As in Section 1, the standard positive semigroup \(([0, \infty), +)\) has the usual Borel \(\sigma\)-algebra, and has Lebesgue measure \(\lambda\) as the invariant measure. Now let \(S\) be an interval of real numbers of the form \([a, b)\) where \(-\infty \lt a \lt b \leq \infty\) or of the form \((a, b]\) where \(-\infty \leq a \lt b \lt \infty\). The \(\sigma\)-algebra \(\ms S\) is the Borel \(\sigma\)-algebra of subsets of \(S\). Let \(\varphi\) be a continuous, one-to-one function from \(S\) onto \([0, \infty)\).
Define the operator \(\oplus\) on \(S\) by \[x \oplus y = \varphi^{-1}[\varphi(x) + \varphi(y)], \quad x, \, y \in S\] and define the measure \(\mu\) on \((S, \ms S)\) by \[\mu(A) = \lambda[\varphi(A)], \quad A \in \ms S\] With \((S, \ms S, \mu)\) as the underlying measure space, the positive semigroup \((S, \oplus)\) is isomorphic to \(([0, \infty), +)\), and \(\varphi\) is an isomorphism. The endpoint of \(S\) is the identity. The order \(\preceq\) associated with \((S, \oplus)\) is the ordinary order \(\le\) if \(S = [a, b)\) and the reverse \(\ge\) if \(S = (a, b]\).
From the assumptions on \(\varphi\), the inverse function \(\varphi^{-1}\) is also continuous and maps \([0, \infty)\) one-to-one onto \(S\), so that \(\varphi\) is a topological homeomorphism. The operation \(\oplus\) makes sense and is measurable. In particular, \[\varphi(x \oplus y) = \varphi(x) + \varphi(y), \quad x, \, y \in S\] Note that \(\varphi\) is strictly increasing if \(S = [a, b)\) with \(\varphi(a) = 0\), and \(\varphi\) is strictly decreasing if \(S = (a, b]\) with \(\varphi(b) = 0\).
We use \(\oplus\) for the operator to emphasize the isomorphism with the standard space \(([0, \infty), +)\) and in particular because the operator is commutative. If \(x, y \in S\) and \(x \preceq y\) then \[y \ominus x = \varphi^{-1}[\varphi(y) - \varphi(x)]\] replacing our generic notation \(x^{-1} y\). We can extend the isomorphism to products by the rule \(c \odot x = \varphi^{-1}[c \varphi(x)]\) for \(x \in S\) and \(c \in [0, \infty)\). This generalizes our generic notation \(x^n\) for \(x \in S\) and \(n \in \N\). The measure \(\mu\) is invariant for \((S, \oplus)\) and is unique up to multiplcation by positive constants. In fact, \(\mu\) is the Lebesgue-Stieltjes measure associated with \(\varphi\). If \(\varphi\) is differentiabale with then \[d\mu(x) = |\varphi^\prime(x)| \, dx\] That is, \(\mu\) is absolutely continuous with respect to \(\lambda\) with density \(|\varphi^\prime|\).
For the graph \((S, \preceq, \mu)\)
The \(\sigma\)-algebra associated with \((S, \preceq)\) is the reference \(\sigma\)-algebra \(\ms S\).
This follows since the associated order is either the ordinary order \(\le\) or its reverse \(\ge\).
Suppose now that \(X\) is a random variable in \(S\) so that \(Y = \varphi(X)\) is a random variable in \([0, \infty)\). Because the spaces are isomorphic, all of the basic functions and concepts for \(X\) relative to \((S, \oplus, \mu)\) are equivalent to those of \(Y\) relative to \(([0, \infty), +, \lambda)\). The following proposition gives a summary. For clairty, we will explicitly give the structures that are necessary.
Suppose that \(X\) is a random variable in \(S\) and that \(Y = \varphi(X)\).
Ordinarily, part (d) requires a reference to the underlying measures, but in this setting we can avoid that since the cumulative rate function of \(Y\) for \(([0, \infty), \le)\) is \(R = - \ln F\) and similarly, the cumulative rate function of \(X\) for \((S, \preceq)\) is \(R \circ \varphi = (- \ln F) \circ \varphi\). In the context of (b), if \(\varphi\) is differentiable then \(X\) has density \((f \circ \varphi) |\varphi^\prime|\) with respect to Lebesgue measure \(\lambda\) on \((S, \ms S)\). The following result gives the equivalence between various aging (improvement) properties: increasing (decreasing) failure rate, increasing (decreasing) failure rate average, new better (worse) than used, and the ageless exponential property.
