In this section we study an interesting collection of graphs on \(\R^n\) that are induced (in the sense of Section 1.7 ) by the standard graph \(([0, \infty), \le)\) and the norm functions. For \(n \in \N_+\), we give \(\R^n\) the usual Borel \(\sigma\)-algebra \(\ms R^n\) and \(n\)-dimensional Lebesgue measure \(\lambda^n\) as the reference measure. For \(m \in \N\) and \(m \le n\), we also use \(\lambda^m\) to denote Lebesgue measure (or more technically Hausdorff measure) on an \(m\)-dimensional Riemannian (smooth) manifold of \(\R^n\). In particular, recall that \(\lambda^0\) is counting measure.
Recall the standard norms on \(\R^n\) for \(n \in \N_+\):
When \(n = 1\), the norms are all the same: \(\|x\|_k = |x|\) for \(k \in [1, \infty]\) and \(x \in \R\). When \(n \in \{2, 3, \ldots\}\), the norms are all different, of course. For \(n \in \N_+\) and \(k \in [1, \infty]\), the mapping \(\|\cdot\|_k\) is a measurable function from \(\R^n\) onto \([0, \infty)\).
For \(n \in \N_+\) and \(k \in [1, \infty]\), let \((\R^n, \Rta_k)\) denote the graph induced by \(\|\cdot\|_k\) and the standard graph \(([0, \infty), \le)\), so that \(\bs x \Rta_k \bs y\) if and only if \(\|\bs x\|_k \le \|\bs y\|_k\) for \(\bs x, \, \bs y \in \R^n\).
Note that the relation \(\Rta_k\) is reflexive and transitive.
For \(n \in \N_+\) and \(k \in [1, \infty]\), define
In this section \(\Gamma\) denotes the ordinary gamma special function not the path generating function (which will be given in ). The following result is well known (see for example the paper by Slavik).
Volumes of the unit ball.
Note that \( \lim_{k \to \infty} b_{n,k} = b_{n, \infty}\), justifying the notation.
For \(n \in \N_+\), \(k \in [1, \infty]\), and \(t \in [0, \infty)\),
As functions of \(t \in [0, \infty)\), the surface area is a multiple of the derivative of the volume, or equivalently, for \(n \in \N_+\) and \(k \in [1, \infty]\) there exists \(a_{n,k} \in (0, \infty)\) such that \(c_{n,k} = n a_{n,k} b_{n,k}\)
For most values of \(n \in \N_+\) and \(k \in [1, \infty]\), there is no simple closed formula for \(a_{n,k}\), but there are a few exceptions:
Special cases:
The basic assumption in Section 1.7 is satisfied:
For \(n \in \N_+\) and \(k \in [1, \infty]\), let \(\R^n_k(t) = \{\bs x \in \R^n: \|x\|_k = t\}\), the surface of the ball of radius \(t \in [0, \infty)\) relative to the \(k\) norm, and the inverse image at \(t\) of the mapping \(\bs x \mapsto \|\bs x\|_k\) from \(\R^n\) onto \([0, \infty)\). Then \[\lambda^n(A) = \int_0^\infty \frac{1}{a_{n, k}} \lambda^{n - 1}[A \cap \R^n_k(t)] dt, \quad A \in \ms R^n\]
This is essentially the co-area formula for computing \(\lambda^n(A)\). Again in the notation Section 1.7, note also that for \(n \in \N_+\) and \(k \in [1, \infty]\), the function \(\beta_{n,k}\) is given by \[\beta_{n,k}(t) = \frac{1}{a_{n,k}} \lambda^{n-1}[\R^n_k(t)] = \frac{c_{n,k}}{a_{n,k}} t^{n-1} = n b_{n,k} t^{n-1}, \quad t \in [0, \infty)\] Going forward, we will surpress the dependence on \(n\) and \(k\) except when necessary. Our first result is the path function.
