Infinitely Divisible Random Probability Distributions with an Application to a Random Motion in a Random Environment

The infinite divisibility of probability distributions on the space P(R) of probability distributions on R is defined and related fundamental results such as the Lévy-Khintchin formula, representation of Itô type of infinitely divisible RPD, stable RPD and Lévy processes on P(R) are obtained. As an application we investigate limiting behaviors of a simple model of a particle motion in a random environment


Introduction
Let w be a random parameter defined on a probability space (W, P ), which corresponds to an environment.Suppose that for each w we have a sequence of families of random variables {X w nk (ω), 1 ≤ k ≤ n}, n = 1, 2, • • • defined on a probability space (Ω, P w ).We are interested in the limiting behavior of n k=1 X w nk (ω) under suitable centering.The usual problem is to find a limit distribution of n k=1 X w nk (ω), after suitable centering, under the joint probability law P (dwdω) = P (dw)P w 0 (dω).On the other hand, for each w let Ξ w n denote the distribution of n k=1 X w nk (ω) under the probability law P w .Then Ξ w n is a random variable with values in the space P(R) of probability distributions on R (shortly, random probability distributions and abbreviated as RPD).The problem is now to investigate the law convergence of Ξ w n as n → ∞, after suitable shift of location, with respect to P .Obtaining this latter result requires a finer structure of the model and more delicate analysis than the former.If X w nk (ω), n ≥ 1, are independent for each fixed w and n, and if we denote by q w nk , 1 ≤ k ≤ n the probability distribution of X w nk (ω) under P w , then Ξ w n = q w n1 * • • • * q w nn .Suppose that q w nk , 1 ≤ k ≤ n are also independent.In this case, if the law convergence of Ξ w n to some Ξ w holds, after suitable shift of location, then the limit Ξ w in law is expected to be an infinitely divisible RPD in the sense described later provided that a certain infinitesimal condition for {q w nk }, 1 ≤ k ≤ n is satisfied.Such an example appears in the study of limiting behavior of a simple model of a particle motion in a random environment, which was considered probably as the simplest model that exhibits many of limiting behaviors similar to those of Solomon (7), Kesten-Kozlov-Spitzer (5), Tanaka (9) and Hu-Shi-Yor (1).The analysis of this simple model was one of the motivation of the present investigation, We thus formulate several notions concerning the infinite divisibility of RPDs and then find fundamental results related to them.Next we investigate limiting behavior of the simple model of a particle motion in a random environment in the framework of our general theory.
A P(R)-valued random variable Ξ is called infinitely divisible if for any n ≥ 2 there exist i.i.d.
= means equivalence in distribution.
Let P(P(R)) denote the space of probability distributions on P(R).For Q and R in P(P(R)) we define Q ⊛ R ∈ P(P(R)) by An element Q of P(P(R)) is called infinitely divisible if for any n ≥ 2 there exists a Q n ∈ P(P(R)) For Q ∈ P(P(R)) we define a moment measure M Q ∈ P(R N ) by where N is the set of positive integers and Then M Q is an exchangeable probability measure on R N (shortly, M Q ∈ P ex (R N )).
Conversely any M ∈ P ex (R N ) can be represented as with a unique Q ∈ P(P(R)).Thus there is a 1 − 1 correspondence between P(P(R)) and P ex (R N ).Clearly M Q⊛R = M Q * M R , and Q is infinitely divisible if and only if M Q is infinitely divisible (in the sense that any finite dimensional projection of M Q has the property).The characteristic function Φ Q of M Q is defined by where µ is the characteristic function of µ and z = {z j } ∈ R N with z j = 0 except for finitely many j ∈ N.
One of the main theorems is to represent Φ Q as the 'Lévy-Khintchin formula'; naturally it resembles the 'classical Lévy-Khintchin formula'.There are three parts, a deterministic part, a Gaussian part and a Poisson-type part; those are connected by convolution.
Another main interest is to obtain a representation of Lévy-Itô type for an infinitely divisible RPDs.To do this we need to define two types of integrals by means of a Poisson random measure Π on P * (R) = P(R) \ {δ 0 }.The first one is defined simply by µ 1 * µ 2 * • • • for a realization j δ µ j of Π, which is convergent almost surely under a certain condition for Π (see (3.1)), and denoted by µ * Π(dµ).The second one called a renormalized Poisson integral and denoted by µ * Π reno (dµ), is defined for a wider class of Π than the first one (see the condition (3.2)).In a case where µ * Π(dµ) is defined, the renormalized Poisson integral is defined as µ * Π(dµ) * δ −α where α = P * (R) [|x|≤1] xµ(dx)m(dµ), m being the intensity measure of Π.
For the definition in a general case we have to take a limit procedure starting from those already defined.This definition is a highlight in our arguments.It corresponds to a compensated Poisson integrals in an ordinary case and also corresponds exactly to the third part of the Lévy-Khintchin formula.By virtue of renormalized Poisson integrals and the Lévy-Khinchin formula our tasks go somewhat in a smooth way.In fact, the representation of Lévy-Itô type of an infinitely divisible RPD is obtained.It consists of the deterministic part, the Gaussian part and the renormalized Poisson integral part, all being connected by convolution.
The definition of a stable RPD and its general structure are given.Examples of (not necessarily stable) RPDs are also given, some of which are useful in the section 7.
We now go to Lévy processes on P(R).First we define a P(R)-valued Lévy process (Ξ(t)) t≥0 in law and then give its representation of Lévy-Itô type.But at this moment the representation is realized on some probability space which, in general, differs from the one on which (Ξ(t)) t≥0 is defined.The represented process is continuous in probability but lacks the right continuity with left limits (cadlag) of the paths.So we have to make a cadlag modification of the process represented as Lévy-Itô type.We can do this by partly imitating [( 6): Lemma 20.2, Lemma 20.4].Thus we arrive at what is appropriate to be called a Lévy process on P(R).It is a P(R)valued cadlag process with a sort of 'stationary independent increments' in the sense described in the section 6.
Finally we apply the above general method to the study of limiting behavior, as x → ∞, of the hitting time σ w (x) to x > 0 for a particle performing a simple movement in a random environment.Let w = {w n } n≥0 be an increasing sequence with w 0 = 0 and set N (x, w) = max{n ≥ 0 : w n ≤ x}.We consider a Markov process (X(t), P w x ) on R + with generator A visual description of the movement of the particle will be explained in the section 7. Now suppose that w = {w n } n≥0 is a random sequence with the property that w ′ n = w n − w n−1 , n ≥ 1 are i.i.d.random variables with P (w ′ n > t) = e −λt , t ≥ 0, for some parameter λ > 0. Then w is called a random environment under which the particle moves randomly.Thus randomeness is two-fold, one is for the random environment and the other is for the random movement of the particle under a frozen environment.T! he limiting behavior of σ w (x) as x → ∞ is usually studied under the probability law P (dwdω) = P (dw)P w 0 (dω) (see (7) and ( 5) for more complicated models).When λ > 2 it can be proved that for almost all frozen environment w the distribution of {x −1/2 (σ w (x) − E w (σ w (x)))} under P w converges to a normal distribution N (0, c λ ) as x → ∞ with some constant c λ > 0 in Theorem 7.1 (iv) of the section 7 (see [(9), p.374] for a similar model).But this type of result does not hold for 0 < λ < 2. So we pose the following problem: Denote by Ξ σ w (x) the probability distribution of σ w (x) under the probability law P w 0 and regard it as an RPD.The problem is to find, depending on λ, various scaling limits in law of Ξ σ w (x) as x → ∞.In the special case λ > 2 in the above Ξ = N (0, c λ ) a.s., i.e., non-random.In general case, scaling limits in law are stable RPD taking values of infinitely divisible distributions on R. In the method and stating the results we mainly use Poisson integrals and renormalized Poisson integrals.Owing to the simplicity of our model some explicit computations are possible and so we can obtain finer limiting results in the case of hitting times.The detailed results depend on λ, and will be stated in several cases.

Infinitely divisible RPDs
We begin by stating some fundamental notions concerning infinitely divisible RPD, some of which may overlope with those already given in the introduction.
Let P(R) be the set of all probability distributions on R, which is equipped with the topology of weak convergence.We note that the topology is compatible with the following metric which makes (P(R), d) a complete separable metric space.On the other hand P(R) is a commutative semigroup with convolution * , i.e. for µ, ν ∈ P(R) We define a shift operator θ b (b ∈ R) and a scaling operator τ c (c > 0) on P(R) by and Let Ξ be a random probability distribution (shortly, RPD), namely, Ξ is a P(R)-valued random variable.Ξ is called infinitely divisible if for any n ≥ 2 there exist i. and Let Q ∈ P(P(R)).We say that Q is infinitely divisible if for any n ≥ 2 there exists a Q n ∈ P(P(R)) such that (1.2) holds.
Let Ξ be an RPD with the distribution Q ∈ P(P(R)).It is clear that Ξ is infinitely divisible if and only if Q is infinitely divisible.
Let denote by ℓ 0 (N) the totality of z = {z j } ∈ R N with z j = 0 except for finitely many j ∈ N.
We use the following notations: and for µ ∈ P(R) and bounded measureble functions F (x) and G(x) defined on R and R N respectively, we use the same notation which will make no confusion with z, x .
For Q ∈ P(P(R)) we define the moment measure M Q ∈ P(R N ) by (1.3), and the characteristic function Φ Q (z) of M Q by (1.4) for z ∈ ℓ 0 (N).
When Φ Q (z) can be expressed as exp Ψ(z) with a unique function Ψ(z) with Ψ(0) = 0 and such that Ψ(z) is continuous in z of length n for each n = 1, 2, • • • , we call Ψ(z) the moment characteristics of Q or of an RPD Ξ with distribution Q and write Ψ(z) = log Φ Q (z).For example if Q is infinitely divisible, its moment characteristics log Φ Q (z) exists.

Example(Gaussian distributions).
To know what distribution Q ∈ P(P(R)) is to be called a Gaussian distribution, we start from the corresponding moment measure M Q .We call M Q is Gaussian if any finite dimensional projection of M Q is Gaussian.We call Q a Gaussian distribution if the corresponding M Q is Gaussian.If a P(R)-valued random variable Ξ has a Gaussian distribution Q, then Ξ is also called Gaussian.A general form of a Gaussian RPD Ξ will be shown to be identical in law to N (c 1/2 ξ +b, a), where N (m, a) stands for the normal distribution with mean m ∈ R and variance a ≥ 0, and ξ is a Gaussian random variable with mean 0 and variance 1.The corresponding moment characteristics is These will be clarified soon.
We first establish the Lévy-Khintchin representation for the moment characteristics of arbitrary infinitely divisible RPDs.
Theorem 2.1.Q ∈ P(P(R)) is infinitely divisible if and only if its moment characteristics of Q has the following representation: for z ∈ ℓ 0 (N) where α ≥ 0, β ≥ 0 and γ ∈ R are constants, ρ is a σ-finite measure on R \ {0} satisfying and m is a σ-finite measure on P * (R) = P(R) \ {δ 0 } satisfying We remark that in the Lévy-Khintchin representation (2.2), the first term corresponds to a deterministic part, the second term to a Gaussian random part and the third term to a non-Gaussian random part, which will be seen in Theorem 3.3.
The proof of Theorem 2.1 is divided into several steps.Suppose that Q ∈ P(P(R)) is infinitely divisible.
Step 1.For each n ≥ 1 let Since M n is an infinitely divisible distribution on R n , its characteristic function has the following Lévy-Khintchin representation: where From the exchangeability of M n and the uniqueness of the Lévy-Khintchin representation it follows that A (n) , γ (n) and ν (n) have the following properties; The consistency condition among {M n } implies that a n , b n , γ n are constants in n, i.e.
Moreover by the non-negative definiteness of A (n) , a and b satisfy Step 2. For n > m ≥ 1 denote by π n,m the projection from R n to R m , i.e.
and by π ∞,m the projection from Then the consistency condition among {M n } and the uniqueness of the representation (2.5) imply that for n > m ≥ 1 By modifying the Kolmogorov extension theorem, one can show that there exists a unique exchangeable σ-finite measure ν (∞) on R N \ {0} satisfying that and Step 3.
Then there exists a Radon measure ρ on R \ {0} satisfying (2.3) and a σ-finite measure m on P * (R) satisfying (2.4) such that where δ k x stands for the point mass at x for the k-coordinate, and the integrand of the first integral of (2.8) is an infinite product of δ k x and {δ j 0 } j∈N\{k} . (Proof.)Let by the exchangeability of ν (∞) we have Hence there exists an exchangeable function ξ For each integer p ≥ 1, define a finite measure ν p on R N \ {0} by and define a σ-finite measure ν 0 on R N \ {0} by Note that ν p is an exchangeable finite measure on R N \ {0}.Hence by De Finetti's theorem (cf.(Ch)) there exists a finite measure m p on P * (R) such that On the other hand, noting that ν 0 is an exchangeable σ-finite measure on R N and ξ = 0 (ν 0 −a.e.), we have lim and from which and the exchangeability of ν 0 it follows (2.9) This means that ν 0 is concentrated on the union of the x j -axis over all j ∈ N excluding 0. Now we set ρ = π ∞,1 • ν 0 , then ρ satisfies (2.3).(2.9) and (2.7) imply that Finally, setting we see that m satisfies (2.4), hence completeing the proof of Lemma 2.2.

Poisson integrals on P(R)
In order to construct an infinitely divisible RPD we introduce a Poisson integral on P(R).
We consider a Poisson random measure.For its definition see (6), p.119.A basic formula for a Poisson random measure Π is for any non-negative measurable function f on the state space of Π where m is the intensity measure of Π.It determines the law of Π and characterizes a Poisson random measure in the sense that if Π is a counting random measure satisfying the above equation, then Π is a Poisson random measure.Our use of a Poisson random measure is mainly to represent various quantities in terms of Π.
Let m be a σ-finite measure on P * (R), and let Π m be a Poisson random measure on P * (R) with intensity measure m, which is defined on a complete probability space (Ω, F, P ).
We now introduce the following two conditions; and Let us first define a stochastic integral as a P(R)-valued random variable under the condition (3.1) by where is convergent uniformly in z on each compact interval.Hence * j µ j is convergent in the metric space (P(R), d).Thus Ξ of (3.3) is well-defined as a P(R)-valued random variable.

Renormalized Poisson integral
Next, we define a stochastic integral on P(R) under the condition (3.2).Let A ∈ B(P then m| A , the restriction of m on A, satisfies (3.1), so we set and This integral also defines a P(R)-valued random variable.Noting that for ε > 0, Lemma 3.2.Under the condition (3.2), there exists an RPD Ξ defined on (Ω, F, P ) such that Ξ ε converges to an RDP Ξ in probability as ε → 0+.The limit is denoted by Then the moment characteristics of Ξ is given by (Proof.)Recall that P(R) is a complete separable metric space with the metric d of (2.1).For 0 Since, by Lemma 3.1 Thus, by (3.8) Hence there exists an RPD Ξ defined on (Ω, F, P ) such that which proves the first part of Lemma 3.2.(3.7) can be easily verified using (3.4) for Ξ ε .
Remark 3.1 Althought Lemma 3.2 asserts that Ξ ε converges to the limit in probability, using the arguments of the section 6 we can show that the convergence holds P -almost surely.
Using this Poisson integral we obtain Itô-type representation for arbitrary infinitely divisible RPDs.
Theorem 3.3.Let Ξ be an infinitely divisible RPD of which moment characteristics is given by (2.2).Then where ξ is a normal random variable with mean 0 and variance 1, independent of Π m , and ν is an infinitely divisible distribution on R with characteristic function We remark that the Itô-type representation (3.11) consists of three terms, and each term corresponds to that of the moment characteristics (2.2) in Theorem 2.1.
Let P id (R) denote the totality of infinitely divisible probability distributions on R. It should be noted that an arbitrary RPD is not a P id (R)-valued RPD in general.However, it is true if the measure m of (2.2) is supported in P id (R).
Theorem 3.4.Let Ξ be an infinitely dibisible RPD.Suppose that the correponding m of (2.2) is supported in P id (R).Then Ξ is an infinitely divisible RPD, taking values in P id (R).
(Proof.)It is obvious that for any ε > 0, Ξ ε of (3.6) is P id (R)-valued since Ξ ε is defined by convolutions of finite number of elements of P id (R) or δ 0 .Moreover Ξ is a limit of such Ξ ε P -a.s. by (3.10), which implies the conclusion of Theorem 3.4.
Let (U, B U , α) be a σ-finite measure space, and let U ∋ u → µ u ∈ P * (R) be a measurable mapping.Suppose that Then for a Poisson random measure Π α with intensity measure α, one can define a Poisson integral U µ u * α(du) on P(R) analogously to (3.3).Moreover, under the condition where

Stable RPDs
Let Ξ be an RPD on R. Ξ is called stable if for any n ≥ 2 and n independent copies Ξ 1 , • • • , Ξ n of Ξ there exists constants b n ∈ R and c n > 0 such that In particular, if b n = 0 in (4.1) for any n ≥ 1, Ξ is called strictly stable.