Suppose again that \(X\) is a random variable in \(S\) and that \(Y = \varphi(X)\).
Note that if \(S = (a, b]\) so that \(\preceq\) is the order \(\ge\), the terms increasing and decreasing in parts (a) and (b) actually mean decreasing and increasing, respectively, in terms of the ordinary order.
The graph \((S, \preceq)\) is stochastic. That is, if \(P\) and \(Q\) are probability measures on \((S, \ms S)\) with the same reliability function \(F\), then \(P = Q\).
This follows since the measurable graph \((S, \ms S, \preceq)\) is isomorphic to the standard continuous graph \(([0, \infty), \ms B, \le)\) (where \(\ms B\) is the \(\sigma\)-algebra of Borel sets). The standard continuous graph is stochastic.
As usual, exponential distributions are of special importance.
Suppose again that \(X\) is a random variable in \(S\) and that \(Y = \varphi(X)\). Then \(X\) has an exponential distribution on \((S, \oplus)\) if and only if \(Y\) has an exponential distribution on \(([0, \infty), +)\)
From , we have the following results:
Suppose again that \(X\) is a random variable in \(S\)
In particular, note that \(\alpha |\varphi^\prime|\) is the failure rate function of \(X\) for \((S, \le)\). Every continuous distribution on \(S\) is exponential relative to some semigroup.
Suppose that \(X\) is a random variable with a continuous distribution on \(S\). Then \(X\) has an exponential distribution with rate parameter 1 for the semigroup \((S, \oplus)\), corresponding to the isomorphism \(\varphi\) defined as follows:
This is a variation of a well-known result from elementary probability. For part (a), let \(F\) denote the reliability function \(F\) for \((S, \le)\) so that \(F(x) = \P(X \ge x)\) for \(x \in S\). Then \(F\) is continuous (since the distribution of \(X\) is continuous) and strictly increasing, mapping \(S\) onto \([0, 1)\) (since \(X\) is supported by \(S\)). Moreover, \(F(X)\) is uniformly distributed on \([0, 1)\). Hence, with \(Y = \varphi(X) = -\ln F(X)\) we have \[ \P(Y \ge y) = \P[-\ln F(X) \ge y] = \P[F(X) \le e^{-y}] = e^{-y}, \quad y \in [0, 1) \] The proof for part (b) is similar.
This idea will be studied further in Section 3 on relative aging. The following result follows from the discussion of entropy in Section 1.5
Suppose that \(X\) is exponential on \((S, \oplus)\) with rate parameter \(\alpha \in (0, \infty)\) and reliability function \(F\) given in . Then \(X\) maximizes entropy over all random variables \(Y\) in \(S\) with \[\E[\varphi(Y)] = \E[\varphi(X)] = 1 / \alpha\] The maximum entropy is \(1 - \ln \alpha\).
Naturally, random walks are also preserved under the isomorphism.
Suppose again that \(X\) is exponential on \((S, \oplus)\) with rate parameter \(\alpha \in (0, \infty)\). Let \(\bs {Y} = (Y_1, Y_2, \ldots)\) is the random walk on \((S, \preceq)\) or equivalently the random walk on \((S, \oplus)\) associated with \(X\). Then \((\varphi(Y_1), \varphi(Y_2), \ldots)\) is the random walk on \(([0, \infty), \le)\) or equivalently the random walk on \(([0, \infty), +)\), respectively, associated with \(\varphi(X)\). In particular,
In the next several subsections we explore a number of specific examples.
Our first example is a rather trivial modification of the standard space, but still leads to some helpful insights.
Let \(S = [a, \infty)\) where \(a \in (0, \infty)\), and define \(\varphi: [a, \infty) \to [0, \infty)\) by \(\varphi(x) = x - a\) so that \(\varphi\) is a homeomorphism from \([a, \infty)\) onto \([0, \infty)\).
For the graph \( ([a, \infty), \le) \),
Suppose now that \(X\) is a random variable in \([a, \infty)\). The reliability function \(F\) of \(X\) for \(([a, \infty), \le)\) is just the standard one, so that \(F(x) = \P(X \ge x)\) for \(x \in [a, \infty)\).