The path function \(\gamma_m\) of order \(m \in \N_+\) for the graph \((\R^n, \Rta_k)\) is given by \[\gamma_m(\bs x) = \frac{b^m_{n,k}}{m!} \|\bs x\|^{m n}_k, \quad \bs x \in \R^n\]
This follows from \(\gamma_0 = 1\) and the recursion relation
\[\gamma_{m+1}(\bs x) = \int_{\bs y \Rta \bs x} \gamma_m(\bs y) \, d\lambda^n(\bs y), \quad \bs x \in \R^n\]
The integral can be evaluated using the co-area formula in . But there is also a simple combinatorial
argument. For \(\bs x \in \R^n\), the measure of the set
\[\{(\bs y_1, \bs y_2, \ldots \bs y_m): \bs y_i \in \R^n, \, \|\bs y_i\|_k \le \|\bs x\|_k \text{ for each } i \in \{1, 2, \ldots, m\}\}\]
is of course \(b^m_{n,k} \|\bs x\|_k^{m n}\). But another algorithm for constructing this set is to select a path \((\bs x_1, \bs x_2, \ldots, \bs x_m)\) of length \(m\) ending in \(\bs x\) for the graph \((\R^n, \Rta_k)\) and then select an ordering \((\bs y_1, \bs y_2, \ldots, y_m)\) of \((\bs x_1, \bs x_2, \ldots \bs x_m)\). By definition, the measure of the set of paths is \(\gamma_m(\bs x)\), and there are \(m!\) permutations of each path, so it follows that
\[m! \gamma_m(\bs x) = b^m_{n,k} \|\bs x\|_k^{m n}\]
Note that \(\gamma_m(\bs x)\) is the path function \(t \mapsto t^m / m!\) of order \(m\) for the standard graph \(([0, \infty), \le)\), evaluated at \(b_{n,k} \|\bs x\|_k^n\) which is the volume of the ball of radius \(\|\bs x\|_k\) in the \(k\) norm. We will see this pattern repeatedly. In particular, it follows that the generating function of \((\R^n, \Rta_k)\) is the generating function of \(([0, \infty), \le)\) evaluated at \(b_{n,k} \|\bs x\|_k^n\):
The path generating function \(\Gamma_0\) of \((\R^n, \Rta_k)\) is given by \[\Gamma_0(\bs x, t) = \exp\left(b_{n,k} \|\bs x\|^n_k t\right), \quad \bs x \in \R^n, \, t \in \R\]
This follows easily from the definition: \[\Gamma_0(\bs x, t) = \sum_{m=0}^\infty \gamma_m(\bs x) t^m, \quad \bs x \in \R^n, \, t \in \R\]
Suppose now that \(\bs X = (X_1, X_2, \ldots, X_n)\) is a random variable in \(\R^n\), so that \(\|\bs X\|_k\) is the corresponding induced variable in \([0, \infty)\).
Basic functions:
We assume that \(\hat f(t) \gt 0\) for almost all \(t \in [0, \infty)\), so that \(\hat F(t) \gt 0\) for almost all \(t \in [0, \infty)\) and \(F(\bs x) \gt 0\) for almost all \(\bs x \in \R^n\). For \(t \in [0, \infty)\), the conditional distribution of \(\bs X\) given \(\|\bs X\|_k = t\) has density function \(\bs x \mapsto f(\bs x \mid t) = f(\bs x) / \hat f(t)\) on \(\R^n_k(t)\). The general results of Section 1.7 apply, of course, but as usual we are most interested in constant rate distributions. Here are the main results:
For \(n \in \N_+\) and \(k \in [1, \infty]\), the graph \((\R^n, \Rta_k)\) has a unique distribution with constant rate \(\alpha\) for each \(\alpha \in (0, \infty)\). If random variable \(\bs X\) has this distribution then
The results follow from Section 1.7. Specifically, \(\bs X\) has constant rate \(\alpha \in (0, \infty)\) for \((\R^n, \Rta_k)\) if and only if \(\|\bs X\|_k\) has rate function \(\hat r\) for \(([0, \infty), \le)\) where \(\hat r(t) = \alpha \beta_{n,k}(t) = \alpha n b_{n,k} t^{n-1}\) for \(t \in [0, \infty)\). So \(\|\bs X\|_k\) has reliability function \(\hat F\) for \(([0, \infty), \le)\) given by \[\hat F(t) = \exp\left(-\int_0^t \hat r(s) \, ds\right) = \exp\left(-\alpha b_{n,k} t^n\right), \quad t \in [0, \infty)\] which we recognize as the Weibull distribution in (a). And then \(\bs X\) has reliability function \(F\) for \((\R^n, \Rta_k)\) given by \(F(\bs x) = \hat F(\|\bs x\|_k)\) for \(\bs x \in \R^n\), and has density function \(f\) given by \(f(\bs x) = \alpha \hat F(\|\bs x\|_k)\) for \(\bs x \in \R^n\).
Note that \(f(\bs x)\) is the ordinary exponential density function for the standard graph \(([0, \infty), +)\), with rate \(\alpha\), evaluated at \(b_{n,k} \|\bs x\|_k^n\), which again is the volume of the ball of radius \(\|\bs x\|_k\) in the \(k\) norm.
Let \(n \in \N_+\) and \(k \in [1, \infty]\), and suppose that \(\bs X = (X_1, X_2, \ldots, X_n)\) has the distribution with constant rate \(\alpha \in (0, \infty)\) for the graph \((\R^n, \Rta_k)\).
These results follow from the density function in part (b) of Proposition .
Let \(n \in \N_+\) and \(k \in [1, \infty]\) and consider the family of constant rate distributionn on \((\R^n, \Rta_k)\).
Suppose again that \(\bs X\) has the distribution with constant rate \(\alpha \in (0, \infty)\) for the graph \((\R^n, \Rta_k)\). Then \(\bs X\) maximizes entropy over all random variables \(\bs Y\) in \(\R^n\) with \(\E(\|\bs Y\|_k^n) = 1 / (\alpha b_{n,k})\).
The case where the norm index is the same as the dimension is particularly interesting. In this case, abbreviate \(b_{n,n}\) by \(b_n\) so that \[b_n = \left[\frac{2}{n} \Gamma\left(\frac{1}{n}\right)\right]^n = 2^n \Gamma^n\left(1 + \frac{1}{n}\right), \quad n \in \N_+\] where again \(\Gamma\) is the ordinary gamma special function.
Let \(n \in \N_+\) and suppose that \(\bs X = (X_1, X_2, \ldots, X_n)\) has the distribution with constant rate \(\alpha \in (0, \infty)\) for the graph \((\R^n, \Rta_n)\). Then \(\bs X\) is a sequence of independent, identically distributed variables with common density function \(g\) on \(\R\) define by \[g(x) = \alpha^{1/n} \exp(-\alpha b_n |x|^n), \quad x \in \R\]
The distributions on \(\R\) defined by the density function in (b) are generalized normal distributions (see the papers by Goodman and by Sinz et al.) The distribution actually makes sense for any \(n \in (0, \infty)\), not just positive integers, and forms an interesting class of special distributions. We will use terminology that is appropriate for our setting:
For \(n, \, \alpha \in (0, \infty)\), the distribution defined in is the generalized normal distribution with index \(n\) and rate \(\alpha\).
For \(n = 1, \, 2\) the generalized normal distributions reduce to standard special distributions:
Suppose that \(X\) has the generalized normal distribution on \(\R\) with index \(n \in (0, \infty)\) and rate \(\alpha \in (0, \infty)\).
