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Cantor distribution

The Cantor distribution is the probability distribution on the unit interval [0,1] whose cumulative distribution function is the Cantor function, a continuous, strictly increasing map from [0,1] to [0,1] that is constant on each of the countably many open intervals complementary to the ternary Cantor set and differentiable almost everywhere with derivative zero.^[1] This distribution is singular continuous, meaning it has no point masses (purely atomic part) and is mutually singular with respect to Lebesgue measure, concentrating all its mass on the ternary Cantor set—a compact, totally disconnected, perfect set of Lebesgue measure zero but with the cardinality of the continuum.^[2] It serves as a canonical example in measure theory and probability of a distribution that is continuous yet lacks a probability density function, illustrating the existence of pathological phenomena in the classification of measures into discrete, absolutely continuous, and singular continuous components.^[2] The Cantor function, upon which the distribution is based, was introduced by Georg Cantor in his 1884 paper "De la puissance des ensembles parfaits de points," where it arose in the study of the topological properties of perfect sets.^[1] Also known as the devil's staircase due to its flat steps resembling a staircase, the function can be explicitly constructed using the ternary (base-3) expansions of points in [0,1]: Points in the Cantor set have ternary expansions using only the digits 0 and 2; the Cantor function G(x) is obtained by replacing each 2 in this expansion with a 1 and interpreting the resulting sequence as a binary expansion. The function is constant on each interval removed during the iterative construction of the Cantor set (middle thirds of [0,1] at each stage), and surjective onto [0,1], thereby defining a valid distribution function for the Cantor distribution.^[2] The ternary Cantor set itself is generated iteratively: starting with C_0 = [0,1], remove the open middle third (1/3, 2/3) to obtain C_1, then remove the open middle thirds of the two remaining intervals to obtain C_2, and continue indefinitely, yielding C = \bigcap_{n=0}^\infty C_n as the support of the distribution.^[2] Each C_n consists of $2^n closed intervals of length $3^{-n}, so the Lebesgue measure of C is \lim_{n \to \infty} 2^n \cdot 3^{-n} = 0, yet C is uncountable and homeomorphic to \{0,1\}^\mathbb{N}.^[2] The Cantor distribution Q arises as the weak limit of the uniform distributions Q_n on C_n, and equivalently as the Lebesgue-Stieltjes measure induced by G, satisfying Q((a,b]) = G(b) - G(a) for intervals (a,b].^[2] Key properties of the Cantor distribution include its self-similarity: Q satisfies the functional equation Q = \frac{1}{2} Q \circ T_0 + \frac{1}{2} Q \circ T_2, where T_0(x) = x/3 and T_2(x) = (x+2)/3 are the left and right similarity maps of the Cantor set.^[2] It has finite moments of all orders, reflecting its fractal nature. In applications, the distribution appears in examples of singular measures in ergodic theory, spectral theory of operators, and as a model for certain fractal phenomena in physics and finance, though its primary significance remains in pure mathematics as a bridge between topology, analysis, and probability.^[2]

Historical Background

Origins with the Cantor set

The Cantor set was constructed by Georg Cantor between 1879 and 1883 as part of his pioneering work in point-set topology, serving as a prototypical example of a nowhere dense perfect set within the unit interval [0,1] that possesses Lebesgue measure zero.^[3] This uncountable set emerged from Cantor's efforts to formalize infinite linear point manifolds, highlighting pathological structures in the real line that defied intuitive notions of density and continuity.^[4] Cantor's motivation stemmed from his analysis of Fourier series, particularly the questions of pointwise convergence and uniqueness for trigonometric representations of functions during the early 1870s.^[5] He sought to identify sets where such series might fail to converge uniquely, leading to the construction of the Cantor set as an exemplar of a compact, perfect set devoid of isolated points—essential for understanding regions of non-convergence in analytic settings.^[5] Earlier explorations of similar nowhere dense sets appeared in the work of contemporaries, such as Hermann Hankel's 1870 conjecture linking Riemann integrability to discontinuity sets of "loose order" (nowhere dense with zero outer content).^[6] Hankel's ideas influenced subsequent constructions, including Henry John Stephen Smith's 1874 generalized iterative removal process for creating such sets to study discontinuous integrable functions.^[4] Cantor's formalization, however, crystallized these concepts in his 1883 paper "Über unendliche, lineare Punktmannigfaltigkeiten, V," where he explicitly defined the ternary version and proved its key topological properties.^[3] The construction proceeds iteratively: begin with the closed interval [0,1] and remove the open middle third (1/3, 2/3), yielding two closed intervals of length 1/3. Repeat the middle-third removal on each remaining interval at every stage, continuing infinitely. The intersection of all these stages forms the Cantor set, which is uncountable yet has total length (measure) zero due to the summed lengths of removed intervals equaling 1.^[4] This process underscores the set's nowhere dense nature—no interval is entirely contained within it—while its perfection ensures every point serves as a limit point.^[6]

