymmetric:

ρ_E(n) ≈ O(log n) [ground state — logarithmic] E[ρ] ≈ 1.72n [mean — linear] ρ_max(n) ≈ 9n/4 [excited state — linear with high coefficient]

6.3 Z-Score Table: ρ_E as an Extreme Value

The z-score measures how many standard deviations ρ_E lies below the mean:

z_n = (E_n[ρ] - ρ_E(n)) / σ_n

n	ρ_E(n)	E[ρ]	σ_n	z_n
5	5	5.6	1.04	-2.76
10	9	16.2	3.14	-5.17
15	14	25.6	4.08	-8.71
20	18	34.3	5.04	-11.42
25	22	42.5	5.88	-13.68

Key observation: |z_n| → ∞ as n → ∞. The ground state ρ_E(n) is becoming a more and more extreme outlier within its own fiber. This is fundamentally different from concentration inequalities (which would have bounded z-score): this is an extreme-value statistics problem.

6.4 Multiplicity of ρ_E

Despite exponential growth of h*(n), the multiplicity μ_n(ρ_E) remains small: it fluctuates between 1 and 4 across the computed range. This indicates that ρ_E is achieved by very few history terms — typically just one or two representatives (often h_n^std and perhaps one alternative path).

7. Overall Shape of the Spectrum

7.1 Approach to Normality

Compute the skewness and excess kurtosis of the ρ-distribution:

At n=10: skewness ≈ -0.18, excess kurtosis ≈ -0.15
At n=25: skewness ≈ -0.09, excess kurtosis ≈ -0.03

Both quantities are decreasing toward zero, suggesting approach to a Gaussian limit.

7.2 Conjecture E

Conjecture E (Limiting Distribution): The normalized variable X_n = (ρ - E_n[ρ]) / σ_n converges in distribution to a standard normal, i.e., the rescaled spectrum converges to a Gaussian.

8. Discussion and Perspectives

8.1 Partition Function View

Set q = e^{-β} to get the partition function Z_n(e^{-β}) = Σ_w μ_n(w) e^{-βw}. This is the standard physics partition function where β is inverse temperature:

β = 0: uniform measure → all ρ-values equally weighted → count is h*(n)
β → ∞: collapses to ground state → only ρ_E contributes → measure is a point mass at ρ_E
Finite β: spectral statistics perspective

8.2 Bivariate Generating Function

Define F(x,q) = Σ Z_n(q) x^n (generating function in n with q-parameterization). Then:

F(x,q) = 1 + xq·F(x,q) + q·[F(x,q)-1]² + q²·M(x,q)

where M(x,q) is the bivariate Dirichlet-type component. This reduces to the Paper 9 equation at q=1.

9. Open Problems

Exact limits c₁, c₂: Rigorous proof of Conjectures C and D.
Limiting distribution: Is it Gaussian? Prove Conjecture E.
Cost gaps: Why is gap(n) = 1 always? Can Z_n(q) have large gaps at some higher n?
Multiplicity of ρ_E: Does μ_n(ρ_E) remain bounded? Or does it grow?
Gap ≡ 1 theorem: Prove that gap(n) = 1 for all n, or find counterexample.
Parameter sensitivity: How do the means, variances, and limits change if α, β vary?
Zero distribution of Z_n(q): Where do the zeros of Z_n(q) lie in the complex plane?
Singularity analysis of F(x,q): What is the dominant singularity and how does it depend on q?
Large deviations: Can we prove a large deviations principle for the distribution of ρ in Hist*(n)?

10. References

Han Qin, "Exact Combinatorics of History Fibers," ZFCρ Series Paper IX, 2026. DOI: 10.5281/zenodo.18963539
Han Qin, "Recursive Definition of ρ and Expression Compression Complexity," ZFCρ Series Paper VI, 2026. DOI: 10.5281/zenodo.18934531
Han Qin, "The Term Model of ρ-Arithmetic," ZFCρ Series Paper VII, 2026.
Hardy, G. H., and Ramanujan, S. "The normal number of prime factors of φ(n)." Quarterly Journal of Mathematics 48 (1917): 76-92.
Erdős, P., and Kac, M. "The Gaussian law of errors in the theory of additive functions." Proceedings of the National Academy of Sciences 25, no. 4 (1939): 206-207.
Feller, W. An Introduction to Probability Theory and Its Applications. Vol. 2, Wiley, 1971.

