Applications of the perturbation formula for Poisson processes to elementary and geometric probability

TL;DR

This paper develops a unified probabilistic framework using perturbation formulas for Poisson processes to derive integral representations, identities, and extensions relevant to various probability distributions and geometric formulas.

Contribution

It introduces a unified approach to derive integral representations and identities for distributions and geometric formulas using perturbation techniques for Poisson processes.

Findings

Derived integral representations for several distributions.

Obtained new integro-differential identities for stable densities.

Extended Crofton's derivative formula to Poisson processes.

Abstract

The binomial, the negative binomial, the Poisson, the compound Poisson and the Erlang distribution do all admit integral representations with respect to its (continuous) parameter. We use the Margulis-Russo type formulas for Bernoulli and Poisson processes to derive these representations in a unified way and to provide a probabilistic interpretation for the derivatives. By similar variational methods, we obtain apparently new integro-differential identities which the density of a strictly $\alpha$-stable multivariate density satisfies. Then, we extend Crofton's derivative formula known in integral geometry to the case of a Poisson process. Finally we use this extension to give a new probabilistic proof of a version of this formula for binomial point processes.

Equations

$$N_A^{+}:=\sum _{i=1}^{m}1\{X^{(i)}\in A,X_{(i)}\notin A\},N_A^{-}:=\sum _{i=1}^{m}1\{X_{(i)}\in A,X^{(i)}\notin A\}.$$

$$\frac{d}{d\theta }{\mathbb{P}}_{\theta }(A)=E_{\theta }[N_A^{+}-N_A^{-}],$$

$$\frac{d}{dp}{\mathbb{P}}_p(A)=E_p{N}_A^{+},$$

$$\mathsf{Bin}(n,p;k):=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k},k\in \{0,\dots ,n\}$$

$$N_A^{+}=\{\begin{array}{ll}n-k+1,& \text{if Sn=k−1},\\ k,& \text{if Sn=k},\\ 0,& \text{otherwise}.\end{array}$$

$$\frac{d}{dp}{\mathbb{P}}_p(A)$$

$$=\frac{n!}{(k-1)!(n-k)!}{p}^{k-1}(1-p{)}^{n-k}.$$

$$\sum _{j=k}^n\mathsf{Bin}(n,p;j)=\frac{n!}{(k-1)!(n-k)!}{\int }_0^p{t}^{k-1}(1-t{)}^{n-k}dt,k\in \{1,\dots ,n\}.$$

$$\mathsf{NB}(r,p;k):=\left(\genfrac{}{}{0ex}{}{k+r-1}{k}\right)p^r(1-p{)}^k,k\in {\mathbb{Z}}_{+},$$

$$N_A^{+}=\{\begin{array}{ll}k,& \text{if Sk+r−1=r−1},\\ 1,& \text{if Sk+r−1=r},\\ 0,& \text{otherwise},\end{array}$$

$$\frac{d}{dp}{\mathbb{P}}_p(A)$$

$$=\frac{(k+r-1)!}{(k-1)!(r-1)!}{p}^{r-1}(1-p{)}^{k-1}.$$

$$\sum _{j=0}^k\mathsf{NB}(r,p;j)=\frac{(k+r-1)!}{(k-1)!(r-1)!}{\int }_0^p{t}^{r-1}(1-t{)}^{k-1}dt,k\in \mathbb{N}.$$

$$N_A^{+}:=\int 1\{\eta +{\delta }_z\in A,\eta \notin A\}\lambda (dz),N_A^{-}:=\int 1\{\eta \in A,\eta +{\delta }_z\notin A\}\lambda (dz),$$

$$E_{\theta }{N}_A^{+}=\frac{1}{\theta }{E}_{\theta }\int 1\{\eta \in A,\eta +{\delta }_z\notin A\}\eta (dz),$$

$$Ef({\eta }_{\lambda +\theta \nu })=Ef({\eta }_{\lambda })+\sum _{k=1}^{\infty }\frac{{\theta }^k}{k!}\int E{D}_{x_1,\dots ,x_k}^{k}f({\eta }_{\lambda }){\nu }^k(d(x_1,\dots ,x_k)),$$

$$\frac{d^k}{d{\theta }^k}Eg({\eta }_{\theta \lambda })=\int \cdots \int E{D}_{z_1,\dots ,z_k}^{k}g({\eta }_{\theta \lambda })\lambda (d{z}_1)\cdots \lambda (d{z}_k),\theta \le {\theta }_0.$$

$$\mathsf{Po}(\theta ;k):=\frac{{\theta }^k}{k!}{e}^{-\theta },k\in {\mathbb{Z}}_{+},$$

$$\sum _{j=k}^{\infty }\mathsf{Po}(\theta ;j)={\int }_0^{\theta }\frac{t^{k-1}}{(k-1)!}{e}^{-t}dt,k\in \mathbb{N}.$$

$$\mathsf{Er}(n,\theta ;x):=\frac{{\theta }^n}{(n-1)!}{x}^{n-1}{e}^{-\theta x},x\ge 0.$$