Suppose that \(X\) has the exponential distribution on \(([a, \infty), \oplus)\) with constant rate \(\beta \in (0, \infty)\).
So \(X\) has a shifted exponential distribution with parameters \(a\) and \(\beta\).
Suppose again that \(X\) is exponential for \(([a, \infty), \oplus)\) with rate parameter \(\beta \in (0, \infty)\) and let \(\bs Y = (Y_1, Y_2, \ldots)\) denote the random walk corresponding to \(X\). The density function \(f_n\) of \(Y_n\) is given by \[f_n(x) = \beta^n \frac{(x - a)^{n-1}}{(n - 1)!} e^{-\beta (x - a)}, \quad x \in [a, \infty)\]
Of course \(f_n\) is the shifted version of the standard gamma distribution with parameters \(n\) and \(\beta\). But we can also consider \([a, \infty)\) with the standard addition operator.
\(([a, \infty), +)\) is a strict positive semigroup, and the associated strict partial order \(\prec\) is given by \(x \prec y\) if and only if \(x + a \le y\).
Once again we use Lebesgue measure \(\lambda\) as the invariant reference measure on the \(\sigma\)-algebra of Borel subsets of \([a, \infty)\). Now we are outside of the setting of the general theory above, but it's interesting to compare the two spaces.
The \(\sigma\)-algebra associated with \(([a, \infty), \prec)\) is \(\ms B \cup \{[a, 2 a) \cup B: B \in \ms B\}\) where \(\ms B\) is the collection of Borel subsets of \([2 a, \infty)\).
We will prove a more general result: Suppose that \(S\) is a nonempty set, \(T\) is a nonempty proper subset of \(S\), and \(\ms A\) a collection of subsets of \(T\). Let \(\ms T\) denote the \(\sigma\)-algebra of subsets of \(T\) generated by \(\ms A\). Then the \(\sigma\)-algebra of subsets of \(S\) generated by \(\ms A\) is \[\ms T \cup \{(S - T) \cup B: B \in \ms T\}\] For the proof, let \(\ms S\) denote the collection of subsets of \(S\) in the displayed equation above. We first show that \(\ms S\) is a \(\sigma\)-algebra of subsets of \(S\). Note that \(S \in \ms S\) since \(S = (S - T) \cup T\) and \(T \in \ms T\). Next, suppose that \(A \in \ms S\). If \(A \in \ms T\) then \(S - A = (S - T) \cup (T - A) \in \ms S\) since \(T - A \in \ms T\). On the other hand, if \(A = (S - T) \cup B\) where \(B \in \ms T\) then \(S - A = (T - B) \in \ms S\) since \(T - B \in \ms T\). Next, suppose that \(A_i \in \ms S\) for \(i\) in a countable index set \(I\). If \(A_i \in \ms T\) for each \(i \in I\) then \(\bigcup_{i \in I} A_i \in \ms T\) and hence \(\bigcup_{i \in I} \in \ms S\). On the other hand, if \(A_j = (S - T) \cup B_j\) for some \(j \in I\) where \(B_i \in \ms T\) then \(\bigcup_{i \in I} A_i\) has this same form and so is in \(\ms S\). For the second part of our proof, note that if \(A \in \ms A\) then \(A \in \ms T\) and hence \(A \in \ms S\). So the \(\ms S\) contains the \(\sigma\) algebra of subsets of \(S\) generated by \(\ms A\). On the other hand, any \(\sigma\)-algebra of subsets of \(S\) the contains \(\ms A\) must contain \(\ms T\) and sets of the form \((S - T) \cup B\) where \(B \in \ms T\).
For the particular theorem here, the set of right neighbors of \(x \in [a, \infty)\) for \(([a, \infty), \prec)\) is \([x + a, \infty) \subseteq [2 a, \infty)\). The \(\sigma\)-algebra of subsets of \([2 a, \infty)\) generated by these right neighbor sets is the Borel \(\sigma\)-algebra \(\ms B\). So the result follows from the general result in the previous paragraph.
So the \(\sigma\)-algebra associated with the graph is a proper subset of the usual reference \(\sigma\)-algebra of all Borel subsets of \([a, \infty)\).
The left walk function \(u_n\) of order \(n \in \N\) for \(([a, \infty), \prec)\) is given by \(u_n(x) = 0\) for \(x \lt (n + 1)a\) and \[u_n(x) = \frac{[x - (n + 1)a]^n}{n!}, \quad x \ge (n + 1) a\]
A combinatorial
proof is best. Let \(n \in \N_+\) and note first that \(u_n(x) = 0\) for \(x \lt (n + 1)a\). Suppose that \(x \ge (n + 1) a\) and let \(\bs t = (t_1, t_2, \ldots, t_n)\) satisfy
\[(n + 1) a \le t_1 \le t_2 \le \cdots \le t_n \le x\]
so that \((t_1, t_2, \ldots, t_n, x)\) is a walk of length \(n\) in \(([a, \infty), \le)\) terminating in \(x\). Define \(\bs x = (x_1, x_2, \ldots, x_n)\) by \(x_k = t_k - (n - k + 1) a\) for \(k \in \{1, 2, \ldots, n\}\). Then
\[x_1 \prec x_2 \prec \cdots \prec x_n \prec x\]
so \((x_1, x_2, \ldots, x_n, x)\) is a walk (actually a path) of length \(n\) in \(([a, \infty), \prec)\) terminating in \(x\). Conversely, given such a walk \(\bs x\) we can recover the walk \(\bs t\) by \(t_k = x_k + (n - k + 1)a\) for \(k \in \{1, 2, \ldots, n\}\). The measure of the set of walks \(\bs x\) is the same as the measure of the set of walks \(\bs t\), which is \([(x - (n + 1) a]^n / n!\).
Suppose now that \(P\) is a probability measure, either on the reference \(\sigma\)-algebra or the associated \(\sigma\)-algebra in . The reliability function \(F\) of \(P\) for \(([a, \infty), \prec)\) is given by \(F(x) = P([a + x, \infty))\) for \(x \in [a, \infty)\).
The graph \(([a, \infty), \prec)\) is stochastic.
This follows from a more general result in Section 1.3, but we give an independent proof. Suppose that \(P\) and \(Q\) are probability measures on the associated \(\sigma\) algebra given in , with the same reliability function \(F\). That is, \(P([x + a, \infty)) = Q([x + a, \infty)) = F(x)\) for \(x \in [a, \infty)\). Restricted to \(\ms B\) (the Borel subsets of \([2 a, \infty)\)), \(P\) and \(Q\) are sub-probability measures and have the same (right) distribution function. Hence they are equal. That is \(P(B) = Q(B)\) for \(B \in \ms B)\). On the other hand, \(P([a, 2 a) = Q([a, 2 a) = 1 - F(2 a)\).
On the other hand, if \(P\) is a probability measure on the reference \(\sigma\)-algebra, then \(P\) is not determined by the reliability function \(F\) since \(F\) gives no information about \(P\) on Borel subsets of \([a, 2 a)\). If \(P\) has density \(f\) (with respect to Lebesgue measure) then \[F(x) = \int_{x + a}^\infty f(t) dt, \quad x \in [a, \infty)\] and hence \(F^\prime(x) = - f(x + a)\) for \(x \in [a, \infty)\).
Suppose that random variable \(X\) has the shifted exponential distribution on \([a, \infty)\) with parameter \(\beta \in (0, \infty)\). Then \(X\) has an exponential distribution for \(([a, \infty), +)\) with constant rate \(\beta e^{\beta a}\).
As before, \(X\) has density function \(f\) given by \(f(x) = \beta e^{-\beta(x - a)}\) for \(x \in [a, \infty)\). The reliability function \(F\) of \(X\) for \(([a, \infty), \prec)\) is given by \[F(x) = \P(X \ge x + a) = \P(X - a \ge x) = e^{-\beta x}, \quad x \in [a, \infty)\] So the distribution of \(X\) is memoryless for \(([a, \infty), +)\) and has constant rate \(\beta e^{\beta a}\).