As in , \(\alpha^{-1 / n}\) is a scale parameter of the family of distributions. That is, if \(Z\) has the standard distribution with index \(n\) and rate \(1\), and with density function \(z \mapsto e^{-b_n |z|^n}\) then \(X = \alpha^{-1 / n} Z\) has the distribution with index \(n\) and rate \(\alpha\) and with density function \(g\) in . The following theorems gives some basic properties:
Let \(g\) denote the density function of the generalized normal distribution on \(\R\) with index \(n \in (0, \infty)\) and rate \(\alpha \in (0, \infty)\), given in .
Part (a) is clear. Parts (b) and (c) follow from standard calculus.
Note that the inflection points in (c) converge to \(\pm 1 / 2\) as \(n \to \infty\).
Suppose again that \(X\) has the generalized normal distribution on \(\R\) with index \(n \in (0, \infty)\) and rate \(\alpha \in (0, \infty)\), with density function \(g\).
The results follow from known results for the generalized normal distribution, but we will give separate proofs for completeness. Suppose that \(Z\) has the standard distribution with index \(n\) and rate \(1\) so that \(X = \alpha^{-1 / n}\) has the generalized normal distribution with index \(n\) are rate \(\alpha\).
In particular, the variance and kurtosis are given by \[\var(X) = \alpha^{-2 / n} \left(\frac{n}{2}\right)^2 \frac{\Gamma(3 / n)}{\Gamma^3(1 / n)}, \; \kurt(X) = \frac{\Gamma(5 / n) \Gamma(1 / n)}{\Gamma^2(3 / n)}, \quad n \in \N_+\] This class of distributions on \(\R\) is interesting because it generalizes the Laplace, the 0 mean normal, and because of the surprising convergence result in part (d). Since the distribution of \(X\) is symmetric about \(0\), it also follows that \(|X|\) has density function \(2 g\) on \([0, \infty)\), \(\sgn(X)\) is uniformly distributed on \(\{-1, 1\}\), and that \(|X|\) and \(\sgn(X)\) are independent.
Open the simulation of the generalized normal distribution. Vary \(n\) and \(\alpha\) with the scrollbars and note the shape and location of the probability density function. For various values of the parameters, run the simulation and compare the empirical density function to the probability density function.
The (ordinary) distribution function can be expressed in terms of the gamma and (upper) incomplete gamma functions, but is not particularly helpful:
Suppose again that \(X\) has the generalized normal distribution on \(\R\) with index \(n \in (0, \infty)\) and rate \(\alpha \in (0, \infty)\). The distribution function \(G\) is given by \[G(y) = 1 - \frac{\Gamma\left[1 / n, 2^n y^n \Gamma^n(1 + 1 / n)\right]}{2 \Gamma(1 / n)}, \quad y \ge 0\] and by symmetry, \(G(y) = 1 - G(-y)\) for \(y \lt 0\).
We can rephrase parts (a) and (b) of :
Constant rate distributions for \(n = 1, \, 2\).
Open again the simulation of the generalized normal distribution. Set \(n = 1\) and \(n = 2\) and note the shape of the graph. Run the simulation in both cases and compare the empirical density function to the probability density function.
Part (d) of suggests that the uniform distribution on \([-\frac 1 2, \frac 1 2]\) has constant rate for the graph corresponding to \(n = \infty\). That is indeed true, properly understood. Let \(I = [-\frac 1 2, \frac 1 2]\) and consder \(I^\infty\) with norm \[\|\bs x\|_\infty = \sup\{|x_i|: i \in \N_+\}, \quad \bs x = (x_1, x_2, \ldots) \in I^\infty\] The graph \((I^\infty, \Rta_\infty)\) induced by the function \(\bs x \mapsto \|\bs x\|_\infty\) and the standard graph \(([0, \infty), \le)\) is defined in the usual way: \(\bs x \Rta_\infty \bs y\) if and only if \(\|\bs x\|_\infty \le \|\bs y\|_\infty\) for \(\bs x, \, \bs y \in I^\infty\). In terms of the measure structure, we give \(I\) the usual Borel \(\sigma\)-algebra \(\ms I\) and \(I^\infty\) the corresponding product \(\sigma\)-algebra \(\ms I^\infty \), namely the \(\sigma\)-algebra generated by sets of the form \(\prod_{i = 1}^\infty A_i\) where \(A_i \in \ms I\) for each \(i \in \N_+\) and \(A_i = I\) for all but finitely many \(i \in \N_+ \). Finally, for the reference measures, let \(\lambda\) denote Lebesgue measure on \((I, \ms I)\) (which is simply the uniform probability distribution) and let \(\lambda^\infty\) the corresponding product measure on \((I^\infty, \ms I^\infty)\).