$${\int }_0^x\mathsf{Er}(n,\theta ;y)dy=\frac{x^n}{(n-1)!}{\int }_0^{\theta }{t}^{n-1}{e}^{-tx}dt,x\ge 0.$$

$$1_A(\eta +{\delta }_z)-1_A(\eta )=1\{\eta \{z\}=k-1\}.$$

$${\int }_0^x\mathsf{Er}(n,\theta ;y)dy=\sum _{j=n}^{\infty }\mathsf{Po}(\theta x;j).$$

$$\mathsf{CPo}(\theta ,\mathbb{Q}):=\sum _{n=0}^{\infty }\frac{{\theta }^n}{n!}{e}^{-\theta }{\mathbb{Q}}^{\ast n},$$

$$\frac{d}{d\theta }F(\theta ,\mathbb{Q};x)=\int F(\theta ,\mathbb{Q};x-z)\mathbb{Q}(dz)-F(\theta ,\mathbb{Q};x).$$

$$\frac{d}{d\theta }\mathsf{CPo}(\theta ,\mathbb{Q};k)=\sum _{j\ne k}{q}_{k-j}\mathsf{CPo}(\theta ,\mathbb{Q};j)-(1-q_0)\mathsf{CPo}(\theta ,\mathbb{Q};k),k\in \mathbb{Z},$$

$$\frac{d}{d\theta }\mathsf{CPo}(\theta ,\mathbb{Q};k)=\sum _{j\in \mathbb{Z}}{q}_{k-j}\mathsf{CPo}(\theta ,\mathbb{Q};j)-\mathsf{CPo}(\theta ,\mathbb{Q};k),k\in \mathbb{Z}.$$

$$1_A(\eta +{\delta }_z)-1_A(\eta )=1\{Z>x,Z+z\le x\}-1\{Z\le x,Z+z>x\},$$

$$\frac{d}{d\theta }{\mathbb{P}}_{\theta }(A)=$$

$$-E_{\theta }{\int }_{(0,\infty )}(1-1\{Z+z\le x\})1\{Z\le x\}\mathbb{Q}(dz)$$

$$=$$

$$\mathsf{CPo}(\theta ,\mathbb{Q};k)=e^{-(1-q_0)\theta }\sum _{j\ne k}{q}_{k-j}{\int }_0^{\theta }{e}^{(1-q_0)t}\mathsf{CPo}(t,\mathbb{Q};j)dt,k\in \mathbb{Z}.$$

Full text

2000 Mathematics Subject Classification. 60E05, 60G55.

Key words and phrases. Margulis-Russo formula, binomial distribution, negative binomial distribution, Poisson distribution, compound Poisson distribution, Erlang distribution, multivariate strictly stable distribution, Poisson process, binomial process, Crofton’s derivative formula

1 Introduction

The aim of the current article is to demonstrate how the fundamental derivative formulas based on the principle of pivotality could be applied to derive various distributional identities. Some of them, notably for classical univariate distributions are not new. The other, for multivariate strictly stable distributions and the Crofton’s formula has not, to our knowledge, been known in the considered generality. Our approach not only shows that all these formulas are based on the same principle, but also provides a probabilistic interpretation of the derivatives as the expected (signed) number (resp. measure) of certain pivotal elements.

The key components of the variational method we use are the following two formulas for Bernoulli systems and for Poisson point processes.

Consider a vector $X=(X_{1},\dots,X_{m}),m\in{\mathbb{N}},$ of independent Bernoulli random variables, all having the same success probability $\theta\in[0,1]$ defined on its canonical probability space $\Omega=\{0,1\}^{m}$. For an $i\in\{1,\dots,m\}$, let $X_{(i)}$ (resp. $X^{(i)}$) be a vector whose entries coincide with those of $X$ except at the $i$-th coordinate, where the entry is [math] (resp. $1$). Let $A\subseteq\Omega$ be an event and let

[TABLE]

The coordinates $i$ which contribute non-zero terms to $N_{A}^{+}$ (resp., to $N_{A}^{-}$) are called $(+)$-pivotal (resp., $(-)$-pivotal) for even $A$. The so-called Margulis–Russo formula (see e.g. [3, Lemma 2.9]) states that

[TABLE]

where $\operatorname{\mathbb{E}}_{\theta}$ denotes expectation with respect to the distribution $\mathbb{P}_{\theta}$ of $X$. A function $g:\Omega\mapsto{\mathbb{R}}$ is increasing, if $g(x_{(j)})\leq g(x^{(j)})$ for all $x\in\Omega$ and $1\leq j\leq n$. An event is increasing if its indicator function is increasing. In this case, there is no (-)-pivotal elements and (1.2) takes the form

[TABLE]

which is the original Margulis–Russo formula. The idea of using pivotal (also called essential) components can be traced back to at least [4]. Subsequently it has been independently exploited in various forms for monotone events in [11] and [16]. The above formulation is due to [12]: an introduction of ($\pm$)-pivotality, suggested by M. Menshikov, allowed to express the derivative of probability for general cylinder events and the derivative of the expectation of random variables, in general.