Development of the Cantor function

The Cantor function was first defined by Georg Cantor in his 1883 paper "Über unendliche, lineare Punktmannigfaltigkeiten, V," where it appears as a continuous, non-decreasing function mapping the closed interval [0,1] onto itself and remaining constant on each of the open intervals removed during the iterative construction of the Cantor set.^[7] This definition arose in the context of Cantor's investigations into point set topology and the structure of continuous functions, building on his earlier work in the series of papers on infinite linear point manifolds.^[3] Cantor emphasized several fundamental properties of the function in his analysis: its strict monotonicity overall, despite being constant on complementary intervals; its uniform continuity on [0,1]; and its surjectivity onto [0,1], which underscores the topological equivalence between the Cantor set (of Lebesgue measure zero) and the full unit interval.^[8] These attributes were motivated by Cantor's broader research on transfinite cardinalities and set theory during the early 1880s, with the function explicitly constructed as a homeomorphism relating ternary expansions (using digits 0 and 2) of points in the Cantor set to binary expansions (using digits 0 and 1) in [0,1], thereby illustrating the uncountable nature and perfect structure of the set.^[7] The function's significance was quickly recognized and extended by Ludwig Scheeffer in his 1884 paper "Eine neue Klasse von continuirlichen Functionen," which highlighted its utility as a counterexample to Axel Harnack's attempted generalization of the fundamental theorem of calculus to discontinuous integrands, demonstrating that continuity alone does not guarantee representation as an integral of a derivative. Scheeffer's analysis thereby popularized the function within studies of real analysis and variation.^[8] By the early 20th century, the Cantor function had become a cornerstone example in advanced integration theory, as noted by Henri Lebesgue in his 1904 monograph Leçons sur l'intégration et la recherche des fonctions primitives, where it exemplified a continuous function of bounded variation whose derivative vanishes almost everywhere on [0,1], yet achieves a total increase of 1.^[9] Giuseppe Vitali further reinforced this interpretation in his 1905 work on functions of bounded variation, using the function to illustrate that such derivatives do not recover the original via integration, a key insight into the limitations of classical differentiation.^[10]

Mathematical Prerequisites

The ternary Cantor set

The ternary Cantor set, often simply called the Cantor set, is constructed through an iterative process of removing middle-third intervals from the unit interval [0,1]. This construction begins with C_0 = [0,1]. At the first stage, the open middle-third interval (1/3, 2/3) is removed, leaving two closed intervals: C_1 = [0,1/3] \cup [2/3,1], each of length $1/3. In the subsequent stage, the open middle-third interval is removed from each of these, yielding C_2 = [0,1/9] \cup [2/9,1/3] \cup [2/3,7/9] \cup [8/9,1], consisting of four intervals each of length $1/9. Generally, C_{n+1} is obtained by removing the open middle-third subinterval of length $1/3^{n+1} from each of the $2^n closed intervals comprising C_n, resulting in $2^{n+1} closed intervals each of length $1/3^{n+1}.^[11] The ternary Cantor set C is then defined as the intersection

C = \bigcap_{n=0}^\infty C_n.

^[11] Topologically, C is compact, as it is closed and bounded in \mathbb{R}.^[11] It is perfect, meaning it is closed and every point is a limit point with no isolated points, and totally disconnected, with the only connected components being singletons.^[11] Moreover, C is uncountable, possessing the cardinality of the continuum, $2^{\aleph_0}.^[11] In terms of measure, the Lebesgue measure of C is zero, since the total length removed across all stages sums to 1: at stage n, the removed length is $2^{n-1}/3^n, and the infinite sum \sum_{n=1}^\infty 2^{n-1}/3^n = 1.^[11] The Hausdorff dimension of C is \log 2 / \log 3 \approx 0.6309, reflecting its fractal structure where the scaling factor is 3 and the number of self-similar copies is 2.^[12] Every point in C admits a ternary (base-3) expansion of the form \sum_{k=1}^\infty a_k / 3^k where each a_k \in \{0, 2\}, and conversely, every such expansion corresponds to a unique point in C (noting that endpoints of removed intervals have dual representations, but the {0,2}-digit form selects the appropriate one).^[13]

Construction of the Cantor function

The Cantor function, also known as the Cantor-Lebesgue function or devil's staircase, can be constructed iteratively in conjunction with the ternary Cantor set. Begin with the initial interval C_0 = [0,1] and define the zeroth approximation c_0(x) = x for x \in [0,1].^[14] At the first stage, form C_1 = [0,1/3] \cup [2/3,1] by removing the open middle-third interval (1/3, 2/3). Define c_1(x) to be linear on each component of C_1, mapping [0,1/3] onto [0,1/2] and [2/3,1] onto [1/2,1], while setting c_1(x) = 1/2 constantly on the removed interval (1/3, 2/3). Specifically, c_1(x) = (3/2)x for x \in [0,1/3] and c_1(x) = (3/2)(x - 2/3) + 1/2 for x \in [2/3,1].^[15] Proceed iteratively: at stage n+1, for each closed interval I of length $3^{-(n+1)} in C_{n+1}, map the left subinterval of I linearly onto the left half of the image under c_n of the parent interval in C_n, and the right subinterval onto the right half, while remaining constant (equal to the value at the endpoints) on the newly removed middle-third open subintervals. This process scales the range by $1/2 at each stage, ensuring the approximations c_n converge uniformly to a limit function c: [0,1] \to [0,1] defined on the Cantor set C = \bigcap_{n=0}^\infty C_n and its complement.^[3]^[14] An explicit formula for c on the Cantor set arises from ternary expansions. Every x \in C admits a ternary expansion x = \sum_{k=1}^\infty a_k 3^{-k} where each a_k \in \{0, 2\} (with endpoints having dual representations resolved by avoiding 1's). The Cantor function is then given by

c(x) = \sum_{k=1}^\infty \frac{a_k / 2}{2^k},

which interprets the sequence (a_k / 2)_{k=1}^\infty \in \{0,1\}^\mathbb{N} as a binary expansion. This mapping replaces 2's with 1's and interprets the result in base 2.^[15]^[14] The resulting function c is continuous on [0,1], as the uniform convergence of the continuous piecewise linear approximations c_n preserves continuity.^[15] It is non-decreasing, since each c_n is non-decreasing and the limit inherits this property.^[3] Moreover, c'(x) = 0 for Lebesgue-almost every x \in [0,1], because c is constant on the open complementary intervals removed during construction, which have total Lebesgue measure 1.^[15] On these complementary intervals, c takes constant values equal to those at their endpoints. Despite the Cantor set C having Lebesgue measure zero, c restricted to C is surjective onto [0,1], as every binary expansion corresponds uniquely to a ternary expansion with digits 0 and 2 via the formula above.^[14]

Definition

Cumulative distribution function

The cumulative distribution function (CDF) of the Cantor distribution is given by the Cantor function c(x) for x \in [0,1], satisfying c(0) = 0, c(1) = 1, right-continuity, and non-decreasing monotonicity.^[16] This CDF is continuous everywhere on [0,1], with no jumps, but remains constant on the complement of the Cantor set and thus increases solely on the Cantor set, a subset of Lebesgue measure zero.^[16]^[17] The Cantor distribution denotes the probability law of a random variable X on [0,1] such that P(X \leq x) = c(x).^[17] The Cantor function relates to the uniform distribution on [0,1] via ternary expansions of points in the Cantor set, which use only the digits 0 and 2; substituting 0 for 0 and 1 for 2 yields a binary expansion that maps the Cantor set onto [0,1] uniformly in a measure-preserving sense.^[16]^[17]