Let Q be the distribution of Ξ. Ξ is a stable RPD if and only if Q ∈ P(P(R)) is stable, i.e. for any n ≥ 2 there exists b n ∈ R and c n > 0 such that Since (4.1) implies that E(Ξ) is a stable distribution on R, it holds that c n = n 1/α with an 0 < α ≤ 2. (cf.(Fe)) In this case Ξ and Q are called α-stable.
(1) Let Ξ be a 2-stable RPD.Then its moment characteristics is of the following form: where α ≥ 0, β ≥ 0 and γ ∈ R are constants.Moreover it holds that where N (m, a) denotes the normal distribution with mean m and variance a ≥ 0, and ξ is a random variable with a standard normal distribution N (0, 1).

.5)
In this case it holds that where ν ∈ P id (R) has the characteristic function and Π m is a Poisson random measure on P * (R) with the intensity measure m.

Examples
Throughout this section we always take z ∈ ℓ 0 (N).
Example 1 (non-random case) Let µ 0 be an infinitely divisible distribution on R with characteristic exponent and set Ξ = µ 0 .
Then Ξ is an infinitely divisible RPD with the moment characteristics which is the first term of the Lévy-Khintchin representation (2.2).
Example 2 Let X a random variable with the distribution µ 0 of (5.1), and set Then Ξ is an infinitely divisible RPD with the moment characteristics The last term corresponds to the third term of the Lévy-Khintchin representation (2.2) with m = R\{0} δ δx ρ(dx).
Example 3. Let e t be the exponential distribution with mean t > 0, and for λ > 0 set Let Π ρ λ be a Poisson random measure on (0, ∞) with intensity measure ρ λ .If 0 < λ < 1, (3.12) is satisfied and Ξ = (0,∞) defines an infinitely divisible RPD which takes values in P id (R) by Theorem 3.4, having the characteristic function we have the Lévy-Khintchin representation for Ξ where ν Ξ is given by which satisfies that if 0 < λ < 1 Note that by (3.4) the moment characteristics and the measure m in (2.2) is given by Let us denote by F λ the distribution function of the first moment E(Ξ) ∈ P(R).Then using (5.7) and we obtain the Laplace transform of F λ with c λ = λΓ(λ + 1)Γ(1 − λ).Thus F λ is a λ-stable distribution on (0, ∞).The stable RPD (5.3) and the distribution function F λ will appear in Theorem 7.1 and Corollary 7.3.
The stable RPD of (5.10) and the stable distribution function G λ will appear in Theorem 7.2 and Corollary 7.3.
Example 4. For t > 0 let θ t • e t be the shifted exponential distribution of e t with mean 0, i.e.
Let ρ λ be of (5.2).Then it satisfies the condition (3.12) if and only if 0 < λ < 1, which is easily verified using However, even in the case 1 ≤ λ < 2, it is possible to define the Poisson integral (5.15) Namely, for ε > 0, we set Then the characteristic function is it holds that P -a.s.
uniformly in each bounded interval.Thus there exists an RPD Ξ, taking values in P id (R), such that lim and the characteristic function of Ξ is given by (5.16) We identify the Poisson integral (5.15) with this limit Ξ.
Note that Ξ is a λ-stable RPD, taking values in P id (R), with the moment characteristics log Φ(z) = (0,∞) and the corresponding m of (2.2) is given by m = (0,∞) Since by (5.5), (5.16) turns to with ν Ξ of (5.6), by (5.13), (5.14) and (5.18) we obtain the following relation between Example 3 and Example 4, Let H λ denote the distribution function of the first moment E(Ξ).Then H λ is a λ-stable distribution on R, and after tedious computations we have the characteristic function where c 1 = πe −1 , γ < 0, c λ > 0 and β ∈ (−1, 1) are given by It may be of interest that the Lévy measure ν of H λ is supported fully by R \ {0} as follows, although the Lévy measure ν Ξ of Ξ is supported in (0, ∞) by (5.6).
The stable RPD of (5.15) with 1 ≤ λ < 2 and the stable distribution function H λ will appear in Theorem 7.1 and Corollary 7. (ii) for a given Lévy process, the Lévy-Itô representation for the sample path is valid, that is, (a) a collection of jump times and jump sizes forms a Poisson point process, and the jump part of the process is recovered from the Poisson point process.
(b) the continuous part is nothing but a Gaussian Lévy process independent of the the jump part.
For Lévy processes on P(R) in law, the first one holds essentially in the same way as for the classical case.However for Lévy processes on P(R), the second one seems difficult.The reason is that for the sample path Ξ(t) the jump size µ should be defined as but existence of such µ does not follow automatically, and the uniqueness of µ does not hold in general.
In this section for a given infinitely divisible probability distribution Q on P(P(R)), we will obtain a Lévy process (Ξ(t)) on P(R) such that Ξ(1) has the distribution Q, by constructing a Lévy-Itô-type representation.
To do this the main task is to define a Poisson integral as a P(R)-valued cadlag process.
(Proof of Theorem 6.1.) By Lemma 6.3 and Lemma 6.5 it holds that for every η > 0 lim Thus it holds that P -a.s.lim Hence there exists a P(R)-valued cadlag process Ξ = (Ξ t ) such that lim completing the proof of Theorem 6.1.
Let Q be the distribution of an infinitely divisible RPD, of which moment characteristics Φ Q is represented by Theorem 2.1 in terms of the characteristic quantities (α, γ, ρ, β, m).Now we construct a Lévy process Ξ = (Ξ(t)) on P(R) such that Ξ(1) has the distribution Q.
The proof is immediate from Theorem 6.1 and Lemma 3.2.

A simple particle motion in a random environment
Given a strictly increasing sequence w = {w n } n≥0 with w 0 = 0, we introduce a Markov process (X(t) = X(t, ω), P w x )) on R + , generated by where N (x, w) = max{n ≥ 0 : and {P w x } x≥0 is a Markov family of probability measures on (Ω, F).The sequence w = {w n } n≥0 is called an environment and each w n a ladder point.
A visual description of the path of the Markov process is as follows.A particle moves to the right with constant speed 1 and at an exponential random time T 1 with mean 1 it jumps back to the left adjacent ladder point w N (X(T 1 −),w) .Here X(t) denotes the position of the particle at t.After jumping back to w N (X(T 1 −),w) the particle starts afresh and continues the same uniform motion with speed 1 until the next exponential random time, at which the particle again continues similar movements.
We now consider the environment to be random.Specifically we consider the case where the randomness is introduced by the assumption that w ′ n = w n − w n−1 , n ≥ 1 are i.i.d.random variables with exponential distribution of parameter λ > 0 defined on a probability space (W, P ).In other words, {w n } n≥1 forms a Poisson point process on (0, ∞) with intensity λ > 0.
Let σ w (x) be the hitting time to x > 0 for (X(t), P w x ).For x > 0, let Ξ σ w (x) be the distribution of the hitting time σ w (x) under the probability law P w 0 , so Ξ σ w (x) is an RPD.Our problem is to investigate a scaling limit of Ξ σ w (x) as x → ∞.
In order to describe scaling limit results of the RPD Ξ σ w (x) , we first introduce a Poisson random measure.For λ > 0, let Π ρ λ be a Poisson random measure on (0, ∞) with intensity measure and we denote by Πρ λ the compensated Poisson random measure of Π ρ λ , i.e.
For a sequence of RPDs {Ξ n } and an RPD Ξ ∞ , let denote the distributions of Ξ n and Ξ ∞ by Recall that for t > 0, e t is the exponential distribution with mean t > 0, and θ t • e t is the shifted exponential distribution with mean 0, which appears in Example 4 of the section 5.
Denote by m w (x) the expectation of σ w (x) with respect to P w 0 .Then we obtain the following results. Then where Let λ > 1.Then by (8.3) in Lemma 8.1 below it holds that So we next discuss another scaling limit for the RPD Ξ σ w (x) , replacing the random centering where ξ is a normal random variable with mean 0 and variance 1.
where c λ is of (7.7) and ξ λ is a normal random variable with mean 0 and variance We remark that the limiting RPDs in Theorems 7.1 and Theorem 7.2 are stable RPDs taking values in P id (R) as seen in Examples 3 and 4 of the section 5.