The power of this formula lies in the fact that it links the variation of an event’s probability to the geometry of configurations comprising the event, i.e. the number of pivotal components for its occurence. To see it in action, consider the binomial distribution

[TABLE]

with parameters $n\in{\mathbb{N}}=\{1,2,\dots\}$ and $p\in[0,1]$. Take $k\in\{1,\dots,n\}$, and consider the increasing event $A:=\{S_{n}\geq k\}$, where $S_{n}:=X_{1}+\dots+X_{n}$. We then have

[TABLE]

Invoking (1.3), an easy calculation shows that

[TABLE]

Since $\mathbb{P}_{0}(A)=0$, we obtain the following integral representation:

[TABLE]

Another example concerns the negative binomial distribution

[TABLE]

with parameters $r\in{\mathbb{N}}$ and $p\in[0,1]$. Fix natural numbers $r,k$ and consider the increasing event $A:=\{S_{k+r-1}\geq r\}$. We then have

[TABLE]

and (1.3) yields that

[TABLE]

Since $\mathbb{P}_{0}(A)=0$, we get the identity

[TABLE]

Both (1.4) and (1.5) could be checked directly by differentiation, but the variation formula (1.2) provides a probabilistic insight into the variation of the probability.

The aim of this paper is to demonstrate the power of an analogous to (1.2) variation formula for Poisson processes. Let $\lambda$ be a finite measure on some measurable space ${\mathbb{X}}$ and $\theta\geq 0$. Consider a point process $\eta$ on ${\mathbb{X}}$ and a probability measure $\mathbb{P}_{\theta}$ such that $\eta$ is (under $\mathbb{P}_{\theta}$) a Poisson process with intensity measure $\theta\lambda$. It was shown in [18] that, if $A$ is an event defined in terms of $\eta$, then (1.2) holds, provided that (1.1) is modified as follows:

[TABLE]

where, generically, $\delta_{z}$ is the Dirac measure at $z$. Notice, that by Mecke’s formula (see, e.g. [9, Th. 4.1]),

[TABLE]

So, analogously to the Bernoulli case, the process points $z_{i}\in\eta$ such that $\eta\in A$, but $\eta-\delta_{z_{i}}\notin A$ maybe called pivotal points while $z\in{\mathbb{X}}$ such that $\eta\in A$, but $\eta+\delta_{z}\notin A$ are called pivotal locations.

In the next section we provide a review of the variation formulas for Poisson processes and to prove with their help distributional identities for some classical one-dimensional distributions: Poisson, Erlang and compound Poisson. We then consider multivariate strictly stable distributions in Section 3. Theorem 3.1 provides formulas which are apparently new and could possibly be used for an effective computation of the stable density which is not known explicitly apart for a very few particular values of its parameters. In Section 4, we extend Crofton’s derivative formula. In the final Section 5 we use this extension to give a new probabilistic proof of a version of this formula for binomial point processes.

2 A perturbation formula for Poisson processes

In this section we review a perturbation formula for general Poisson processes. Let $({\mathbb{X}},\mathcal{X})$ be a measurable space and let ${\mathbf{N}}({\mathbb{X}})\equiv{\mathbf{N}}$ be the space of integer-valued $\sigma$-finite measures $\varphi$ on ${\mathbb{X}}$, equipped with the smallest $\sigma$-field $\mathcal{N}$ making the mappings $\varphi\mapsto\varphi(B)$ measurable for all $B\in\mathcal{X}$.

For any $g\colon{\mathbf{N}}\rightarrow{\mathbb{R}}$ and $z\in{\mathbb{R}}^{n}$, introduce a function $D_{z}g\colon{\mathbf{N}}\to{\mathbb{R}}$ by means of $D_{z}g(\varphi)=g(\varphi+\delta_{z})-g(\varphi)$. The mapping $g\mapsto D_{z}g$ is known as difference operator. For $k\in{\mathbb{N}}$ the $k$-th iteration $D^{k}_{z_{1},\ldots,z_{k}}g:{\mathbf{N}}\rightarrow{\mathbb{R}}$, of this operator is inductively defined by $D^{k}_{z_{1},\ldots,z_{k}}g=D_{z_{k}}D^{k-1}_{z_{1},\ldots,z_{k-1}}g$ for $(z_{1},\ldots,z_{k})\in({\mathbb{R}}^{n})^{k}$.

Given any $\sigma$-finite measure $\rho$ on ${\mathbb{X}}$, we let $\eta_{\rho}$ denote a Poisson process with this intensity measure. The following perturbation formula is a special case of Theorem 19.3 in [9].