Suppose that \(\bs X = (X_1, X_2, \ldots)\) is a sequence of independent variables, each uniformly distributed on \([-\frac 1 2, \frac 1 2]\). Then \(\bs X\) has constant rate 1 for the graph \((I^\infty, \Rta_\infty)\).
Random variable \(X_i\) has constant density function \(1\) on \(I\) and similarly \(\bs X\) also has constant density function \(1\) on \(I^\infty\). On the other hand, the reliability function \(F\) of \(\bs X\) for \((I^\infty, \Rta_\infty)\) is \[F(\bs x) = \P(\bs x \Rta_\infty \bs X) = \P(\|\bs x\|_\infty \le \|\bs X\|_\infty), \quad \bs x \in I^\infty\] But \(\P(\|\bs X\|_\infty = \frac 1 2) = 1\). To see this, note that if \(\epsilon \gt 0\) then with probability \(1\), \(|X_i| \gt \frac 1 2 - \epsilon\) for some (and in fact infinitely many) \(i \in \N_+\). Hence, \(F\) is also the constant function 1 on \(I^\infty\) so \(F = f\).
Open again the simulation of the generalized normal distribution. Starting with \(n = 1\), increase \(n\) to the maximum value and note the change in shape of the graph.
There are two interesting and tractible cases in the bivariate setting (\(n = 2\)) corresponding to the extreme values of the norm parameter (\(k = 1\) and \(k = \infty\)). In both cases, various distribution functions are best expressed in terms of the error function and its complement.
The error function \(\erf\) is defined by \[ \erf(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} \, dt, \quad x \in \R \] and the complementary error function is \(\erfc = 1 - \erf\).
The error function is closely related to the standard normal distribution function \(\Phi\): \[ \Phi(x) = \frac 1 2 + \frac 1 2 \erf\left(\frac{x}{\sqrt{2}}\right), \quad x \in \R \]
Suppose that \((X, Y)\) has the distribution with constant rate \(\alpha \in (0, \infty)\) for the graph \((\R^2, \Rta_1)\). Then \((X, Y)\) has density function \(f\) given by \[f(x, y) = \alpha \exp[-2 \alpha (|x| + |y|)^2], \quad (x, y) \in \R^2\]
Of course, the density is constant on the the squares \(|x| + |y| = t \) for \(t \in (0, \infty)\) in contrast to a bivariate nomral density, for example, which is constant on ellipses. From the general results in Proposition , \(X\) and \(Y\) are uncorrelated and identically distributed, and the distribution is symmetric about 0. So in particular, the variables have mean 0 and skewness 0. The family of distributions is a scale family with scale parameter \(1 / \sqrt{\alpha}\).
Suppose again that \((X, Y)\) has the distribution with constant rate \(\alpha \in (0, \infty)\) for the graph \((\R^2, \Rta_1)\). The common density \(g\) of \(X\) and \(Y\) is given by \[g(x) = \sqrt{\frac{\pi \alpha}{2}} \erfc\left(\sqrt{2 \alpha} |x|\right), \quad x \in \R \] The variance is \(1 / 6 \alpha\) and the kurtosis is \(18 / 5\).
The density function \(g\) in has the following properties:
This follows from standard calculus:
So this distribution might be considered a second-order version of the Laplace distribution.
