A limit theorem for the number of sign changes for a sequence of one-dimensional diffusions

The (normalized) number of sign changes for a weakly convergent sequence of onedimensional diffusion processes is considered. The limit theorem for this number is established.

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Datum:	2008
1. Verfasser:	Kulik, A.M.
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spelling	irk-123456789-45552009-12-07T12:00:28Z A limit theorem for the number of sign changes for a sequence of one-dimensional diffusions Kulik, A.M. The (normalized) number of sign changes for a weakly convergent sequence of onedimensional diffusion processes is considered. The limit theorem for this number is established. 2008 Article A limit theorem for the number of sign changes for a sequence of one-dimensional diffusions / A.M. Kulik // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 2. — С. 79–92. — Бібліогр.: 14 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4555 519.21 en Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	The (normalized) number of sign changes for a weakly convergent sequence of onedimensional diffusion processes is considered. The limit theorem for this number is established.
format	Article
author	Kulik, A.M.
spellingShingle	Kulik, A.M. A limit theorem for the number of sign changes for a sequence of one-dimensional diffusions
author_facet	Kulik, A.M.
author_sort	Kulik, A.M.
title	A limit theorem for the number of sign changes for a sequence of one-dimensional diffusions
title_short	A limit theorem for the number of sign changes for a sequence of one-dimensional diffusions
title_full	A limit theorem for the number of sign changes for a sequence of one-dimensional diffusions
title_fullStr	A limit theorem for the number of sign changes for a sequence of one-dimensional diffusions
title_full_unstemmed	A limit theorem for the number of sign changes for a sequence of one-dimensional diffusions
title_sort	limit theorem for the number of sign changes for a sequence of one-dimensional diffusions
publisher	Інститут математики НАН України
publishDate	2008
url	http://dspace.nbuv.gov.ua/handle/123456789/4555
citation_txt	A limit theorem for the number of sign changes for a sequence of one-dimensional diffusions / A.M. Kulik // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 2. — С. 79–92. — Бібліогр.: 14 назв.— англ.
work_keys_str_mv	AT kulikam alimittheoremforthenumberofsignchangesforasequenceofonedimensionaldiffusions AT kulikam limittheoremforthenumberofsignchangesforasequenceofonedimensionaldiffusions
first_indexed	2025-07-02T07:46:15Z
last_indexed	2025-07-02T07:46:15Z
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fulltext	Theory of Stochastic Processes Vol. 14 (30), no. 2, 2008, pp. 79–92 UDC 519.21 ALEXEY M. KULIK A LIMIT THEOREM FOR THE NUMBER OF SIGN CHANGES FOR A SEQUENCE OF ONE-DIMENSIONAL DIFFUSIONS The (normalized) number of sign changes for a weakly convergent sequence of one- dimensional diffusion processes is considered. The limit theorem for this number is established. 1. Introduction In this paper, we consider a sequence of one-dimensional diffusion processes satisfying SDE’s dXn(t) = an(Xn(t)) dt+ σn(Xn(t)) dW (t), t ∈ R+, n ∈ N, (1) weakly convergent to a diffusion process satisfying SDE dX(t) = a(X(t)) dt+ σ(X(t)) dW (t), t ∈ R+. (2) We take some discretization parameter α > 0 and consider the processes φt n = n−α 2 ∑ 0≤k<tnα 1IXn(kn−α)·Xn((k+1)n−α)<0, t ∈ R+, n ∈ N. (3) We call φt n the (normalized) number of sign changes for the diffusion Xn, corresponding to the time discretization {kn−α, k ∈ Z+}. The question under discussion is whether the sequence {φ·n} converges weakly and what is the structure of the limiting process. This question has a long history. In the 