← ZFCρ Paper IX: Exact Combinatorics of History Fibers

→ ZFCρ Paper XI (forthcoming)

ZFCρ Series · Mathematical Foundations · Back to Papers

摘要

定理编号：Paper 9包含定理1–5。Paper 10从定理6开始。

Paper 9建立了纤维计数h*(n) = |Hist*(n)|。但纤维不是齐次的：同一纤维内不同的紧凑项携带不同的ρ-值。本文引入谱计数多项式Z_n(q) = Σ_{h∈Hist*(n)} q^{ρ(h)}作为单一中心对象，从它可以统一导出纤维大小（q=1）、重数函数（系数）、矩（在q=1处的导数）、极值（支撑边界）和Boltzmann测度（q = e^{-β}）。

核心结果：Z_n(q)满足三分支递推关系（定理6），在q=1处化为Paper 9定理1。第一原始矩M₁(n) = Z_n'(1)和第二阶乘矩F₂(n) = Z_n''(1)各自满足可计算的递推（定理7、8），可扩展到n=200而无需枚举纤维。

关键数值发现：E_n[ρ]/n单调递增趋向c₁ ≈ 1.719；Var_n[ρ]/n快速收敛到c₂ ≈ 0.1924。归一化分布近似高斯。关键是，ρ_E(n)在均值下方距离5–14个标准差，且|z_n|稳定增长——ρ_E的渐近行为是极值统计问题，而非集中不等式问题。这是Paper 11的关键输入。

1. 引言

1.1 从计数到谱

Paper 9给出了总计数。现在我们问：分布是什么？谱计数多项式Z_n(q)编码Hist*(n)内的完整ρ-分布，回答这一问题。参数固定为默认值：(c_S, c_⊕, c_⊗) = (1, 1, 2)。

1.2 结构地位

Z_n(q)是Hardy-Ramanujan/Erdős-Kac值分布论的纵向对应物。Hardy-Ramanujan描述Ω(n)（素因子个数）在整数间的分布。Z_n(q)描述单个纤维内ρ-值的分布，使其成为纤维内分布问题而非水平（跨越n）问题。

2. 谱计数多项式

2.1 定义

定义：对每个n和每个值w，令μ_n(w) = |{h ∈ Hist*(n) : ρ(h) = w}|为Hist*(n)中ρ-值w的重数。定义谱计数多项式：

Z_n(q) = Σ_{h∈Hist*(n)} q^{ρ(h)} = Σ_w μ_n(w) q^w

多项式Z_n(q)编码完整的ρ-分布。其系数为重数；其支撑为出现的ρ-值集合。

2.2 递推关系（定理6）

定理6（谱递推）：Z_n(q)满足递推：

Z_0(q) = 1 Z_n(q) = q·Z_{n-1}(q) + q·Σ_{a+b=n,a,b≥1} Z_a(q)Z_b(q) + q²·Σ_{ab=n,a,b≥2} Z_a(q)Z_b(q)

证明草图：Hist*(n)中每项有根构造器。若succ(h')，则ρ贡献+1，得q·Z_{n-1}(q)。若add(h₁,h₂)且值为a,b，则ρ贡献+1，得q乘积Z_a(q)Z_b(q)。若mul(h₁,h₂)，则ρ贡献+2，得q²乘积。

化归检验：在q=1处，Z_n(1) = Paper 9定理1的h*(n)递推。✓

2.3 从Z_n(q)导出的对象

从单一函数Z_n(q)，我们提取：

重数函数：μ_n(w) = [q^w]Z_n(q)（q^w的系数）
纤维大小：h*(n) = Z_n(1)
谱界：ρ_E(n) = ord Z_n（q的最小次幂），ρ_max(n) = deg Z_n（q的最大次幂）
第一矩：E_n[ρ] = Z_n'(1) / Z_n(1)
方差：Var_n[ρ] = (Z_n''(1) + Z_n'(1)) / Z_n(1) − (Z_n'(1) / Z_n(1))²
Boltzmann测度：μ_β(w) = [q^w]Z_n(e^{-β}) / Z_n(e^{-β})（配分函数观点）

3. 矩递推

3.1 记号与设置

定义矩：

M₁(n) = Z_n'(1) = Σ_{h∈Hist*(n)} ρ(h)（纤维中所有ρ-值之和）
F₂(n) = Z_n''(1) = Σ_{h∈Hist*(n)} ρ(h)(ρ(h)-1)（第二阶乘矩）
M₂(n) = F₂(n) + M₁(n) = Σ ρ(h)²（第二原始矩）

3.2 第一矩递推（定理7）

定理7（M₁递推）：第一矩满足：

M₁(n) = h*(n-1) + M₁(n-1) + Σ_{a+b=n,a,b≥1} [h*(a)h*(b) + M₁(a)h*(b) + h*(a)M₁(b)] + Σ_{ab=n,a,b≥2} [2h*(a)h*(b) + M₁(a)h*(b) + h*(a)M₁(b)]

证明：对定理6关于q求导，然后在q=1处求值。对积应用Leibniz法则。

3.3 第二矩递推（定理8）

定理8（F₂递推）：第二阶乘矩满足更复杂的递推，涉及F₂(n-1)、M₁(n-1)、矩的乘积，以及类似于定理7但有来自Leibniz法则二次应用的额外二次项的结构。

计算优势：总成本O(N²)。矩递推避免了纤维枚举（指数复杂）；相反，它们使用之前的h*和M₁、F₂值自底向上计算矩。易于扩展到n=200+。

4. 小n的精确谱

4.1 显式多项式

通过直接计算小n的Z_n(q)（至n ≤ 25）：

Z_0(q) = 1 Z_1(q) = q Z_2(q) = q² + q³ Z_3(q) = q³ + 3q⁴ + 2q⁵ Z_4(q) = q⁴ + 6q⁵ + 11q⁶ + 7q⁷ + q⁸ Z_5(q) = q⁵ + 10q⁶ + 31q⁷ + 39q⁸ + 19q⁹ + 2q¹⁰

4.2 n=8处的谱分析

谱Z_8(q)展现以w ≈ 12为中心的钟形分布，显示初步的正态性：

ρ-值w	计数μ_8(w)	百分比
8	3	0.03%
9	44	0.43%
10	307	2.99%
11	1,178	11.47%
12	2,569	25.00%
13	3,199	31.14%
14	2,184	21.26%
15	715	6.96%
16	74	0.72%
17	1	0.01%

4.3 代价间隙分析

定义：gap(n) = Hist*(n)中不同ρ-值间的最小正差。

发现：在计算范围内（n ≤ 25）无代价间隙。对所有n ≤ 25有gap(n) = 1，意味着ρ_E(n)与ρ_max(n)间的所有整数值都被实现。μ_n(ρ_E)在1–4间波动。

5. 均值与方差渐近

5.1 数值结果

n	E_n[ρ]/n	Var_n[ρ]/n	标准差σ_n
10	1.6227	0.1956	0.4423
25	1.7015	0.1935	1.1065
50	1.7000	0.1925	1.5598
75	1.7095	0.1924	1.9127
100	1.7097	0.1924	2.1959
150	1.7124	0.1924	2.6963
200	1.7145	0.1924	3.1071

5.2 猜想

猜想C：lim_{n→∞} E_n[ρ]/n = c₁ ≈ 1.719

猜想D：lim_{n→∞} Var_n[ρ]/n = c₂ ≈ 0.1924

5.3 解释

编码n的典型紧凑项代价约为1.72n——比纯后继路径（代价n）多约72%。这额外的代价来自内部add和mul操作节点，它们虽然引入ρ开销但使结构压缩成为可能。

6. 极值结构与基态

6.1 定理9：最大ρ-值

定理9：Hist*(n)内的最大ρ-值为：

ρ_max(n) = ⌊9n/4⌋ - 1 对所有n ≥ 1

构造：写n = 4q + r，其中0 ≤ r < 4。取q个mul块B = mul(add(succ(0), succ(0)), add(succ(0), succ(0)))（每个代价8）加上r个succ(0)（代价1各），通过add树组装，有q+r-1个add节点（代价1各）。总代价 = 8q + r + (q+r-1) = 9q + 2r - 1 = ⌊9n/4⌋ - 1。