Probability measure

The Cantor distribution is defined as a probability measure \mu on the Borel \sigma-algebra of [0,1], fully supported on the ternary Cantor set C \subseteq [0,1], with \mu(C) = 1 and \mu([0,1] \setminus C) = 0.^[18] This measure assigns zero mass to the complement of C, which consists of the open intervals removed during the iterative construction of the Cantor set, reflecting the concentration of probability entirely on the uncountable, measure-zero set C.^[18] One standard construction of \mu is as the pushforward of the Lebesgue measure \lambda on [0,1] under the continuous surjection \phi: [0,1] \to C that maps a point t \in [0,1] with binary expansion t = \sum_{k=1}^\infty b_k 2^{-k} (where b_k \in \{0,1\}) to the point \phi(t) = \sum_{k=1}^\infty 2 b_k 3^{-k} \in C, interpreting the binary digits as ternary digits scaled by 2.^[18] Thus, \mu = \phi_* \lambda, ensuring that \mu is a Borel probability measure with the desired support. Equivalently, for basic intervals I arising in the Cantor set construction (such as the $2^n closed intervals of length $3^{-n} at stage n), \mu(I) equals the Lebesgue length of the image c(I) under the Cantor function c, which scales these lengths uniformly by $2^{-n}.^[18] The measure \mu exhibits self-similarity, satisfying the invariance relation \mu(E) = \frac{1}{2} \mu(3E) + \frac{1}{2} \mu(3E - 2) for Borel sets E \subseteq [0,1], corresponding to the contractions by $1/3 on the left and right subintervals.^[19] This equation arises from the iterated function system (IFS) consisting of the maps S_1(x) = x/3 and S_2(x) = (x+2)/3, each with equal probability $1/2, where \mu is invariant under the weighted average of the pushforwards induced by these maps.^[19] Under the open set condition (satisfied by the IFS for the Cantor set, as the images S_1([0,1]) and S_2([0,1]) are disjoint), \mu is the unique Borel probability measure invariant for this IFS.^[19] The cumulative distribution function of \mu is the Cantor function F(x) = \mu([0,x]).^[18]

Properties

Support and continuity

The support of the Cantor distribution is the ternary Cantor set C, defined as the intersection of a nested sequence of closed sets obtained by iteratively removing the open middle-third intervals from [0,1], and it is the smallest closed set carrying the entire probability measure of 1.^[20] The Cantor set C is compact as a closed and bounded subset of \mathbb{R}, nowhere dense in [0,1] since its complement is dense, perfect in that every point is a limit point of C, uncountable, and has Lebesgue measure zero.^[21]^[20] The cumulative distribution function F(x) of the Cantor distribution is continuous everywhere on \mathbb{R}, monotonically increasing from F(-\infty) = 0 to F(\infty) = 1, and exhibits no jumps, which implies that the probability mass at any single point is zero, i.e., P(X = x) = 0 for all x \in \mathbb{R}, confirming the absence of atoms.^[1]^[20] This continuity distinguishes the Cantor distribution from discrete distributions, as its support C is uncountable rather than finite or countable.^[21] Visually, the graph of F(x) resembles a "devil's staircase," remaining constant (flat) on each open interval in the complement of C within [0,1]—the intervals removed during the construction of the Cantor set—and strictly increasing only over the points in C, thereby concentrating all probabilistic rise within this fractal subset despite its zero Lebesgue measure.^[1]