Suppose again that \(X\) has the shifted exponential distribution on \([a, \infty)\) with parameter \(\beta \in (0, \infty)\), and that \(\bs Y = (Y_1, Y_2, \ldots)\) is the random walk on \(([a, \infty), +)\) associated with \(X\). For \(n \in \N_+\), The density function \(f_n\) of \(Y_n\) is given by \[f_n(x) = (\beta e^{\beta a})^n u_{n - 1}(x) F(x) = \beta^n \frac{(x - n a)^{n - 1}}{(n - 1)!} e^{-\beta (x - n a)}, \quad x \ge n a\]
It's interesting to compare the two spaces. The positive semigroup \(([a, \infty), \oplus)\) corresponds to the standard order \(\le\) but has the non-standard operator \(\oplus\). The strict positive semigroup \(([a, \infty), +)\) has the standard operator \(+\) but has the non-standard order \(\prec\). The shifted exponential distribution on \([a, \infty)\) is an exponential distribution for both spaces, but with different rate parameters.
Let \(S = (0, 1]\) and let \(\varphi(x) = -\ln x\) for \(x \in (0, 1]\) so that \(\varphi\) is a homeomorphism from \((0, 1]\) onto \([0,\infty)\).
For the graph \( ((0, 1], \ge, \mu) \),
Suppose that \(X\) has the exponential distribution for \( ((0, 1], \cdot) \) with rate parameter \( \alpha \in (0, \infty) \).
The distribution of \( X \) is the beta distribution with left parameter \(\alpha\) and right parameter 1, and the special case \(\alpha = 1\) gives the uniform distribution on \((0, 1]\). Random variable \(X\) maximizes entropy over all random variables \(Y\) taking values in \((0, 1]\) with \(\E(-\ln Y) = 1 / \alpha\).
The app below is a simulation of the beta distribution. The probability density function displayed is the ordinary one, with respect to Lebesgue measure. The rate parameter \(\alpha\) can be varied with the scroll bar.
Suppose again that \(X\) has the exponential distribution for \( ((0, 1], \cdot) \) with rate parameter \( \alpha \in (0, \infty)\) and that \(\bs Y = (Y_1, Y_2, \ldots)\) is the random walk associated with \( X \). For \(n \in \N_+\),
Let \(S = [1, \infty)\) and let \(\varphi(x) = \ln x\) for \(x \in [1, \infty)\) so that \(\varphi\) is a homeomorphism from \([1, \infty)\) onto \([0,\infty)\).
For the graph \( ([1, \infty), \le, \mu) \),
Suppose that \(X\) has the exponential distribution for \( ([1, \infty), \cdot) \) with rate parameter \( \alpha \in (0, \infty) \).
The distribution of \( X \) is the Pareto distribution with shape parameter \(\alpha\). Random variable \(X\) maximizes entropy over all random variables \(Y\) taking values in \([1, \infty)\) with \(\E(\ln Y) = 1 / \alpha\).
The app below is a simulation of the Pareto distribution. The probability density function displayed is the ordinary one, with respect to Lebesgue measure. The rate parameter \(\alpha\) can be varied with the scrollbar.
Suppose again that \(X\) has the exponential distribution for \( ([1, \infty), \cdot) \) with rate parameter \( \alpha \in (0, \infty)\) and that \(\bs Y = (Y_1, Y_2, \ldots)\) is the random walk associated with \( X \). For \(n \in \N_+\),
Let \(S = [0, 1/2)\) and let \(\varphi(x) = x / (1 - 2x)\) for \(x \in [0, 1/2)\) so that \(\varphi\) is a homeomorphism from \([0, 1/2)\) onto \([0,\infty)\).
The positive semigroup \(([0, 1/2), \oplus)\) occurs in a study of generalized Brownian functionals by Hida.
For the graph \(([0, 1/2), \le, \mu)\),
Suppose that \(X\) has the exponential distribution on \( ([0, 1/2), \oplus) \) with rate parameter \(\alpha \in (0, \infty) \).
We will refer to the distribution of \(X\) as the Hida distribution with parameter \(\alpha\). Random variable \(X\) maximizes entropy over all random variables \(Y\) in \([0, 1/2)\) with \(\E[Y / (1 - 2 Y)] = 1 / \alpha\).
The app below is a simulation of the Hida distribution. The probability density function displayed is the ordinary one, with respect to Lebesgue measure. The rate parameter \(\alpha\) can be varied with the scrollbar.
Suppose again that \(X\) has the exponential distribution on \( ([0, 1/2), \oplus) \) with rate parameter \(\alpha \in (0, \infty) \) and that \(\bs Y = (Y_1, Y_2, \ldots)\) is the random walk associated with \(X\). For \(n \in \N_+\),