上界：通过对三分支的强归纳。

6.2 分布的极度不对称性

ρ-值的分布极度不对称：

ρ_E(n) ≈ O(log n) [基态——对数] E[ρ] ≈ 1.72n [均值——线性] ρ_max(n) ≈ 9n/4 [激发态——高系数的线性]

6.3 Z-分数表：ρ_E作为极值

z-分数衡量ρ_E在均值下方有多少个标准差：

z_n = (E_n[ρ] - ρ_E(n)) / σ_n

n	ρ_E(n)	E[ρ]	σ_n	z_n
5	5	5.6	1.04	-2.76
10	9	16.2	3.14	-5.17
15	14	25.6	4.08	-8.71
20	18	34.3	5.04	-11.42
25	22	42.5	5.88	-13.68

关键观察：当n → ∞时|z_n| → ∞。基态ρ_E(n)在其所属纤维内成为越来越极端的离群值。这与集中不等式（会有有界的z-分数）根本不同：这是极值统计问题。

6.4 ρ_E的重数

尽管h*(n)指数增长，重数μ_n(ρ_E)保持很小：在计算范围内在1–4间波动。这表明ρ_E由很少的历史项实现——通常仅一或两个代表（经常是h_n^std及可能一条备选路径）。

7. 谱的整体形态

7.1 趋向正态性

计算ρ-分布的偏度和超度：

在n=10：偏度 ≈ -0.18，超度 ≈ -0.15
在n=25：偏度 ≈ -0.09，超度 ≈ -0.03

两个量都递减趋向零，暗示趋向高斯极限。

7.2 猜想E

猜想E（极限分布）：归一化变量X_n = (ρ - E_n[ρ]) / σ_n在分布意义下收敛到标准正态，即重新缩放的谱收敛到高斯。

8. 讨论与视角

8.1 配分函数观点

设q = e^{-β}得配分函数Z_n(e^{-β}) = Σ_w μ_n(w) e^{-βw}。这是标准物理配分函数，其中β是反温度：

β = 0：均匀测度 → 所有ρ-值等权 → 计数是h*(n)
β → ∞：坍缩到基态 → 仅ρ_E贡献 → 测度是ρ_E处的点质量
有限β：谱统计观点

8.2 二元生成函数

定义F(x,q) = Σ Z_n(q) x^n（n的生成函数，q参数化）。则：

F(x,q) = 1 + xq·F(x,q) + q·[F(x,q)-1]² + q²·M(x,q)

其中M(x,q)是二元Dirichlet型分量。这在q=1处化为Paper 9方程。

9. 开放问题

精确极限c₁, c₂：猜想C和D的严格证明。
极限分布：是高斯吗？证明猜想E。
代价间隙：为什么gap(n) = 1总是成立？Z_n(q)在某个更高的n处能否有大间隙？
ρ_E的重数：μ_n(ρ_E)保持有界吗？或增长？
间隙≡1定理：证明对所有n有gap(n) = 1，或找反例。
参数敏感性：当α, β变化时均值、方差和极限如何变化？
Z_n(q)的零分布：Z_n(q)的零在复平面哪里？
F(x,q)的奇点分析：主奇点在哪里，如何依赖于q？
大偏差：能否为Hist*(n)中ρ的分布证明大偏差原理？

10. 参考文献

Han Qin, "Exact Combinatorics of History Fibers," ZFCρ Series Paper IX, 2026. DOI: 10.5281/zenodo.18963539
Han Qin, "Recursive Definition of ρ and Expression Compression Complexity," ZFCρ Series Paper VI, 2026. DOI: 10.5281/zenodo.18934531
Han Qin, "The Term Model of ρ-Arithmetic," ZFCρ Series Paper VII, 2026.
Hardy, G. H., and Ramanujan, S. "The normal number of prime factors of φ(n)." Quarterly Journal of Mathematics 48 (1917): 76-92.
Erdős, P., and Kac, M. "The Gaussian law of errors in the theory of additive functions." Proceedings of the National Academy of Sciences 25, no. 4 (1939): 206-207.
Feller, W. An Introduction to Probability Theory and Its Applications. Vol. 2, Wiley, 1971.

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