Singular continuous nature

The Cantor distribution serves as a canonical example of a singular continuous probability distribution, occupying a position intermediate between discrete and absolutely continuous distributions. It is singular with respect to the Lebesgue measure \lambda, denoted \mu \perp \lambda, because the distribution \mu concentrates its entire mass on the Cantor set C, satisfying \mu(C) = 1, while \lambda(C) = 0. This mutual singularity implies that \mu and \lambda have disjoint supports in the sense of the Radon-Nikodym theorem, underscoring the pathological nature of measures that evade absolute continuity.^[22] The continuity aspect arises from the cumulative distribution function (CDF) of the Cantor distribution, which is the Cantor function itself—a continuous, non-decreasing function mapping [0,1] onto [0,1]. However, this CDF is not absolutely continuous, as it remains constant on the complementary intervals removed during the Cantor set construction (which have total Lebesgue measure $1$) and only increases on the Cantor set of measure zero, thereby failing the defining condition for absolute continuity.^[22] Historically, the Cantor distribution provided one of the earliest concrete examples of such a singular continuous measure, illuminating key pathologies in the theory of monotonic functions and integration during the late 19th and early 20th centuries. Defined by Georg Cantor in his 1884 paper on point sets, it was promptly recognized by L. Scheeffer as a counterexample to Axel Harnack's proposed extension of the fundamental theorem of calculus to all monotonic functions. This discovery influenced the development of modern measure theory, particularly Henri Lebesgue's 1904 work on integration and differentiation.^[23]^[22] Unlike atomic (discrete) distributions, which concentrate probability on countable points, or diffuse (absolutely continuous) distributions, which admit densities with respect to Lebesgue measure, the Cantor distribution exhibits continuity without atoms or a density, fully supported on the uncountable Cantor set.^[22]

Absence of density and atoms

The Cantor distribution is non-atomic, meaning that for any point x \in \mathbb{R}, the probability measure \mu assigns zero mass to the singleton \{x\}, i.e., \mu(\{x\}) = 0. This follows directly from the continuity of its cumulative distribution function (CDF) F, which is the Cantor function: since F has no jumps, the probability at any single point is given by P(X = x) = F(x) - \lim_{y \to x^-} F(y) = 0.^[24] The Cantor distribution also lacks a probability density function with respect to Lebesgue measure \lambda, as it is not absolutely continuous. Specifically, the derivative F'(x) exists and equals zero \lambda-almost everywhere on [0,1], yet F(1) - F(0) = 1. Consequently, no Radon-Nikodym derivative d\mu / d\lambda exists, confirming that \mu is singular with respect to \lambda.^[24]^[25] To see why F'(x) = 0 almost everywhere, note that F is constant on each open interval in the complement of the Cantor set C (the middle-third intervals removed iteratively), so its derivative is zero there. The complement [0,1] \setminus C has Lebesgue measure 1, while C has measure zero; by Lebesgue's differentiation theorem for monotone functions, F is differentiable almost everywhere, and the derivative is zero on a set of full measure. On C itself, F' may be undefined or infinite at some points, but this occurs on a null set, ensuring \int_0^1 F'(x) \, d\lambda(x) = 0.^[25] This property makes the Cantor distribution a canonical counterexample to the intuition that a continuous CDF must admit a density function, illustrating the existence of singular continuous measures that concentrate probability on sets of Lebesgue measure zero without point masses.^[24]

Characterization

Iterative construction of the measure

The iterative construction of the Cantor measure proceeds by defining a sequence of approximating probability measures \mu_n on the pre-Cantor sets C_n, which are the finite-stage approximations to the ternary Cantor set C. At stage n = 0, C_0 = [0, 1] and \mu_0 is the uniform (Lebesgue) measure on this interval, normalized to have total mass 1. For each subsequent stage n \geq 1, C_n consists of $2^n closed intervals, each of length $3^{-n}, obtained by iteratively removing the open middle-third subinterval from each component of C_{n-1}. The measure \mu_n is then defined to be uniform on C_n, assigning equal mass $2^{-n} to each of these $2^n intervals, while placing zero mass on the complementary intervals removed up to stage n.^[26] This stage-wise definition incorporates a consistent probability allocation rule: at each iteration, the total mass of 1 from the previous stage is split equally, with half (mass $1/2) allocated to the union of the leftmost subintervals and half to the rightmost subintervals across all components of C_{n-1}. Within each pair of subintervals receiving mass $1/2, the mass is distributed uniformly, ensuring that the approximation \mu_n remains a probability measure supported precisely on C_n.^[27] The finite approximations \mu_n preserve the self-similar structure inherent to the Cantor set, as the construction at each stage mirrors the overall fractal geometry: the support C_n is a scaled and translated copy of C_{n-1} by factors of $1/3 and probabilities of $1/2, reflecting the self-similarity dimension \log 2 / \log 3. This consistency ensures that the measures \mu_n form a coherent sequence aligned with the iterative removal process defining C = \bigcap_{n=0}^\infty C_n.^[26] As n \to \infty, the sequence \mu_n converges weakly to the Cantor measure \mu, the unique Borel probability measure supported on C that extends the mass assignments from all finite stages. This weak convergence implies that for any continuous bounded function f on [0,1], \int f \, d\mu_n \to \int f \, d\mu, and in particular, the cumulative distribution function of \mu_n converges pointwise to the Cantor function F, the continuous, non-decreasing CDF of \mu.^[27]