Open the simulation of the norm distribution with \(n = 2\) and \(k = 1\). Note the shape and location of the probability density function and the mean \(\pm\) standard deviation bar. Run the simulation 1000 times and compare the empirical density and moments with the distribution density and moments.
Suppose again that \((X, Y)\) has the distribution with constant rate \(\alpha \in (0, \infty)\) for the graph \((\R^2, \Rta_1)\). The common distribution function \(G\) of \(X\) and \(Y\) is given by \[G(y) = 1 - \frac{1}{2} \exp\left(-2 \alpha y^2\right) + \sqrt{\frac{\alpha \pi}{2}} y \, \erfc\left(\sqrt{2 \alpha} \, y\right), \quad y \in [0, \infty)\] and \(G(y) = 1 - G(-y)\) for \(y \in (-\infty, 0)\).
Suppose that \((X, Y)\) has the distribution with constant rate \(\alpha \in (0, \infty)\) for the graph \((\R^2, \Rta_\infty)\). Then \((X, Y)\) has joint density function \(f\) given by \[f(x, y) = \alpha \exp\left[-4 \alpha \left(|x| \vee |y|\right)^2\right], \quad(x, y) \in \R^2\]
This density is constant on the squares \(|x| \vee |y| = t\) for \(t \in (0, \infty)\). As in the general case, \(X\) and \(Y\) are uncorrelated (but dependent) with identical distributions that are symmetric about 0. So in particular, the variables have mean 0 and skewness 0.
Suppose again that \((X, Y)\) has the distribution with constant rate \(\alpha \in (0, \infty)\) for the graph \((\R^2, \Rta_\infty)\). Then \(X\) and \(Y\) have common density function \(g\) given by \[g(x) = 2 \alpha |x| \exp\left(-4 \alpha x^2\right) + \frac{\sqrt{\pi \alpha}}{2} \erfc\left(2 \sqrt{\alpha} |x|\right), \quad x \in \R \] The common variance is \(1 / 6 \alpha\) and the common kurtosis is \(27 / 10\).
The density function \(g\) in has the following properties:
This follows from standard calculus:
It's interesting that the variance is the same as in the case \(n = 2\), \(k = 1\).
Open the simulation of the norm distribution with \(n = 2\) and \(k = \infty\). Note the shape and location of the probability density function and the mean \(\pm\) standard deviation bar. Run the simulation 1000 times and compare the empirical density and moments with the distribution density and moments.
Suppose again that \((X, Y)\) has the distribution with constant rate \(\alpha \in (0, \infty)\) for the graph \((\R^2, \Rta_\infty)\). The common distribution function \(G\) of \(X\) and \(Y\) is given by \[G(y) = 1 - \frac{1}{2} \exp\left(-4 \alpha y^2\right) + \frac{\sqrt{\alpha \pi}}{2} y \, \erfc\left(2 \sqrt{\alpha} \, y\right), \quad y \in [0, \infty)\] and \(G(y) = 1 - G(-y)\) for \(y \in (-\infty, 0)\).
Returning to the general case of \(n \in \N_+\) and \(k \in [1, \infty]\), consider the random walk \((\bs Y_1, \bs Y_2, \ldots)\) on \((\R^n, \Rta_k)\) associated with \(\bs X\), where \(\bs X\) has constant rate \(\alpha \in (0, \infty)\) for \((\R^n, \Rta_k)\). From our general results and the path function in we have
For \(m \in \N_+\), random variable \(\bs Y_m\) has density function \(f_m\) given by \[f_m(\bs x) = \alpha^m \frac{1}{(m - 1)!} (b_{n,k} \|\bs x\|^n_k)^{m - 1} \exp\left(-\alpha b_{n,k} \|x\|^n_k\right), \quad \bs x \in \R^n\]
Note that \(f_m(\bs x)\) is the ordinary gamma density function for the standard graph \(([0, \infty), \le)\), (with order \(m\) and rate \(\alpha\)), evaluated at \(b_{n,k} \|\bs x\|_k^n\).