Weak convergence to isotropic complex \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$S\alpha S$\end{document}SαS random measure

In this paper, we prove that an isotropic complex symmetric α-stable random measure (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0<\alpha<2$\end{document}0<α<2) can be approximated by a complex process constructed by integrals based on the Poisson process with random intensity.

Stroock [] proved that when n → ∞, the law of U n converges weakly in the Banach space C( [, T]) of continuous functions on [, T], to a Wiener measure. Bardina   On the other hand, on approximation to complex Brownian motion, Bardina [] considered the process U θ where i is an imaginary unit. Bardina proved that if θ ∈ (, π) ∪ (π, π), when n → ∞, P θ n the image law of U θ n in the Banach space C([, T], C) converges weakly to the law of a complex Brownian motion in C([, T], C). When θ = π , P θ n , converges weakly to the law of √ } is a standard Brownian motion. Inspired by the above works, in this paper, we will prove similar results about an isotropic complex SαS random variable. Define where {N α (t), t ∈ [, ∞)} is a Poisson process with random intensity  A . In the trivial case, when θ = , the process X θ n (t) is deterministic, and when n tends to infinity, X θ n (t) goes to infinity. When θ = π , the process X θ n (t) is real and () becomes this case was studied by Dai and Li [].
We will prove under certain conditions, when θ ∈ (, π) ∪ (π, π), that the law of X θ n converges weakly in C([, ], C) to the law of isotropic complex SαS random measure.
The rest of the paper is organized as follows. First we give preliminaries and the main result, then we present some lemmas, including proving the tightness and identification of the limit law, to prove the main result.

Preliminaries and the main result
Now we give some preliminary definitions and the main result.
Definition . ([]) Let (  , F  , P  ) be a probability space. Suppose that A is a nonnegative random variable on the probability space (  , F  , P  ). Let ( , F, P) = (  ×  , F  × F  , P  × P  ) be the underlying probability space of this paper. Let N = {N(t), t ≥ } be a counting process on ( , F, P) satisfying the following assumptions: (a) When A >  is given, N is a Poisson process on (  , F  , P  ) with intensity  A ; (b) When A = , N =  a.s. Then we call N as the Poisson process with random intensity  A .
Throughout this paper, we define the Poisson process {N α (t), t ≥ } with random intensity  A , we assume that A is a strictly α  -stable random variable with respect to (  , F  , P  ), totally skewed to the right, with Laplace transform given by Considering the sequences X θ n defined by (), we have the following theorem.
Theorem . Let the process {X θ n (t),  ≤ t ≤ }, considering P θ n the image law of X θ n , in the Banach space C([, ], C). Then, if θ ∈ (, π) ∪ (π, π), when n tends to infinity, P θ n converges weakly to the law in C([, ], C) of isotropic complex SαS random measure The main idea of the proof of the theorem is that, A converges weakly to complex Brownian motion independent of A. Then, according to the idea of [], we need to check that if θ ∈ (, π) ∪ (π, π), the family P θ n is tight and the law of all possible weak limits of P θ n is the law of a complex Brownian motion. We split the proof of Theorem . into two parts. We first prove the tightness of the process X θ n and then identify the limit law of the process X θ n . In this paper, K denotes a positive constant independent of n, it may change value from one expression to another.

Proof of tightness We give an auxiliary process
For any n ≥ , we have and Lemma . There exists a constant K such that, when θ ∈ (, π) ∪ (π, π), for any s, t ∈ [, ] and n > , Proof Without loss of generality, we assume s < t. Then Using the independence increments of the Poisson process, we obtain where · denotes the modulus of the complex number. It is easy to obtain Then Next we calculate B  . Considering the fact that By the independence increments of the Poisson process, we have Then we obtain that This completes the proof.

Lemma . The set of laws of {X
Proof To prove the tightness, we have to prove that the law corresponding to the real part and the imaginary part of the process X θ n is tight. In fact, it is almost sure that Y θ n () =  for all n ≥ . Lemma . and Theorem . of [] show that the set of the law of the real part and the imaginary part of the process {Y θ n } n≥ is tight in C([, ], R) for a fixed constant A. Hence, for any ε > , there exists a compact set S ⊂ C([, ], R) such that Because A is a finite positive random variable, then there exists a bounded set N ε such that Observe that the set ε := {af ; a ∈ N ε , f ∈ S ε } is compact in C([, ], R). Then, combining () and (), for any n ≥ , we have Then, combining () and (), we obtain that, for any ε > , the real part of the process X θ n has P Re X θ n ∈ ε ≥ ε.
Using a similar idea, we obtain that, for any ε > , the imaginary part of the process X θ n has P Im X θ n ∈ ε ≥ ε.
This completes the proof.
Identification of the limit law Let {P θ n k } be a subsequence of {P θ n } weakly convergent to some probability P θ . If θ ∈ (, π) ∪ (π, π), when A >  is given, then the canonical process Z = {Z t (X n ) =: X n (t)} is a complex Brownian motion under the probability P θ , that is, the real part and the imaginary part of the process are two independent Brownian motions with respect to the probability space (  , F  , P  ).
Using Paul Lévy's theorem, it suffices to prove that under P θ , when A >  is given, the real part and the imaginary part of the canonical process are both martingales with respect to the natural filtration {F t }, with quadratic variations Re[Z], Re[Z] t = At, Im[Z], Im[Z] t = At, and covariation Re[Z], Im[Z] t =  with respect to the probability space (  , F  , P  ).
Next, we first prove the martingale property and then prove the quadratic variations and covariation; these proofs are similar to the proof in []. Here, we give a sketch of the proof with some lemmas.
Let θ ∈ (, π) ∪ (π, π), in order to prove that under P θ the real and imaginary parts of the canonical process Z are martingales with respect to its natural filtration {F t }, we have to prove that for any s  ≤ s  ≤ · · · ≤ s k ≤ s, k ≥  and for any bounded continuous function ϕ : C k → R such that and We first consider ().
Since {P θ n } weakly converges to P θ , combining Lemma ., we have So it suffices to prove that E ϕ X θ n (s  ), . . . , X θ n (s k ) I θ ,n (t) -I θ ,n (s) converges to zero when n → ∞. Now, we just need to prove that In fact  -cos θ → , n → ∞.
Using the same idea of proof (), we can obtain the proof of ().
We give the following auxiliary lemma.
Combining Lemma . and that P θ n converges weakly to P θ , we can obtain that when n → ∞, above two integrals converge to A(ts)E[ϕ(Z s  , . . . , Z s k )]. This completes the proof of ().

Conclusions
We prove that an isotropic complex symmetric α-stable random measure ( < α < ) can be approximated by a complex process constructed by integrals based on the Poisson process with random intensity, which gives a new method to construct complex-valued random measure.