Ternary expansions and mapping

Every point x in the Cantor set C admits a ternary expansion of the form x = \sum_{k=1}^\infty a_k 3^{-k}, where each digit a_k belongs to the set \{0, 2\}. This representation holds uniquely except for the endpoints of the intervals removed during the construction of C, which possess dual ternary expansions: one terminating in infinite 0s and another in infinite 2s, but the expansion using only 0s and 2s is selected for consistency with the Cantor set. The Cantor function F, which serves as the cumulative distribution function of the Cantor distribution, establishes a bijection between C and [0, 1] via a mapping from these ternary expansions to binary expansions. Specifically, for x = \sum_{k=1}^\infty a_k 3^{-k} with a_k \in \{0, 2\}, define F(x) = \sum_{k=1}^\infty b_k 2^{-k}, where b_k = a_k / 2 (mapping 0 to 0 and 2 to 1). This transformation effectively reinterprets the ternary digits as binary digits after scaling, yielding a continuous, strictly increasing surjection from C onto [0, 1]. From a probabilistic perspective, a random variable X following the Cantor distribution has the same law as X = \sum_{k=1}^\infty Z_k 3^{-k}, where the Z_k are independent and identically distributed with P(Z_k = 0) = P(Z_k = 2) = 1/2.^[28] This series representation arises directly from the ternary expansion characterization, with the digits Z_k selected randomly according to the uniform measure on C.^[28] The bijection induced by F, which is a homeomorphism from C onto [0,1], implies that the pushforward of the Cantor distribution under F is the uniform distribution on [0,1]. Equivalently, if U is uniform on [0,1], then F^{-1}(U) follows the Cantor distribution. This explains its singular continuous nature, concentrating total mass 1 on the uncountable support C of Lebesgue measure zero.^[2]

Moments and Cumulants

Mean and variance

The mean of the Cantor distribution is \frac{1}{2}, arising from the symmetry of its iterative construction on [0, 1], which is invariant under the reflection x \mapsto 1 - x.^[29] This symmetry centers the distribution at the midpoint of the support interval and also implies zero skewness.^[29] The variance is \frac{1}{8}.^[29] It can be derived using the self-similarity of the distribution or, equivalently, via its characterization as the law of X = \sum_{k=1}^{\infty} Z_k 3^{-k}, where the Z_k are i.i.d. taking values 0 and 2 each with probability \frac{1}{2}. In the latter approach, the independence of the terms gives

\mathrm{Var}(X) = \sum_{k=1}^{\infty} \mathrm{Var}\left( \frac{Z_k}{3^k} \right) = \sum_{k=1}^{\infty} \frac{\mathrm{Var}(Z_k)}{9^k}.

Since \mathrm{Var}(Z_k) = 1,

\mathrm{Var}(X) = \sum_{k=1}^{\infty} \frac{1}{9^k} = \frac{1/9}{1 - 1/9} = \frac{1/9}{8/9} = \frac{1}{8}.

^[29] This value quantifies the spread, capturing the repeated scaling by \frac{1}{3} and equal mass partition of \frac{1}{2} in the construction.

Higher moments

The Cantor distribution is symmetric about its mean of \frac{1}{2}, which implies that all odd-order central moments vanish: \mu_{2k+1} = 0 for every integer k \geq 0.^[30] The even-order central moments \mu_{2k} satisfy a recursive relation arising from the self-similar construction of the distribution, where a random variable X with the Cantor distribution satisfies X \stackrel{d}{=} \frac{U + X'}{3} with U independent of X' (another independent copy of X) and U taking values 0 and 2 each with probability \frac{1}{2}. This yields the recursion