Theorem 2.1**.**

Let $\lambda$ be a $\sigma$-finite and let $\nu$ be a finite measure on ${\mathbb{X}}$. Let $g\colon{\mathbf{N}}\rightarrow{\mathbb{R}}$ be a measurable function such that $\operatorname{\mathbb{E}}|g(\eta_{\lambda+\nu})|<\infty$. Let $\theta\in(-\infty,1]$ such that $\lambda+\theta\nu$ is a measure. Then

[TABLE]

where the series converges absolutely.

The earliest version of Theorem 2.1 (for a bounded function $g$) was proved in [18]. Later this was generalised in [13]. For square integrable random variables the result can be extended to certain (signed) $\sigma$-finite perturbations; see [8].

For later reference we provide the following consequence of Theorem 2.1.

Proposition 2.2**.**

Let $\lambda$ be a finite measure on ${\mathbb{X}}$. Let $g\colon{\mathbf{N}}\rightarrow{\mathbb{R}}$ be a measurable function such that $\operatorname{\mathbb{E}}|g(\eta_{\theta_{0}\lambda})|<\infty$ for some $\theta_{0}>0$. Then $\theta\mapsto\operatorname{\mathbb{E}}g(\eta_{\theta\lambda})$ is analytic on $[0,\theta_{0}]$ and its derivatives are given by

[TABLE]

Using the indicator ${\bf 1}_{A}$ above as the function $g$, implies the Poisson process version of the Margulis-Russo formula (1.2) with notation (1.6).

The Poisson and the Erlang distribution.

We now apply (2.2) to derive a few distributional identities for classical univariate distributions.

Proposition 2.3**.**

The Poisson distribution with parameter $\theta\geq 0$ is given by

[TABLE]

and satisfies

[TABLE]

The Erlang distribution with parameters $n\in{\mathbb{N}}$ and $\theta>0$ has density function

[TABLE]

Its distribution function may be written as

[TABLE]

Proof.

In the notation of the previous section, set ${\mathbb{X}}=\{z\}$ to be a singleton and $\lambda\{z\}=1$. Then the Poisson points process $\eta\{z\}$ is just a Poisson parnom variable with parameter 1. Take $k\in{\mathbb{N}}$ and consider the event $A:=\{\eta\{z\}\geq k\}$. Then $\mathbb{P}_{\theta}(A)$ is given by the left-hand side of (2.3) and we have

[TABLE]

Since $\mathbb{P}_{0}(A)=0$, (2.2) implies (2.3). Eq. (2.4) follows from (2.3) and the identity

[TABLE]

∎

The Compound Poisson distribution.

Let $\mathbb{Q}$ be a probability distribution on ${\mathbb{R}}$. The compound Poisson distribution with parameters $\theta\geq 0$ and $\mathbb{Q}$ is given by

[TABLE]

so it equals the distribution of Poisson ${\mathsf{Po}}(\theta)$ number of independent summands each having distribution $\mathbb{Q}$.

Proposition 2.4**.**

The distribution function $F(\theta,\mathbb{Q};x):={\mathsf{CPo}}(\theta,\mathbb{Q})((-\infty,x])$, $x\in{\mathbb{R}}$, of the Compound Poisson distribution satisfies

[TABLE]

When $\mathbb{Q}$ is concentrated on ${\mathbb{Z}}$,

[TABLE]

where ${\mathsf{CPo}}(\theta,\mathbb{Q};j):={\mathsf{CPo}}(\theta,\mathbb{Q})(\{j\})$, and $q_{j}:=\mathbb{Q}(\{j\})$, $j\in{\mathbb{Z}}$. Equivalently,

[TABLE]

Proof.

To apply (2.2), take ${\mathbb{X}}:={\mathbb{R}}$ and let $\lambda:=\mathbb{Q}$. Under $\mathbb{P}_{\theta}$ the random variable $Z:=\int z\eta(dz)$ has the compound Poisson distribution ${\mathsf{CPo}}(\theta,\mathbb{Q})$. Consider the event $A:=\{Z\leq x\}$ for some $x\in{\mathbb{R}}$. Then, for $z\in{\mathbb{R}}$,

[TABLE]

so that (2.2) writes

[TABLE]

This yields (2.6). If $\mathbb{Q}({\mathbb{Z}})=1$, then (2.8) (and hence also (2.7)) follows upon taking suitable differences. ∎

Remark 2.1*.*

Identity (2.7) is equivalent to

[TABLE]

If, in addition, $q_{j}=0$ for $j<0$, then it follows from the definition of ${\mathsf{CPo}}(\theta,\mathbb{Q})$ (or from ${\mathsf{CPo}}(\theta,\mathbb{Q};0)=e^{-(1-q_{0})\theta}$ and (2.9)) that $e^{(1-q_{0})\theta}{\mathsf{CPo}}(\theta,\mathbb{Q};k)$ is a polynomial in $\theta$ of degree $k$. Equations (2.9) provide a recursion for the coefficients of these polynomials.

