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Summable canonical differentiation

Statement

Let $I, E_{i}, g_{i}, J, Z$ be as in Countable canonical model. Suppose that for every compact interval $K \subset J$ there is a summable sequence $a_{i} \geq 0$ such that $g_{i} (1 + ∣ E_{i} ∣^{2}) \exp (- β E_{i}) \leq a_{i}$ for every $i \in I$ and $β \in K$ . Then $Z$ is positive, finite, and $C^{2}$ on $J$ , with $Z^{'} (β) = - \sum_{i} g_{i} E_{i} \exp (- β E_{i})$ and $Z^{''} (β) = \sum_{i} g_{i} E_{i}^{2} \exp (- β E_{i})$ . The probabilities $p_{i}$ are normalized. Define $\overset{ˉ}{E} : = ⟨ E ⟩_{β}$ and ${Var}_{β} (E) : = \sum_{i} p_{i} (E_{i} - \overset{ˉ}{E})^{2}$ . These quantities are finite and satisfy $- \partial_{β} \log Z = \overset{ˉ}{E}$ and $\partial_{β}^{2} \log Z = {Var}_{β} (E)$ . Moreover $H_{g} (p) = \log Z + β \overset{ˉ}{E}$ , and its defining series is absolutely convergent.

Facts & Assumptions

Given: As stated in the item.

[F1]

Countable canonical model: $Z$ , $p_{i}$ , expectation and $H_{g}$ have the definitions and domains stated there. The centered variance is defined in this item's Statement.

[F2]

The real exponential function and the number $e$ by a power series and The exponential function is smooth and $(\exp)^{'} = \exp$ : $\exp$ is positive and differentiable with $\exp^{'} = \exp$ .

[F3]

The natural logarithm as the inverse of the exponential function and The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t: $\log$ is defined on positive reals and $(\log u)^{'} = u^{'} / u$ for differentiable $u > 0$ .

[F4]

The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ : a function continuous on a closed interval and differentiable in its interior has a point where its secant slope equals its derivative.

[A1]

For every compact $K \subset J$ there is a nonnegative summable sequence $(a_{i})$ such that $g_{i} (1 + ∣ E_{i} ∣^{2}) e^{- β E_{i}} \leq a_{i}$ for all $i$ and $β \in K$ ; the tail $\sum_{i > N} a_{i}$ tends to zero.

[F5]

The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ : differentiating $e^{- β E_{i}}$ with respect to $β$ uses the one-variable chain rule.

Proof

Given: The countable canonical model and the local summable envelope in the statement.

1.1F1A1F2F5

On a compact interval $K \subset J$ , let $Z_{N}$ , $D_{N}$ , and $Q_{N}$ be the first $N$ partial sums of $\sum_{i} g_{i} e^{- β E_{i}}$ , $- \sum_{i} g_{i} E_{i} e^{- β E_{i}}$ , and $\sum_{i} g_{i} E_{i}^{2} e^{- β E_{i}}$ . Since $∣ E_{i} ∣^{r} \leq 1 + ∣ E_{i} ∣^{2}$ for $r = 0, 1, 2$ , each tail is bounded uniformly on $K$ by the corresponding tail of $\sum_{i} a_{i}$ , which tends to zero [A1]. Thus all three sequences converge uniformly; call their limits $Z, D, Q$ . Each finite partial sum has the displayed derivatives by the chain rule [F5] and the exponential derivative [F2]. Each partial sum is continuous, so the uniform limits are continuous by the $ε / 3$ definition of continuity.

2.1F2F4F5step 1.1

For an interior $β$ and sufficiently small $h \neq 0$ , apply [F4] to $Z_{N}$ on the interval with endpoints $β$ and $β + h$ : its difference quotient is $D_{N} (ξ_{N, h})$ for some intermediate $ξ_{N, h}$ . Choose $N = N (h)$ so that $\sup_{K} ∣ Z_{N} - Z ∣ < h^{2}$ and $\sup_{K} ∣ D_{N} - D ∣ < ∣ h ∣$ . The difference quotient for $Z$ then differs from $D (β)$ by at most $2 ∣ h ∣ + ∣ h ∣ + \sup_{∣ t - β ∣ \leq ∣ h ∣} ∣ D (t) - D (β) ∣$ , which tends to zero by continuity of $D$ . Hence $Z^{'} = D$ . Applying the same argument to $D_{N}$ and $Q_{N}$ proves $D^{'} = Q$ , so the displayed derivatives hold.

3.1F1F2F3step 2.1algebra

The summands of $Z$ are positive and at least one is present, hence $0 < Z < \infty$ on $K$ ; summing the definition of $p_{i}$ gives $\sum_{i} p_{i} = 1$ . The envelope gives finite first and second moments, and $\sum_{i} p_{i} (E_{i} - \overset{ˉ}{E})^{2} \leq 2 \sum_{i} p_{i} E_{i}^{2} + 2 {\overset{ˉ}{E}}^{2} < \infty$ . Differentiating $\log Z$ using [F3] yields $- Z^{'} / Z = \overset{ˉ}{E}$ and $Z^{''} / Z - (Z^{'} / Z)^{2} = ⟨ E^{2} ⟩ - {\overset{ˉ}{E}}^{2} = {Var}_{β} (E)$ after expanding the defining variance.

4.1F1F2F3step 3.1∎

Since $p_{i} / g_{i} = e^{- β E_{i}} / Z$ , its logarithm is $- β E_{i} - \log Z$ . The finite first moment makes $\sum_{i} p_{i} ∣ \log (p_{i} / g_{i}) ∣ \leq ∣ β ∣ \sum_{i} p_{i} ∣ E_{i} ∣ + ∣ \log Z ∣ \sum_{i} p_{i} < \infty$ ; summing the identity gives $H_{g} = β \overset{ˉ}{E} + \log Z$ .

Summable canonical differentiation

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources