Gerhard Tintner, The Distribution of the Variances of Variate Differences in the Circular Case (1955)

Tintner’s mathematical statistics article derives the sampling distribution of the variance of a finite-difference series under circular boundary conditions. Its central move is to replace the difficult endpoint structure of ordinary differencing with a periodic arrangement: observations whose indices differ by the sample size are identified, so differences wrap around the series. This makes the relevant quadratic form circulant and permits an explicit characteristic function, cumulants, and distributional formulae. The article proceeds from this algebraic simplification to large-sample approximation, then to exact inversion for even and odd sample sizes, concluding with two numerical examples.

The starting point is a normal population with a common variance:

The individual items of the population are normally distributed with mean zero and variance $\sigma^2$.

The displayed joint-density expression indicates the intended independent Gaussian framework, although its transcription is malformed. Tintner defines (V_k) from the sum of squared circular (k)-th differences, normalized by the sample size and the central binomial coefficient (\binom{2k}{k}). This normalization relates the dispersion of the differenced observations to the variance of the original population. Circularity does not make overlapping differences independent; instead, it supplies a symmetry through which their dependence can be handled.

The first part of the derivation expands the squared differences into sums of squares and products of observations at successive lags. Combining this quadratic form with the Gaussian density reduces the characteristic function to the inverse square root of a determinant. Because the matrix is circulant, its determinant factors through the (N)-th roots of unity. Tintner then converts the paired complex terms into cosines and uses a trigonometric identity to express the resulting weights through powers of (\sin(\pi j/N)). Equation (12), the principal distributional result, is consequently a product over circular frequencies. The conceptual achievement is to turn a statistic built from dependent differences into a tractable product of frequency-specific factors.

Taking the logarithm of that product gives the cumulant-generating expression. Tintner differentiates it and presents a general cumulant formula, followed by the first four cumulants and the corresponding central moments. The first cumulant supplies the estimator’s basic justification:

This shows that the mathematical expectation of $V_k$ is $\sigma^2$, and hence $V_k$ is unbiased.

Unbiasedness here means that the normalized difference statistic recovers the population variance in expectation, not that its finite-sample distribution resembles that of an ordinary sample variance. The higher cumulants describe the remaining sampling uncertainty. In the formulae presented, the variance decreases inversely with sample size, while the third and fourth cumulants decrease more rapidly. Their dependence on the order of differencing remains explicit through ratios of central binomial coefficients.

The discussion then passes from moments to distributional shape. Tintner gives two measures of skewness and two of kurtosis, using their large-sample limits to motivate a normal approximation:

As the sample size $N$ increases we see that $\beta_2$ tends to 3 and $\gamma_2$ to zero. Hence for large samples $V_k$ approximates the normal distribution with mean $\sigma^2$ and variance (18).

The approximation preserves the variance appropriate to the differenced statistic; it does not substitute the variance of a conventional estimator. This distinction links the asymptotic argument to the preceding exact calculation. The normal law is offered as a practical simplification whose adequacy depends on sample size.

For finite samples, Tintner inverts the characteristic function, separating even from odd (N). The distinction arises because circular frequencies generally occur in pairs, but an even-sized sample has an additional unpaired frequency. For odd (N), the factors reduce to simple inverse linear terms, whose Fourier inverses are exponential densities. For even (N), a remaining square-root factor introduces an error-function expression. Partial-fraction expansion then produces explicit finite sums. Although the notation (F_1) and (F_2) is described as giving “the distribution,” the inversion formulae represent density functions rather than cumulative probabilities.

The article’s practical relevance extends beyond the circular model, but Tintner states that extension cautiously:

Since the sample variances of finite differences have a very complicated distribution in the non-circular case it is believed that we may use the above formulae as approximations even in the non-circular case, if $N$ is somewhat large and the effect of the assumption of circularity small.

Circularity is thus both an exact mathematical assumption and a possible approximation strategy. The latter claim is conditional, not accompanied by an error bound. Its plausibility rests on endpoint effects becoming relatively unimportant in sufficiently long series.

The concluding examples take second differences with population variance one, first for (N=7), then for (N=8). They instantiate the odd- and even-sample formulae and compare the resulting densities with normal curves having the derived means and variances. Their role is to make the inversion procedure concrete and distinguish exact finite-sample shape from its approximation.

The supplied transcription requires checking before numerical reuse. Several equations contain visibly malformed notation, and the even-sample factor in equation (27) lacks a power-of-two multiplier present in equation (12). These problems do not obscure the article’s central method, but they prevent treating every displayed coefficient as verified. Tintner’s lasting contribution here is the circulant reduction: periodic boundaries make the sampling law of a difference-based variance accessible through spectral factorization, cumulants, and explicit inversion.