Remark 2.2*.*

The characteristic function of ${\mathsf{CPo}}(\theta,\mathbb{Q};k)$ is given by

[TABLE]

where $\mathbf{i}$ is the imaginary unit and $G_{\mathbb{Q}}$ is the characteristic function of $\mathbb{Q}$. The recursion (2.9) can also be obtained by differentiating (2.10) with respect to $\theta$. Differentiation of (2.10) with respect to $s$ and assuming $q_{j}=0$ for $j<0$, yields the widely used Panjer recursion [14]:

[TABLE]

Remark 2.3*.*

Consider a compound Poisson process $(X_{t})_{t\geq 0}$ driven by a unit rate Poisson process and jump size distribution $\mathbb{Q}$, see e.g. [6, Ch. 12]. Then $X_{t}$ has distribution ${\mathsf{CPo}}(t,\mathbb{Q})$ and (2.8) and (2.6) are two examples for the Kolmogorov forward equation, see e.g. [6, Ch. 19].

Remark 2.4*.*

Take ${\mathbb{X}}:=[0,\infty)$ and $\lambda$ as the Lebesgue measure on ${\mathbb{X}}$. Let $T_{1}<T_{2}<\dots$ be the atoms of $\eta$ arranged in increasing order. Let $n\in{\mathbb{N}}$ and consider the event $A:=\{T_{n}\leq x\}$ for $x\geq 0$. It is well-known that $\mathbb{P}_{\theta}(A)$ coincides with the left-hand side of (2.4). On the other hand we have for all $z\geq 0$ (with obvious notation)

[TABLE]

so that (2.3) yields

[TABLE]

Since $\mathbb{P}_{0}(A)=0$, we again obtain (2.4).

3 Strictly $\alpha$-stable laws

A random vector $\xi$ (or its distribution) is called strictly $\alpha$-stable (St$\alpha$S), if the following equality in distribution holds:

[TABLE]

where $\xi^{\prime},\xi^{\prime\prime}$ are independent distributional copies of $\xi$. In Euclidean spaces St$\alpha$S laws exist only for $0<\alpha\leq 2$ and $\alpha=2$ corresponds to the Gaussian distribution centred at the origin. Symmetrical St$\alpha$S random vectors in ${\mathbb{R}}^{n}$ with $\alpha<2$ and all St$\alpha$S random vectors with $\alpha<1$ admit the following LePage series representation (see [10]):

[TABLE]

where $\Gamma_{1},\Gamma_{2},\dots$ are the successive times of jumps of a homogeneous Poisson process on ${\mathbb{R}}_{+}$ and $\varepsilon_{1},\varepsilon_{2},\dots$ are i.i.d. random vectors on the unit sphere ${\mathbb{S}}^{n-1}$. Thus their distribution is characterised by two parameters: the intensity $\theta$ of the Poisson process and the probability measure $\hat{\sigma}$ on the sphere – the distribution of $\varepsilon_{k}$’s. Considering $\varepsilon_{i}$’s as independent marks of the Poisson process points, we can appeal to the marking and mapping theorems for Poisson processes (see, e.g.  [9, Th. 5.1, 5.6]) to see that

[TABLE]

is a Poisson process on ${\mathbb{R}}^{n}\setminus\{0\}$ with intensity measure

[TABLE]

The right-hand side of (3.2) can be written as a point process integral, that is,

[TABLE]

is a stable random vector with the given parameters. The integrals here in this section are taken over ${\mathbb{R}}^{n}\setminus\{0\}$ unless specified otherwise. The St$\alpha$S distribution is infinitely divisible with Lévy measure $\Lambda_{\theta}$. The measure $\sigma=\theta\hat{\sigma}$ on ${\mathbb{S}}^{n-1}$ is said to be the spectral measure of $\xi_{\theta}$ (or $\Lambda_{\theta}$). The convergence of the integral (3.3) for $\alpha<1$ or for all $\alpha<2$ in the case of a symmetrical spectral measure is guaranteed, for instance, by [6, Lem. 12.13]. However, the spectral measure need not be symmetric in order for the LePage representation (3.2) to hold. For instance, when $\alpha\geq 1$ it is sufficient that a non-symmetric spectral measure satisfies $\int s\,\sigma(ds)=0$, see [2, Th. 2].

By definition of $\Lambda_{\theta}$, we have for each Borel set $B\subset{\mathbb{R}}^{n}\setminus\{0\}$ and each $c>0$ that

[TABLE]

Hence we obtain from the mapping theorem that $\xi_{\theta}\overset{D}{=}\theta^{1/\alpha}\xi_{1}$.

For a $B\subset{\mathbb{S}}^{n-1}$, introduce the set

[TABLE]

Let $S_{\sigma}$ be the support of the spectral measure $\sigma$. The corresponding stable law is non-degenerate if $\operatorname{cone}(S_{\sigma})$ has a positive $n$-volume. It is known that non-degenerate stable laws possess an infinitely differentiable density, see [15, Sec. 3.2.4].

We are now ready to formulate the main result of this section.