Recall the probability measure P on W × Ω, which is defined by P (dwdω) = P (dw)P w 0 (dω).We regard σ w (x) = σ w (x, ω) as a random variable on the probability space (W × Ω, P ).From Theorem 7.1 and 7.2 it follows immediately that Corollary 7.3.(i) Let 0 < λ < 1.Then for any a > 0 where F λ is a λ-stable distribution function with the Laplace transform (5.9).
(ii) Let 1 ≤ λ < 2. Then for any a ∈ R where H λ and G λ are the λ-stable distribution functions which appear in Example 4 and Example 3 of the section 5.
(Proof.)Let 0 < λ < 1.By Theorem 7.1 (i) it holds that lim Hence by Theorem 7.1 we see that and by Example 3 of the section 5, the distribution of E (Ξ λ ) coincides with F λ , completing the proof of (i).The remaining cases can be proved in the same way, so we omit them.
8 Proof of Theorem 7.1 In order to prove Theorem 7.1 we prepare several lemmas.
Lemma 8.1.Let x > 0. (i) Before proceeding to discuss convergence results we review the topology of P(R) and P(P(R)).
Recall that the metric d of P(R) is defined by (2.1), which is compatible with the topology of weak convergence and makes P(R) a complete separable metric space.
Let R = R ∪ {∞} be the one point compactification of R. Then P(R) is compact with respect to the weak topology.Since P(R) is regarded as a subset of P(R) and the topology of P(R) coincides with the relative topology induced by P(R), it holds that Then each Q ∈ P(R) is identified with Q ∈ P(R) defined by so that by Perseval's identity, for every For f ∈ S(R), denote by f the extension of f on R with f (∞) = 0.By the Stone-Weierstrass theorem, the linear hull of the family consisting of functions of µ of the form m j=1 µ, fj ; m ≥ 1, f j ∈ S(R) where f j (1 ≤ j ≤ m) are real and ∈ S(R), m ≥ 1, and of constant functions is dense in C(P(R)).Hence (8.5) implies that Q n converges weakly to Q in P(P(R)) and Q n (P(R)) = Q ∞ (P(R)) = 1.Finally since the topology of P(R) coincides with the relative topology induced by P(R),we obtain For two sequences of random variables {X n } and {Y n } if and Z n converges to 1 in probability, as n → ∞, we use the notation In the sequel we use M (x, w) instead of N (x, w), since M (x, w) is an F w n -stopping time, where F w n is the σ-algebra generated by {w k } 1≤k≤n .The following lemma makes this replacement clear.For example, the lemma justifies to write For each x > 0 we introduce a random counting measure Π (x) on (0, ∞) and its compensated one Π(x) by which will be used later.Furthermore, it holds that for any 0 < λ < 2, has finite expectation, and (1 + θ(x, w ′ k )) = 1 P − a.s.
(i) If f satisfies that for some η > 0 (Proof.) (i) For the proof we may assume that f is nonnegative.For any ξ > 0 and ε > 0, .
The first term vanishes as x → ∞, and by (8.13) the second term turns to On the other hand Thus we obtain = E e −ξ Πρ λ ,f , which yields (8.14).Next suppose that f satisfies (8.15).From (i) it follows that for ε > 0, Moreover using (8.9) and ( 8 = E e iz Πρ λ ,f , completing the proof of (ii).
We use the following facts, which are verified by elementary calculations.
For the proof of (ii) and (iii) let ε(x) = (2x log x) −1/2 for λ = 2 and ε(x) = x −1/2 for λ > 2. Use For n ≥ 1 we set Then Y n is an F w n -martingale with mean 1.Since M (x, w) is an F w n -stopping time for each x > 0 and the optional sampling theorem is applicable, we have i.d.RPDs Ξ 1 , • • • , Ξ n such that (1.1) holds.The operations * , θ b (b ∈ R) and τ c (c > 0) are extended to operators ⊛, Θ b and T b on P(P(R)) by .4) (Proof.)Assuming that m is a finite measure, we show(3.4),since the extension to a general m satisfying (3.1) is straightforward.If m is the zero measure, (3.4) is trivial, so we assume m(P * (R)) > 0. Let {ξ j } be an i.i.d.sequence of P(R)-valued random variables with the common distribution m/m(P * (R)), and let N be a Poisson random variable with parameter m(P * (R)), which is independent of {ξ j }.
and denote their moment characteristic functions by Φ Qn and Φ Q respectively.Then Q n converges to Q ∞ weakly as n → ∞ if and only if