(3^{m} - 1) \, E[X^{m}] = \sum_{j=1}^{m} \binom{m}{j} 2^{j-1} \, E[X^{m-j}]

for m \geq 1, with E[X^{0}] = 1; since odd moments are zero, the relation simplifies for even m = 2k.^[30] The moments can also be computed using the series representation X = \sum_{k=1}^{\infty} Y_{k}, where the Y_{k} = Z_{k}/3^{k} are independent and each Z_{k} takes values 0 and 2 with equal probability \frac{1}{2}. Independence implies that the m-th raw moment is

E[X^{m}] = \sum_{\alpha \in \mathbb{N}^{\infty} : |\alpha| = m} \frac{m!}{\alpha !} \prod_{k=1}^{\infty} E[Y_{k}^{\alpha_{k}}],

where the sum is over multi-indices \alpha = (\alpha_{1}, \alpha_{2}, \dots ) with finitely many nonzero entries summing to m, and E[Y_{k}^{\alpha_{k}}] = 2^{\alpha_{k}-1} / 3^{k \alpha_{k}} for \alpha_{k} \geq 1 (and 1 for \alpha_{k} = 0). Central moments follow by recentering at the mean.^[30] The cumulants of the Cantor distribution provide an alternative characterization, with the first cumulant \kappa_{1} = \frac{1}{2} (the mean) and the second \kappa_{2} = \frac{1}{8} (the variance). All odd cumulants beyond the first vanish due to symmetry, while the higher even cumulants are

\kappa_{2n} = \frac{2^{2n-1} (2^{2n} - 1) B_{2n}}{n (3^{2n} - 1)}, \quad n \geq 2,

where B_{2n} denotes the $2n-th Bernoulli number.^[31] These cumulants reveal the non-Gaussian nature of the tails, as evidenced by the excess kurtosis \frac{\kappa_{4}}{\kappa_{2}^{2}} = -\frac{8}{5}, indicating a platykurtic distribution with lighter tails than the normal.^[31]

Variants of Cantor distributions

Variants of the Cantor distribution arise from modifications to the iterative construction of the standard Cantor set, leading to singular continuous measures supported on generalized Cantor sets with altered geometric and probabilistic properties. These variants maintain the singular continuous nature but exhibit skewness, positive Lebesgue measure supports, or higher-dimensional structures.^[32] Asymmetric variants are constructed by removing unequal intervals during the iterative process, such as the middle-α portion where 0 < α < 1/2, instead of the symmetric middle-third. This yields a skewed singular continuous measure μ_{α,w} supported on the middle-α Cantor set Γ_3(α), where w denotes unequal weights for the self-similar branches. The cumulative distribution function forms a generalized devil's staircase, continuous and non-decreasing, but constant on the removed gaps and increasing only on the support set. For α = 1/3, it recovers the standard Cantor distribution; deviations introduce asymmetry, with the Hausdorff dimension \frac{\log 2}{\log \frac{2}{1-\alpha}} influencing the measure's concentration.^[32] Fat Cantor sets, such as the Smith-Volterra-Cantor (SVC) set, are constructed by removing progressively smaller open intervals from [0,1] while preserving positive Lebesgue measure, typically 1/2 for the standard SVC. A singular continuous distribution on the SVC set is defined via an analogous devil's staircase function G(x), built iteratively as the limit of piecewise linear approximations that are constant on removed intervals and increase monotonically on the remaining closed set. This function is continuous, strictly increasing, but singular with respect to Lebesgue measure, having derivative zero almost everywhere despite the support's positive measure. The SVC function is Lipschitz continuous and exhibits one-sided derivatives, distinguishing it from absolutely continuous measures on sets of positive measure.^[33] Multidimensional analogs include product measures on Cantor dusts, formed as the Cartesian product of one-dimensional Cantor sets, resulting in zero Lebesgue measure supports in higher dimensions. The uniform distribution on a Cantor dust in ℝ^d is the d-fold product of the standard Cantor distribution, yielding a singular continuous measure with Hausdorff dimension d · log(2)/log(3). Salem sets provide further generalizations, constructed as randomized or explicit higher-dimensional Cantor-like sets where the Hausdorff and Fourier dimensions coincide, supporting singular measures with full Fourier dimension for enhanced analytic properties. These sets, often built via iterated function systems with variable contractions, extend the Cantor distribution's singularity to dimensions s ∈ (0,d). Parameterized families, such as the generalized Cantor distribution GC(α, β) for β ∈ (0,1/2) and α ∈ (0,1), are defined as the law of X = ∑_{n=1}^∞ β^{n-1} U_n, where U_n are i.i.d. Bernoulli(α). This self-similar measure is singular continuous on [0,1], with the parameter β controlling the contraction ratio and thus the Hausdorff dimension of the support \frac{\log 2}{\log (1/\beta)}, allowing variation from the standard case (α=1/2, β=1/3). Adjusting β varies the support's dimension continuously between 0 and 1, while ensuring singularity when β < 1/2 to avoid overlaps.^[32]

Applications in probability and analysis

The Cantor distribution serves as a prototypical example of a singular continuous measure in measure theory, illustrating the failure of the Radon-Nikodym theorem to guarantee the existence of a density with respect to Lebesgue measure. Specifically, while the distribution is absolutely continuous with respect to itself, it is mutually singular with Lebesgue measure on [0,1], as its support lies entirely on the Cantor set of Lebesgue measure zero, yet it assigns positive measure to this set. This property highlights the distinction between absolute continuity and singularity, where no Radon-Nikodym derivative exists relative to Lebesgue measure.^[34]^[35] In the Lebesgue decomposition theorem, the Cantor distribution exemplifies the singular continuous component, decomposing into parts that are neither absolutely continuous nor purely atomic with respect to Lebesgue measure. It demonstrates how a measure can be continuous (no point masses) yet singular, providing a concrete test case for the theorem's assertion that any σ-finite measure admits such a unique decomposition. This makes it valuable for verifying the theorem's scope and limitations in finite-dimensional spaces.^[34]^[35] In probability theory, the Cantor distribution arises as the case \lambda = 1/3 of the infinite Bernoulli convolution, defined as the law of \sum_{n=1}^\infty X_n / 3^n where X_n are i.i.d. Bernoulli(1/2) random variables taking values in \{0,2\}. This model is used to study transitions between singular and absolutely continuous measures, particularly in the Erdős problem, which conjectures singularity for certain algebraic \lambda (like reciprocals of Pisot numbers) and absolute continuity for Lebesgue-almost every \lambda \in (1/2, 1). The Cantor case confirms singularity due to non-overlapping iterated function system branches, serving as a baseline for analyzing overlap conditions that lead to density existence in higher \lambda regimes.^[36]^[37] These convolutions inform broader questions on the dimension of exceptional sets where singularity persists; for instance, the set of \lambda \in (1/2,1) yielding singular Bernoulli measures has Hausdorff dimension zero, underscoring the rarity of singularity beyond the Cantor-like cases.^[37] In fractal analysis, the Cantor distribution is a self-similar measure generated by the iterated function system with contractions x \mapsto x/3 and x \mapsto x/3 + 2/3, each with equal probability $1/2. Its Hausdorff dimension equals the similarity dimension \log 2 / \log 3 \approx 0.6309, achieved without overlaps, and it supports studies of dimension preservation under projections and intersections in dynamical systems. For example, in random walks on self-similar sets, it models invariant measures whose dimension equals the Lyapunov dimension H(p)/\lambda(p), where H(p) is the entropy and \lambda(p) the exponent, aiding analysis of ergodic properties and dimension drops in non-trivial overlaps.^[38] Such measures appear in thermodynamic formalisms for dynamical systems, where the Cantor distribution's exact dimensionality informs the Hausdorff measure of attractors in expanding maps.^[38] Numerically and educationally, the Cantor distribution acts as a counterexample to the intuition that continuous distributions on \mathbb{R} admit probability density functions, as its cumulative distribution function—the Cantor function—is continuous and strictly increasing but has derivative zero almost everywhere. This illustrates the existence of singular continuous random variables, emphasizing that continuity of the CDF does not imply absolute continuity or a density in teaching measure-theoretic probability.^[39]^[24] For simulations, samples can be generated via the ternary expansion: for each realization, independently draw bits B_n \in \{0,1\} with equal probability and set X = \sum_{n=1}^\infty 2 B_n / 3^n, approximating the infinite sum with finite truncations for Monte Carlo methods in fractal geometry or dimension estimation. This approach leverages the distribution's self-similar structure for computational exploration of its properties.^[24]

References

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