Theorem 3.1**.**

Let $\xi_{\theta}$ be a St$\alpha$S random vector with LePage representation (3.2) corresponding to the spectral measure $\sigma=\theta\hat{\sigma}$ such that $K:=\operatorname{cone}(S_{\hat{\sigma}})$ has a positive $n$-volume. Then

• (i)

The density $f_{\theta}$ of $\xi_{\theta}$ satisfies

[TABLE]

where $\langle\cdot\,,\cdot\rangle$ is the scalar product in ${\mathbb{R}}^{n}$.

• (ii)

Let $f_{|\xi_{\theta}|}$ denote the p.d.f. of the radius vector $|\xi_{\theta}|$. Then for all $r>0$,

[TABLE]

Corollary 3.2**.**

The c.d.f. $f_{\theta}$ and the p.d.f. $F_{\theta}$ of a positive St$\alpha$S on ${\mathbb{R}}_{+}$ with $0<\alpha<1$ are related through

[TABLE]

Proof.

For a measurable $B\subset{\mathbb{R}}^{n}$, consider the indicator function

[TABLE]

By (3.3), $\operatorname{\mathbb{E}}g_{B}(\eta_{\theta})=\mathbb{P}(\xi_{\theta}\in B)$. Moreover,

[TABLE]

Using (2.2) and noting that $\Lambda_{\theta}=\theta\Lambda_{1}$, we obtain that for any measurable $B\subset{\mathbb{R}}^{n}$,

[TABLE]

Since $\xi_{\theta}\overset{D}{=}\theta^{1/\alpha}\xi_{1}$, the density and its gradient satisfy

[TABLE]

Therefore,

[TABLE]

Take a set $B$ such that its closure is in $\operatorname{Int}(K)$. The density $f_{\theta}$ is bounded on such $B$ and the left-hand-side of (3.8) becomes

[TABLE]

The right-hand-side of (3.9) is

[TABLE]

Equating it to (3.10), we get the identity which holds for all measurable $B\subset\operatorname{Int}(K)$ which implies the identity (3.4) for almost all $x\in\operatorname{Int}(K)$. But the density is continuously differentiable there, so it also holds for all $x\in\operatorname{Int}(K)$.

Recall that all one-dimensional St$\alpha$S laws with $0<\alpha<1$ are totally skewed concentrated on either ${\mathbb{R}}_{+}$ or ${\mathbb{R}}_{-}$. Consider, for definitivness, a positive $\xi_{\theta}$. The spectral measure $\sigma$ is then $\theta\delta_{1}$ and (3.4) becomes (3.6).

Now let $B$ in (3.9) be the ball $B_{r}$ of radius $r$ centred at the origin. Since

[TABLE]

and also

[TABLE]

the relation (3.9) takes the form (3.5).

Notice that its one-dimensional variant (3.7), when differentiated, gives (3.6). ∎

4 Crofton’s derivative formula for Poisson processes

The classical Crofton formula known in integral and stochastic geometry relates the probability of events and, generally, expectation of a random variable defined by configuration of a fixed number of points uniformly distributed in a domain when the domain is infinitesimally expanded. The property, described by the event or the random variable should depend only on the mutual position of points, so it must be rotation and translation invariant once all the points are still in the domain, see, e.g.  [7, Ch.2]. We will revisit this formula in Section 5, but now we establish its counterpart for Poisson processes.

Let $K\subset{\mathbb{R}}^{n}$ be a compact set and define

[TABLE]

where $B^{n}$ is the Euclidean unit ball. This is the so-called parallel set of $K$ at distance $t$. Let $h\colon{\mathbb{R}}^{n}\to[0,\infty)$ be a continuous function and let $\lambda$ be the measure on ${\mathbb{R}}^{n}$ with Lebesgue density $h$. For $t\geq 0$ let $\lambda_{t}$ be the restriction of $\lambda$ to $K_{t}$ and let $\eta_{t}$ be a Poisson process on ${\mathbb{R}}^{n}$ with intensity measure $\lambda_{t}$. Let $g\colon{\mathbf{N}}({\mathbb{R}}^{n})\to{\mathbb{R}}$ be measurable. Under certain technical assumptions on $K$ and $g$ we shall prove that

[TABLE]

where $\partial K$ is the boundary of $K$ and $\mathcal{H}^{n-1}$ is the $(n-1)$-dimensional Hausdorff measure on ${\mathbb{R}}^{n}$.

Our main technical geometrical tool are the support measures from [5]. We recall here briefly their definition and main properties. We put $p(K,z):=y$ whenever $y$ is a uniquely determined point in $K$ with $d(K,z):=\min\{x-z:x\in K\}=|y-z|$, and we call this point the metric projection of $z$ on $K$. If $0<d(K,z)<\infty$ and $p(K,z)$ is defined, then $p(K,z)$ lies on the boundary $\partial K$ of $K$ and we put $u(K,z):=(z-p(K,z))/d(K,z)$. The exoskeleton $\operatorname{exo}(A)$ of $A$ consists of all points of ${\mathbb{R}}^{n}\setminus K$ which do not admit a metric projection on $K$. The normal bundle of $K$ is defined by

[TABLE]

It is a measurable subset of $\partial K\times S^{n-1}$, where $S^{n-1}:=\{x\in{\mathbb{R}}^{n}:\|x\|=1\}$ is the unit sphere in ${\mathbb{R}}^{n}$. The reach function $\delta(K,\cdot):{\mathbb{R}}^{n}\times{\mathbb{S}}^{n-1}\rightarrow[0,\infty]$ of $K$ is defined by

[TABLE]

and $\delta(K,x,u):=0$ for $(x,u)\notin N(K)$. Note that $\delta(K,\cdot)>0$ on $N(K)$.

We write $x\wedge y$ for $\min\{x,y\}$. By Theorem 2.1 in [5], there exist signed measures $\mu_{0}(K;\cdot),\ldots,\mu_{n-1}(K;\cdot)$ on ${\mathbb{R}}^{n}\times{\mathbb{S}}^{n-1}$ satisfying

[TABLE]

and, for each measurable bounded function $f:{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}$ with compact support, we have the following local Steiner formula:

[TABLE]

where $\omega_{j}:=j\kappa_{j}$ and $\kappa_{j}$ is the volume of the unit ball in ${\mathbb{R}}^{j}$. These measures are called support measures of $K$. They are uniquely defined by (4.4) and the requirement $|\mu_{i}|(K;{\mathbb{R}}^{n}\times{\mathbb{S}}^{n-1}\setminus N(K))=0$. In general, the total variation measures $|\mu_{i}|(K;\cdot)$ featuring in (4.3) are not finite. However, it follows from (4.4) that

[TABLE]

Therefore the integrals on the right-hand side of (4.4) are well-defined. An important special case is that of a convex set $K$. Then $\delta(K,x,u)=\infty$ for all $(x,u)\in N(K)$.

We start with the following proposition of independent interest. For $i\in\{1,2\}$ we define $\partial^{i}K$ as the set of all $x\in\partial K$ such that $\operatorname{card}\{u\in{\mathbb{S}}^{n-1}:(x,u)\in N(K)\}=i$.

Proposition 4.1**.**

Let $t_{0}>0$ and let $f\colon{\mathbb{R}}^{n}\to{\mathbb{R}}$ be continuous on $K_{t_{0}}$. Then the right and left derivatives of $t\mapsto\int_{K_{t}\setminus K}f(x)\,dx$ exist on $(0,t_{0})$ and are given by

[TABLE]

Moreover, if

[TABLE]

then equation (4.6) remains valid for $t=0$, that is

[TABLE]

Proof.

Let $t\in(0,t_{0})$ and $r>0$ such that $t+r\leq t_{0}$. By the Steiner formula (4.4)

[TABLE]

Since $f$ is continuous on $K_{t_{0}}$, there exists $c\geq 0$ such that $|f(x+su)|\leq c$ for all $(x,u)\in N(K)$ and $s\leq t_{0}$. Moreover we have for each $i\in\{0,\ldots,n-1\}$ that

[TABLE]

for some $c_{i}\geq 0$ (depending on $t$ but not on $r$). By (4.5) and continuity of $f$ we can apply the dominated convergence theorem to conclude that

[TABLE]

By Corollary 4.4 in [5] the above right-hand side equals (note that $\omega_{1}=2$)

[TABLE]

By Proposition 4.1 in [5] we have for any compact set $A\subset{\mathbb{R}}^{n}$, that

[TABLE]

Since $\partial^{2}K_{t}=\emptyset$ we obtain the first assertion (4.6).

Similarly we obtain for the left derivative

[TABLE]

Writing ${\bf 1}\{t\leq\delta(K,x,u)\}={\bf 1}\{t=\delta(K,x,u)\}+{\bf 1}\{t<\delta(K,x,u)\}$, we can prove (4.1) as before.

Assuming (4.8), the proof of (4.9) again follows from the Steiner formula, dominated convergence and (4.10). Details are left to the reader. ∎

Let us define $I_{K}$ as the set of all $t>0$ such that

[TABLE]

In view of (4.5) the set $(0,\infty)\setminus I_{K}$ is at most countably infinite.

Theorem 4.2**.**

*Let $g\colon{\mathbf{N}}\rightarrow{\mathbb{R}}$ be measurable and $t_{0}>0$ such that $\operatorname{\mathbb{E}}|g(\eta_{t_{0}})|<\infty$ and $x\mapsto\operatorname{\mathbb{E}}g(\eta_{t}+\delta_{x})$ is continuous on $K_{t_{0}}$ for each $t<t_{0}$. Assume also that there exists $c>0$ such that *

[TABLE]

Then $t\mapsto\operatorname{\mathbb{E}}g(\eta_{t})$ is differentiable on $I_{K}\cap(0,t_{0})$ and the derivative is given by

[TABLE]

Moreover, if (4.8) holds, then

[TABLE]

Proof.

Let $t\in[0,t_{0})$ and let $r>0$ be such that $t+r\leq t_{0}$. The intensity measure of the process $\eta_{t+r}$ equals the sum of $\lambda_{t}$ and the restriction $\nu_{r}$ of $\lambda$ to $K_{t+r}\setminus K_{t}$. Thus, by (2.1) (for $\nu=\nu_{r}$ and $\theta=1$)

[TABLE]

where

[TABLE]

We have that

[TABLE]

where $c(t,r):=c\int_{K_{t+h}\setminus K_{t}}r(x)\,dx$. If $t>0$ then Proposition 4.1 shows the convergence $\lim_{r\to 0+}r^{-1}c(t,r)=c(t)$ for some $c(t)\in{\mathbb{R}}$. Therefore

[TABLE]

Under assumption (4.8) we have (4.9) so that the above remains true for $t=0$.

Again by Proposition 4.1 we have that

[TABLE]

first for $t>0$ (then the second term can be skipped) and then under the assumption (4.8) also for $t=0$.

Let us now assume that $t-r>0$. Then it follows as above that

[TABLE]

so that Proposition 4.1 shows that

[TABLE]

Choosing now $t\in I_{K}$, concludes the proof. ∎

A bounded function $g$ satisfies the integrability assumptions of Theorem 4.2 for all $t_{0}>0$, so that (4.13) holds under a rather weak continuity assumption for each compact $K$. Equation (4.14) requires (4.8), constituting a non-trivial assumption on $K$. This assumption is certainly satisfied if $|\mu_{i}|(K;{\mathbb{R}}^{n}\times{\mathbb{S}}^{n-1})<\infty$ for each $i\in\{0,\ldots,n\}$. This is the case, for instance, if $K$ has a positive reach or is a finite union of convex sets, see [5].

5 Crofton’s derivative formula for binomial processes

For $t\geq 0$ we let $K_{t}$ and $\lambda_{t}$ be as in Section 4. We assume that $\lambda(K)>0$. In this section we consider a binomial process $\xi_{t}^{(m)}$ of size $m\in{\mathbb{N}}$ with sample distribution $\lambda_{t}/\lambda_{t}(K_{t})$. This is a point process of the form

[TABLE]

where $X_{1},\ldots,X_{m}$ are independent random vectors in ${\mathbb{R}}^{n}$ with distribution $\lambda_{t}/\lambda_{t}(K_{t})$. It is convenient to let $\xi_{t}^{(0)}:=0$ be the null measure (a point process with no point.) Let $g\colon{\mathbf{N}}\rightarrow{\mathbb{R}}$ be a measurable and bounded function. Under certain assumptions on $K$ and $g$ we wish to prove that

[TABLE]

The heuristic and historic background of this formula is explained in [17]. If the boundaries of the sets $K_{t}$ are smooth, then (5.1) follows from more general results in [1]. Our proof is very different and relies on the Poisson version from Section 4.

Recall the definition of the set $I_{K}$ at (4.11).

Theorem 5.1**.**

Let $g\colon{\mathbf{N}}\rightarrow{\mathbb{R}}$ be measurable and bounded and let $m\in{\mathbb{N}}$ and $t_{0}>0$. Suppose that $x\mapsto\operatorname{\mathbb{E}}g(\xi^{(m-1)}_{t}+\delta_{x})$ is continuous on $K_{t_{0}}$ for each $t<t_{0}$. Then $t\mapsto\operatorname{\mathbb{E}}g(\xi^{(m)}_{t})$ is differentiable on $I_{K}\cap(0,t_{0})$ and the derivative is given by

[TABLE]

Moreover, if (4.8) holds, then

[TABLE]

Proof.

We are using the Poisson process $\eta_{t}$ introduced in the previous section and the well-known distributional identity (see e.g. [9, Proposition 3.8])

[TABLE]

where the function $h_{m}\colon[0,\infty)\rightarrow{\mathbb{R}}$ is defined by $h_{m}(u):=m!e^{u}u^{-m}$. Note that the derivative of $h_{m}$ is given by

[TABLE]

Let $t\in I_{K}\cap(0,t_{0})$. We apply (4.13) to the function $\tilde{g}(\varphi):={\bf 1}\{\varphi({\mathbb{R}}^{n})=m\}g(\varphi)$. Since $\operatorname{\mathbb{E}}g(\xi^{(m)}_{t})=h_{m}(\lambda_{t}(K_{t}))\operatorname{\mathbb{E}}\tilde{g}(\eta_{t})$ and $\tilde{g}(\eta_{t}+\delta_{x})={\bf 1}\{\eta_{t}({\mathbb{R}}^{n})=m-1\}g(\eta_{t}+\delta_{x})$, $x\in{\mathbb{R}}^{n}$, this gives us

[TABLE]

Taking into account (4.6), we obtain that the first summand equals

[TABLE]

By (4.13) the second summand equals

[TABLE]

Hence (5.2) follows. The proof of (5.3) is similar. ∎