1950s, I.I.Gikhman proposed a general method for investigation of the limiting behavior of functionals of such a type, based on the asymptotic study of difference equations for the family of corresponding characteristic functions ([1],[2]). Later on, this method was developed further and used widely by M.I.Portenko and his pupils (see the discussion and overview in [3]). This method appears to be powerful enough to provide the limit theorems for {φ·n} in quite delicate situations where σn converge to σ only in a weak L∞-sense (i.e., where {Xn} is a sequence of diffusions with oscillating coefficients), see the recent preprint [4]. We investigate the limiting behavior of {φ·n} using another approach introduced in [5]. In [5], the general limiting theorem was proved, being in fact a generalization of the Dynkin’s criterion for the L2-convergence of W-functionals of a given Markov process in terms of their characterictics (i.e., expectations). In this paper, we show that this theorem can be applied in order to prove the limit theorem for {φ·n} in a situation where {Xn} is a sequence of diffusions with oscillating coefficients. Our results differ from those obtained in [4], since we do not suppose, in general, the coefficients of (1) to have the 2000 AMS Mathematics Subject Classification. Primary 60J55, 60J60, 60F17. Key words and phrases. The number of sign changes, additive functional, characteristic, Markov approximation, local time. This research has been partially supported by the Ministry of Education and Science of Ukraine, project N GP/F13/0095 79 80 ALEXEY M. KULIK form an(x) = nαâ(nαx), σn(x) = σ̂(nαx). On the other hand, some assumptions of [4] are less restrictive than those made in the current paper. For instance, in [4], non-uniform partitions of the time axis are allowed. 2. The main result We formulate the main statement for processes (1),(2) under supposition that an ≡ 0, a ≡ 0. This allows us to simplify notation but does not restrict generality, since one can reduce the general case to the one indicated before, using the following standard trick. If Xn, X are given by (1),(2) with non-trivial an, a, then the processes X̃n(t) = Sn(Xn(t)), X̃(t) = S(X(t)) with Sn(x) = ∫ x 0 e − y� 0 2an(z) σ2 n(z) dz dy, S(x) = ∫ x 0 e − y� 0 2a(z) σ2(z) dz dy, x ∈ R satisfy SDE’s with the coefficients ãn ≡ 0, ã ≡ 0 and σ̃n(x) = S′ n(S−1 n (x))σn(S−1(x)), σ̃(x) = S′(S−1(x))σ(S−1(x)), x ∈ R, respectively. Since the mappings x !→ Sn(x) preserve the sign, the functionals φn given by (3) coincide for the processes Xn and X̃n. Together with the processes Xn, X , we consider the re-scaled processes Zn(t) = n α 2 Xn(tn−α), Zn(t) = n α 2 X(tn−α), t ∈ R+, n ∈ N. One can easily see that if Xn(t) = Xn(0) + ∫ t 0 σn(Xn(s)) dW (s), X(t) = X(0) + ∫ t 0 σ(X(s)) dW (s), t ∈ R+, then Zn(t) = Zn(0) + ∫ t 0 �n(Zn(s)) dWn(s), Zn(t) = Zn(0) + ∫ t 0 �n(Zn(s)) dWn(s), t ∈ R+, with �n(z) = σn(n−α 2 z), �n(z) = σ(n− α 2 z),Wn(t) = n α 2 W (tn−α). Denote, by Σ, the class of measurable functions b : R → R that are globally bounded and separated from 0 on every finite interval. If σn, σ belong to Σ, then Eqs. (1),(2) (with an ≡ 0, a ≡ 0) uniquely define Feller Markov processes (see [7], Chapter 6 §3). For R ≥ 1, denote, by ΣR, the class of functions b ∈ Σ such that R−1 ≤ b2(x) ≤ R, x ∈ R. For the process X , its local time at the point x is defined via the Tanaka formula: LX(t, x) = \|X(t) − x\| − \|X(0) − x\| − ∫ t 0 sign(X(s) − x) dX(s), t ∈ R+. Denote Kt(x, y) = ∫ t 0 1√ 2πs e− (y−x)2 2s ds, t ∈ R+, x, y ∈ R. Define the weak L∞- convergence of a sequence {fn} ⊂ L∞(R) by the relation fn w→ f df⇔ ∫ R fn(y)g(y) dy → ∫ R f(y)g(y) dy, g ∈ L1(R). The main statement of the paper is given in the following theorem. A LIMIT THEOREM FOR THE NUMBER OF SIGN CHANGES 81 Theorem 1. Suppose that the following conditions hold true: (A) σ ∈ ΣR, σn ∈ ΣR for some R > 0, (B) Equation (2) possesses the path-wise uniqueness property, (C) For every T > 0, sup x∈R,t≤T ∣∣∣∣∫ R [σ−2 n (y) − σ−2(y)]Kt(x, y) dy ∣∣∣∣ → 0, n→ ∞, (D) (D) sup x∈R,t≤T 1 T ∣∣∣∣∫ R [�−2 n (y) − (�n)−2(y)]Kt(x, y) dy ∣∣∣∣ → 0, n, T → +∞, (E) There exists � ∈ Σ such that �−2 n w→ �−2. Then the sequence Xn converges weakly to the process X, and the sequence φ·n con- verges weakly to the process φ· = cLX(·, 0) w.r.t. topologies of uniform convergence on compacts in C(R+) and D(R+), respectively. The constant c is equal to c = ∫ R �−2(z)P ( Z(1) · z < 0 ∣∣∣Z(0) = z ) dz with the diffusion process Z defined by SDE dZ(t) = �(Z(t)) dW (t). One can interpret the statement of Theorem 1 in the following way. The process X represents the ”macroscopic” behavior of the sequence Xn, while the process Z represents the ”microscopic” behavior of the same sequence at the vicinity of the point 0. The ”shape” of the limiting functional φ is completely defined by the macroscopic behavior of the sequence: up to a some constant, nothing but the local time of X can occur at the limit. But this constant, having a natural interpretation as the ”intensity” of φ, depends essentially on the microscopic behavior of the sequence. Examples given in Section 6 below demonstrate that the macro- and microscopic descriptions for the sequence Xn may differ essentially. 3. Weak convergence of additive functionals of a sequence of Markov chains Our proof of Theorem 1 is based on the general theorem on the convergence in distri- bution of a sequence of additive functionals of Markov chains given in [5]. In this section, this theorem is formulated, and the necessary auxiliary notions are introduced. Suppose the processes Xn(·), X(·) to be defined on R+ and to take their values in a locally compact metric space (X , ρ). We say that the process X possesses the Markov property at the time moment s ∈ R+ w.r.t. filtration {Gt, t ∈ R+}, if X is adapted with this filtration and, for every k ∈ N, t1, . . . , tk > s, there exists a probability kernel {Pst1...tk (x,A), x ∈ X , A ∈ B(X k)} such that E[1IA((X(t1), . . . , X(tk)))\|Gs] = Pst1...tk (X(s), A) a.s., A ∈ B(X k). The measure Pst1...tk (x, ·) has a natural interpretation as the conditional finite-dimen- sional distribution of X at the points t1, . . . , tk under condition {X(s) = x}. Below we suppose the discretization parameter α > 0 to be fixed and claim the process X to possess the Markov property (w.r.t. its canonical filtration) at every s ∈ R+, and every process Xn to possess this property (w.r.t. its canonical filtration) at the points of the type in−α, i ∈ Z+; this means that every process Xn is, in fact, a Markov chain with the time scale proportional to n−α. 82 ALEXEY M. KULIK Consider a sequence of non-negative additive functionals {φs,t n , 0 ≤ s ≤ t}, n ≥ 1 of the processes Xn of the form φs,t n = ∑ k:s≤kn−α<t Fn ( Xn(kn−α), . . . , Xn((k + L)n−α) ) , 0 ≤ s ≤ t, (4) where L ∈ Z+ and Fn are non-negative measurable functions on XL+1. For the functional φn, its characteristic fn (the analogue of the characteristic of a W-functional) is defined by the formula fs,t n (x) = E[φs,t n \|Xn(s) = x], s = in−α, i ∈ Z+, t ≥ s, x ∈ X . (5) The process Xn possesses the Markov property w.r.t. its canonical filtration at the time moments s = in−α, i ∈ Z+, and functional (4) is a function of the values of Xn at the finite family of such time moments. Therefore, the mean value in (5) is well defined as the integral w.r.t. family of the conditional finite-dimensional distributions {Pst1...tk (x, ·), t1, . . . , tk > s, k ∈ N} of the process Xn. The following result ([5], Theorem 1) is an analogue of the well-known theorem by E.B.Dynkin that describes the convergence of W-functionals in the terms of their charac- teristics ([7], Theorem 6.4). Denote T = {(s, t) : 0 ≤ s ≤ t} ⊂ R2 and define the random broken lines ψn corresponding to φn by ψs,t n = φ(j−1)n−α,(k−1)n−α n + (nαs− j + 1)φ(j−1)n−α,jn−α n + (nαt− k + 1)φ(k−1)n−α,kn−α n , s ∈ [(j − 1)n−α, jn−α) , t ∈ [(k − 1)n−α, kn−α) . Theorem 2. Let the sequence of the processes Xn be given, providing a Markov ap- proximation for the homogeneous Markov process X (see Definition 1 below), and let the sequence {φn} be defined by (4). Suppose that the following conditions hold true: 1. The functions Fn(·) are non-negative, bounded on XL+1, and uniformly converge to zero: δ(Fn) = sup x0,...,xL∈X Fn(x0, . . . , xL) → 0, n→ ∞. 2. There exists a function f , that is the characteristic of a certain W-functional φ of the limiting process X, such that, for every T ∈ R+, sup s=in−α,t∈(s,T ) sup x∈X \|fs,t n (x) − f t−s(x)\| → 0, n→ ∞. 3. The limiting function f is continuous w.r.t. variable x, locally uniformly w.r.t. time variable, i.e., for every T ∈ R+, sup t≤T ∣∣f t(x) − f t(y) ∣∣ → 0, ‖x− y‖ → 0. Then ψn ≡ {ψs,t n , (s, t) ∈ T} ⇒ φ ≡ {φs,t, (s, t) ∈ T} in a sense of weak convergence in C(T,R+). The notion of Markov approximation introduced in [8] is the key one in Theorem 2. Below we give the slightly modified definition, taking into account that, in the current considerations, the time discretization points have the form in−α, i ∈ Z+. A LIMIT THEOREM FOR THE NUMBER OF SIGN CHANGES 83 Definition 1. The sequence of the processes {Xn} provides the Markov approximation for the Markov process X, if for every γ > 0, S < +∞ there exist a constant K(γ, S) ∈ N and a sequence of two-component processes {Ŷn = (X̂n, X̂ n)}, possibly defined on another probability space, such that (i) X̂n d=Xn, X̂ n d=X; (ii) the processes Ŷn,X̂n, X̂ n possess the Markov property at the points iK(γ, S)n−α, i ∈ N w.r.t. the filtration {F̂n t = σ(Ŷn(s), s ≤ t)}; (iii) lim sup n→+∞ P ⎛⎝ sup i≤ Snα K(γ,S) ρ ( X̂n ( iK(γ, S)n−α ) , X̂n ( iK(γ, S)n−α )) > γ ⎞⎠ < γ. 4. Weak convergence and Markov approximation In this section, we prove that, under conditions of Theorem 1, the processes Xn both converge weakly to X and provide the Markov approximation for X . Lemma 1. Under conditions (A) and (C), Xn converge to X weakly in C(R+). Proof. Let Ŵ be a Wiener process. Consider the processes of the type X̂n(t) = x+ Ŵ (ζn,t), X̂(t) = x+ Ŵ (ζt), where ζn,·, ζ· are inverse functions to the functions ηn,·, η· defined by ηn,t = ∫ t 0 σ−2 n (x+ Ŵ (s)) ds, ηt = ∫ t 0 σ−2(x+ Ŵ (s)) ds. Then X̂n has the same distribution with Xn, and X̂ has the same distribution with X . The processes ηn,·, η· are W-functionals of the Wiener process with their characteristics equal to gt n(x) = Exηn,t = ∫ R Kt(x, y)σ−2 n (y) dy, gt(x) = Exηt = ∫ R Kt(x, y)σ−2(y) dy. Thus, condition (C) provides that gn → g uniformly on R × [0, T ]. Now, the Dynkin’s theorem ([7], Theorem 6.4) provides that, for every T , supt≤T \|ηn,t−ηt\| → 0, n→ +∞ in probability. Since 0 ≤ [ d dtηn,t ]−1 = σ2 n(x + Ŵ (t)) ≤ R2, the convergence in probability supt≤T \|ηn,t − ηt\| → 0, T > 0 implies the convergence in probability supt≤T \|ζn,t − ζt\| → 0, T > 0 and, therefore, the convergence in probability supt≤T \|X̂n(t)− X̂(t)\| → 0, T > 0. The lemma is proved. In order to prove that Xn provide the Markov approximation for X , we need some auxiliary estimates and constructions. Denote d2(ξ, η) = [ inf (ξ̂,η̂), ξ̂ d = ξ,η̂ d = η E(ξ̂ − η̂)2 ] 1 2 , ξ, η ∈ L2, which is the Wasserstein–Kantorovich–Rubinshtein distance between the distributions of ξ and η. Denote, by Xn(t, x), X(t, x), t ∈ R+, the diffusion processes satisfying (1) and (2), respectively, with the initial conditions Xn(0, x) = x,X(0, x) = x. 84 ALEXEY M. KULIK Lemma 2. Under conditions (A) and (D), for every ε > 0, there exist T = Tε ∈ N and N = Nε ∈ N such that d2(Xn(Tn−α, x), X(Tn−α, x)) ≤ ε √ Tn−α, n ≥ N, x ∈ R. (6) Proof. One can easily see that (6) is equivalent to the following estimate for the re-scaled processes Zn, Z n: d2(Zn(T, xn α 2 ), Zn(T, xn α 2 )) ≤ ε √ T , n ≥ N, x ∈ R, that, in turn, follows from the estimate sup z∈Rd d2(Zn(T, z), Zn(T, z)) ≤ ε √ T , n ≥ N. (7) Let us prove (7). Let z be fixed, and let Ŵ be a Wiener process. Consider the processes of the type Ẑn(t, z) = z + Ŵ (θn,t), Ẑn(t, z) = z + Ŵ (θn t ), (8) where θn,·, θn· are inverse functions to the functions ϑn,·, ϑn· defined by ϑn,t = ∫ t 0 �−2 n (z + Ŵ (s)) ds, ϑn t = ∫ t 0 (�n)−2(z + Ŵ (s)) ds. Since Ẑn(t, z) d=Zn(t, z), Ẑn(t, z) d=Zn(t, z) and θn,t, θ n t are a stopping times w.r.t. fil- tration generated by Ŵ , we have d2 2(Zn(T, z), Zn(T, z)) ≤ E(Ẑn(T, z) − Zn(T, z))2 = E(Ŵ (θn,T ) − Ŵ (θn T ))2 = E\|θn,T − θn T \|. Condition (A) provides that \|θn,T − θn T \| ≤ R sups≤TR \|ϑn,s − ϑn s \|. The processes ϑn,·, ϑn · are W-functionals of the process Ŵ with their characteristics fn, f n equal to f t n(z) = ∫ R �2 n(y)Kt(z, y) dy, fn,t(z) = ∫ R (�n)2(y)Kt(z, y) dy. Since ∫ R Kt(z, y) dy = t, z ∈ R, t ∈ R+, ‖fT n ‖ ≤ R2T, ‖fn,T‖ ≤ R2T by condition (A) (here and below, we denote ‖f‖ = supx∈R \|f(x)\|). Then Lemma 3 below provides the estimate E( sup s≤TR \|ϑn,s − ϑn s \|)2 ≤ 8( √ 2 + √ 3)2T 2R2 sups≤TR ‖fs n − fn,s‖ T . (9) By condition (D), 1 T sups≤TR ‖fs n − fn,s‖ → 0, n, T → +∞. This provides inequality (7) for N,T large enough and completes the proof of the lemma. Estimate (9) in the proof above is provided by the following result, that is a general- ization of Lemma 6.5 [7]. Lemma 3. Let Y be a homogeneous Markov process with its phase space Y being a locally compact metric space. Let φ, ψ be W-functionals of Y , and let f, g be their characteristics, respectively. Then E [sup s≤t (φ0,s−ψ0,s)2\|Y (0) = y] ≤ 8( √ 2+ √ 3)2(‖f t‖+‖gt‖) sup s≤t ‖fs−gs‖, t ∈ R+, y ∈ Y. A LIMIT THEOREM FOR THE NUMBER OF SIGN CHANGES 85 Proof. The proof is based on the idea of the proof of Doob’s maximal martingale in- equality ([9], Chapter 7, §3). Since the functions φ0,·, ψ0,· are continuous, it is sufficient to prove that, for every k ∈ Z+,m ∈ N, E [max j≤k (φ0,j2−m − ψ0,j2−m )2\|Y (0) = y] ≤ 8( √ 2 + √ 3)2(‖fk2−m‖ + ‖gk2−m‖) sup s≤k2−m ‖fs − gs‖. We suppose y ∈ Y,m ∈ N to be fixed and omit them in the notation. Denote Mk = maxj≤k(φ0,j2−m − ψ0,j2−m ), k ∈ Z+. We have that Mk ≥ 0 since φ0,0 = ψ0,0 = 0. For u > 0, denote τu = min{k : Mk ≥ u} and write uP (Mk ≥ u) = uP (τu ≤ k) ≤ E(φ0,(τu∧k)2−m − ψ0,(τu∧k)2−m )1Iτu≤k = = E(φ0,k2−m − ψ0,k2−m )1Iτu≤k − E(φ(τu∧k)2−m,k2−m − ψ(τu∧k)2−m,k2−m )1Iτu≤k. (10) The sequence {(Y (k2−m), φ0,k2−m , ψ0,k2−m ), k ∈ Z+} is a Markov chain, and therefore it is strongly Markov. Denote, by G = {Gk}, the corresponding filtration and write −E[(φ(τu∧k)2−m,k2−m − ψ(τu∧k)2−m,k2−m \|Gτu∧k] = = g(k−τu∧k)2−m (Y ((τu ∧ k)2m)) − f (k−τu∧k)2−m (Y ((τu ∧ k)2m)) ≤ sup j≤k ‖f j2−m − gj2−m‖ (11) (here, we have used the strong Markov property and the fact that τu is a stopping time w.r.t. G). One can easily verify that {τu ≤ k} ∈ Gτu∧k, and thus (11) shows that the second summand on the right-hand side of (10) is estimated by max j≤k ‖f j2−m − gj2−m‖P (τu ≤ k) = max j≤k ‖f j2−m − gj2−m‖P (Mk ≥ u). Denote d = (‖fk2−m ‖ + ‖gk2−m ‖) 1 2 sup s≤k2−m ‖fs − gs‖ 1 2 ; then d ≥ maxj≤k ‖f j2−m − gj2−m‖. Inequalities (10), (11) provide the estimate (u− d)P (Mk ≥ u) ≤ E(φ0,k2−m − ψ0,k2−m )1IMk≥u, u ≥ d. Then EM2 k = 2 ∫ ∞ 0 uP (Mk ≥ u) du ≤ 2 ∫ 2d 0 u du+ 4 ∫ ∞ 2d (u− d)P (Mk ≥ u) du ≤ ≤ 4d2 + 4E ∣∣∣φ0,k2−m − ψ0,k2−m ∣∣∣ ∫ ∞ 2d 1Iu≤Mk du ≤ 4d2 + 4E\|φ0,k2−m − ψ0,k2−m \|Mk ≤ ≤ 4d2 + 4 [ E(φ0,k2−m − ψ0,k2−m )2 ] 1 2 · [EM2 k ] 1 2 . By Lemma 6.5 [7], E(φ0,k2−m − ψ0,k2−m )2 ≤ 2d2. Thus, κ ≡ [EM2 k ] 1 2 satisfies the inequality κ2 ≤ 4d2 + 4 √ 2dκ, that means that EM2 k ≤ 4( √ 2 + √ 3)2d2. (12) Completely analogously, one can prove that E[min j≤k (φ0,j2−m − ψ0,j2−m )]2 ≤ 4( √ 2 + √ 3)2d2. (13) 86 ALEXEY M. KULIK Since maxj≤k(φ0,j2−m −ψ0,j2−m )2 ≤ [maxj≤k(φ0,j2−m − ψ0,j2−m )]2 + [minj≤k(φ0,j2−m − ψ0,j2−m )]2, inequalities (12) and (13) provide the required estimate. The lemma is proved. Now we are ready to prove the main statement of this section. Theorem 3. Under conditions of Theorem 1, the sequence {Xn(·, x)} provides the Markov approximation for X(·, x). Proof. For every ε > 0, n ∈ N, we fix T = Tε from Lemma 2 and construct iteratively the process Qn(t) = (X̂n(t), R̂n(t), X̂n(t)) and the sequence (κk, ςk, χk), k ≥ 1 in the following way. For t ∈ [0, Tn−α), put X̂n(t) = n−α 2 Ẑn(tnα, xn α 2 ), R̂n(t) = X̂n(t) = Ẑn(tnα, xn α 2 ), where ẐnẐ n are defined by (9). Put κ1 = n−α 2 Ẑn(nα, xn α 2 ), ς1 = χ1 = n−α 2 Ẑn(nα, xn α 2 ). Next, suppose that Qn(t) is already defined for t ∈ [0,mTn−α), and (κk, ςk, χk) is already defined for k ≤ m. Consider some Wiener process Ŵm independent of the values of the process Qn(t) on t ∈ [0,mTn−α) and consider the processes Ẑn,m(·, z), Ẑm n (·, z), z ∈ R defined by (9) with Ŵ replaced by Ŵm. For t ∈ [mTn−α, (m+ 1)Tn−α), put X̂n(t) = n−α 2 Ẑn,m(tnα −m,κmn α 2 ), R̂n(t) = Ẑn m(tnα −m,κmn α 2 ). The process R̂n satisfies SDE (2) on [mTn−α, (m+1)Tn−α) with a certain Wiener process W̌ . Define the process X̂n on [mTn−α, (m + 1)Tn−α) as the solution to SDE (2) with the same Wiener process W̌ and X̂n(Tn−α) = χm. Such a definition is correct since (2) has a weak solution, possesses the path-wise uniqueness property, and therefore, by the Yamada–Watanabe theorem, possesses the unique strong solution. At last, put κm+1 = X̂n((m+ 1)Tn−α−), ςm+1 = R̂n((m+ 1)Tn−α−), χm+1 = X̂n((m+ 1)Tn−α−). Repeating this construction, we obtain the processes Qn which are defined on R+ and possess the following properties: (i) X̂n d=Xn, X̂ n d=X ; (ii) the processes Qn,X̂n, X̂ n possess the Markov property at the points iTεn −α, i ∈ N w.r.t. the filtration {Fn t = σ(Q̂n(s), s ≤ t)}. Now we are going to prove that, for every γ > 0, S < +∞, there exists ε > 0 such that lim sup n→+∞ P ⎛⎝ sup i≤ Snα Tε ∣∣∣X̂n ( iTεn −α ) − X̂n ( iTεn −α )∣∣∣ > γ ⎞⎠ < γ. (14) This will mean that conditions (i) – (iii) of Definition 1 hold true with K(γ, S) = Tε. Using the fact that the coefficients σn, σ are uniformly bounded, one can verify that, for every a > 0, sup n P (wS(X̂n, δ) > a) + sup n P (wS(X̂n, δ) > a) → 0, δ → 0+, where wS(X, δ) ≡ sup\|s−t\|≤δ,s,t∈[0,S] \|X(t) −X(s)\| (the proof is standard and omitted). Since R̂n(iTεn −α) = X̂n(iTεn −α), i ∈ N, this implies that sup t≤S \|X̂n(t) − R̂n(t)\| → 0, n→ +∞ (15) A LIMIT THEOREM FOR THE NUMBER OF SIGN CHANGES 87 in probability. The process X̂n satisfies SDE X̂n(t) = x+ ∫ t 0 σ(X̂n(s)) dWn(s), t ∈ R+. On the other hand, the process R̂n, by construction, satisfies SDE R̂n(t) = x+ ∫ t 0 σ(R̂n(s)) dWn(s) + Δn(t), t ∈ R+ with Δn(t) = ∑ k≤tT−1 ε nα(κk − ςk). By construction, Δn is a martingale (here, we make use of the supposition made at the beginning of Section 2 that the coefficients an, a in Eqs. (1),(2) are equal to 0). By Lemma 2, E(κk − ςk)2 ≤ ε2Tn−α, k ≥ 1. Therefore, Emax t≤S Δ2 n(t) ≤ 2EΔ2 n(S) = 2 ∑ k≤ST−1 ε nα (κk − ςk)2 ≤ 2ε2S. (16) Now suppose that, for every ε > 0, (14) fails. This means that there exists some subse- quence {nr} such that P ⎛⎝ sup i≤Snα r Tε ∣∣∣X̂nr ( iTεnr −α ) − X̂nr ( iTεn −α r )∣∣∣ > γ ⎞⎠ ≥ γ. (17) The families {X̂n}, {X̂n} are weakly compact in C(R+), and therefore one can sup- pose that the 3-component processes (X̂nr , X̂ nr ,Wnr) converge weakly in C(R+,R3) to some process (X̂∗, X̂∗,W ∗). Relation (15) implies that (X̂nr , X̂ nr ,Wnr , R̂nr) ⇒ (X̂∗, X̂∗,W ∗, X̂∗) in C(R+,R3) × D(R+) with D(R+) endowed with the topology of uniform convergence on every compact. Then( X̂nr , X̂ nr ,Wnr , R̂nr , ∫ · 0 σ(R̂nr (s))dWnr (s) ) ⇒ ( X̂∗, X̂∗,W ∗, X̂∗, ∫ · 0 σ(X̂∗(s))dW ∗(s) ) in C(R+,R3) ×D(R+) × C(R+), and consequently( X̂nr , X̂ nr ,Wnr , R̂nr , ∫ · 0 σ(R̂nr (s))dWnr (s),Δnr ) ⇒ ( X̂∗, X̂∗,W ∗, X̂∗, ∫ · 0 σ(X̂∗(s))dW ∗(s),Δ∗ ) in C(R+,R3)×D(R+)×C(R+)×D(R+). The statement analogous to this one was given in the proof of the Theorem 1 [6, Chapter 5.3], (see also the discussion after Lemma 2.3 in [10]). Now we conclude that, for every ε > 0, there exists the 4-component process H∗ = (X̂∗, X̂∗,W ∗,Δ∗) such that W ∗ is a Wiener process w.r.t. filtration generated by H∗, the processes X∗, X∗ satisfy the relations X∗(t) = x+ ∫ t 0 σ(X∗(s)) ds+ Δ∗(t), X∗(t) = x+ ∫ t 0 σ(X∗(s)) ds, and the estimates E sup s≤S Δ2 ∗(s) ≤ 2ε2S, (18) P (sup s≤S \|X∗(s) −X∗(s)\| ≥ γ) ≥ γ (19) 88 ALEXEY M. KULIK hold true [(18) follows from (16), and (19) follows from (17)]. Once more passing to the limit as ε → 0+, we obtain the 3-component process (X♦, X♦,W♦) such that both X♦ and X♦ satisfy (2) with the same initial condition x and the same Wiener process W♦, but X♦ �≡ X♦ due to (19). This contradicts the condition on (2) to possess the path-wise uniqueness property. Consequently, our supposition that (14) fails for every ε > 0 is false, and the conditions of Definition 1 hold true with K(γ, S) = Tε with some ε > 0. The theorem is proved. 5. Proof of Theorem 1 We reduce the proof of Theorem 1 to the verification of the conditions of Theorem 2. The sequence Xn provides the Markov approximation for the process X due to Theorem 3. Condition 1 of Theorem 2 holds true since δn = n−α 2 . In this section, we prove that conditions 2 and 3 hold true. Recall that fn, n ∈ N denote characteristics for the functionals φn, n ∈ N (see (5)). Lemma 4. Denote LZn(t, z) = \|Zn(t) − z\| − \|Zn(0) − z\| − ∫ t 0 sign(Zn(s) − z) dZn(s), t ∈ R+. Then, for every bounded measurable function Φ with bounded support,∫ t 0 Φ(Zn(s)) ds = ∫ R Φ(z)LZn(t, z)�−2 n (z) dz. (20) Proof. The statement of the lemma is known to hold true for the Wiener process (i.e., for �n ≡ 1, see [11], Chapter 3, §4). The process Zn can be represented in the form (9), and then LZn(t, z) = LŴ (θn,t, z), where Ŵ , θn,· denote the Wiener process and the time change from this representation (the proof is easy and omitted). Now (20) follows from the analogous equality for Φ̃ ≡ Φ�−2 n for the Wiener process by changing the variables in the integral w.r.t. ds. The lemma is proved. Lemma 5. Under conditions (A), (C) – (E), fs,t n (x) → cE ( LX(t− s, 0) ∣∣∣X(0) = x ) , n→ +∞, uniformly on R × {0 ≤ s ≤ t ≤ T } for every T , with the constant c defined in the formulation of Theorem 1. Proof. We have fs,t n (x) = n−α 2 ∑ k<(t−s)nα E ( 1IZn(k)Z(k+1)<0 ∣∣∣Zn(0) = xn α 2 ) = n−α 2 ∑ k<tnα E ( Φn(Zn(k)) ∣∣∣Zn(0) = xn α 2 ) , where Φn(z) = P ( Zn(1) · z < 0 ∣∣∣Zn(0) = z ) . Denote, by Gn(t, x, dy), the transition probability for the process Zn. Due to Theorem 1.2 in [12], there exists a constant μ > 0 depending on R only, such that, for every g,∣∣∣ d dt ∫ R g(y)Gn(t, z, dy) ∣∣∣ ≤ μt− 3 2 ∫ R e− μ(y−z)2 t \|g(y)\| dy. (21) A LIMIT THEOREM FOR THE NUMBER OF SIGN CHANGES 89 Since the diffusion coefficients for Zn are uniformly bounded, for every m ≥ 1, there exists a constant Cm such that Φn(z) ≤ Cm(1∧ \|z\|)−m, n ∈ N. This, together with (21), implies that sup 0≤s≤t≤T,x∈R ∣∣∣∣∣fs,t n (x) − n−α 2 ∫ (t−s)nα 0 E ( Φn(Zn(s)) ∣∣∣Zn(0) = xn α 2 ) ds ∣∣∣∣∣ → 0, n→ +∞. The function Φn does not have a compact support, and Lemma 4 can not be applied to it straightforwardly. But, applying Lemma 4 to the functions Φn,A = Φn1I[−A,A], using the estimate of Φn given before, and then passing to the limit as A → +∞, one can obtain the estimate n−α 2 ∫ (t−s)nα 0 E ( Φn(Zn(s)) ∣∣∣Zn(0) = xn α 2 ) ds = n−α 2 ∫ R Φn(z)�−2 n (z)E ( LZn((t− s)nα, z) ∣∣∣Zn(0) = xn α 2 ) dz = ∫ R Φn(z)�−2 n (z)E ( \|Xn(t− s, x) − n−α 2 z\| − \|x− n−α 2 z\| ) dz. The estimates given in the proof of Lemma 1 provide that, for every A > 0, sup x∈R,\|z\|≤A ∣∣∣Φn(z)E ( \|Xn(t− s, x) − n−α 2 z\| − \|x− n−α 2 z\| ) − Φ(z)E ( \|X(t, x)\| − \|x\| )∣∣∣ → 0, n→ +∞, with Φ(z) = P ( Z(1) · z < 0 ∣∣∣Z(0) = z ) . Then the estimate on Φn given before provides that sup x∈R,t−s≤T ∫ R �−2 n (z) ∣∣∣Φn(z)E ( \|Xn(t− s, x) − n−α 2 z\| − \|x− n−α 2 z\| ) − Φ(z)E (\|X(t, x)\| − \|x\|) ∣∣∣ dz → 0, n → +∞. Since �−2 n are uniformly bounded and converge weakly to �−2, this implies that fs,t n (x) → lim N→+∞ ∫ R Φ(z)�−2 N (z) dz ·E ( \|X(t− s, x)\| − \|x\| ) = ∫ R Φ(z)�−2(z) dz · E ( LX(t− s, 0) ∣∣∣X(0) = x ) , n→ +∞, uniformly for x ∈ R, 0 ≤ s ≤ t ≤ T . The lemma is proved. The function (x, t) !→ LX(t, x) is continuous in mean square. In order to prove this, one should write LX(t, z) = LW (θt, z) (as in the proof of Lemma 4) and then use the same property for the local time of the Wiener process. Then LX(t, 0) = L2 − lim ε→0+ 1 2ε ∫ ε −ε LX(t, x) dx = L2 − lim ε→0+ 1 2ε ∫ t 0 1I\|X(s)\|<εσ 2(X(s)) ds, and, therefore (see [7] Chapter 6, §2,3), the process φs,t = LX(t, 0) − LX(s, 0) is a W- functional of the process X . Hence, condition 2 of Theorem 2 holds true. At last, one can use the arguments from the proof of Lemma 2 in order to prove that E(X(t, x) − X(t, y))2 → 0, \|x− y\| → 0 uniformly for t ≤ T . Since \|f t(x) − f t(y)\| = \|E (LX(t, 0) \| X(0) = x) − E (LX(t, 0) \| X(0) = x)\| ≤ \|E\|X(t, x)\| − \|X(t, y)\| − \|x\| + \|y\|\| ≤ \|x− y\| + ( E(X(t, x) −X(t, y))2 ) 1 2 , 90 ALEXEY M. KULIK this provides condition 3 of Theorem 3. Therefore, all conditions of Theorem 3 hold true. Applying this theorem to functionals (3), we obtain the statement of Theorem 1. 6. Examples Example 1. Let σn(x) = �(nαx), and let the function � ∈ Σ satisfy the condition R−1 ≤ � ≤ R. Suppose that 1 v − u ∫ v u �−2(z) dz → κ+, 1 v − u ∫ −u −v �−2(z) dz → κ−, u, v → +∞, v − u→ +∞, (22) and put σ = σ−1IR− + σ+1IR+ with σ± = κ − 1 2± . Then �n = �, �n = σ for every n, and conditions (C) and (D) are, in fact, equivalent one to another and follow from condition (22) (we omit the detailed exposition here, since analogous estimates are given, in a more delicate situation, in the next example). Condition (E) is trivial. At last, condition (B) holds true by the Nakao theorem [13]. Suppose that σ− + σ+ = 1 (one can reduce the general case to this one by making the time change t !→ (σ− + σ+)− 1 2 t), and put q = σ− − σ+. Then (see [14]) the process X defined by SDE (2) is the image of the skew Brownian motion W q with the skewing parameter q under the phase transformation x !→ x 1 + q 2 1IR− + x 1 − q 2 1IR+ . Moreover, the local time of X at the point 0 is equal to the local time of W q at the same point. Thus, in the example under consideration, the number of sign changes (3) converges weakly to the local time of the skew Brownian motion W σ−−σ+ at the point 0 multiplied by c = ∫ R �−2(z)P ( X1(1) · z < 0 ∣∣∣X1(0) = z ) dz. Example 2. Let σn(x) = ς(x) exp cos(nαx), x ∈ R, with the function ς ∈ ΣR that is uniformly continuous on R. We state that conditions (C) and (D) hold true with σ = C− 1 2 · ς, where C = 1 2π ∫ 2π 0 e−2 cos y dy. Let us prove (D); the proof for (C) is analogous and more simple. After the change of variables x̂ = x√ s , ŷ = y√ s , we have∫ R [�−2 n (y) − (�n)−2(y)]Kt(x, y) ds = ∫ t 0 1√ 2πs ∫ R (e−2 cos y − C)ς(yn−α)e− (y−x)2 2s dy ds = = 1√ 2π ∫ t 0 ∫ R (e−2 cos(ŷ √ s) − C)ς(ŷn−α√s)e− (ŷ−x̂)2 2 dŷ ds. Therefore, in order to prove (D), it is enough to show that sup x 1√ 2π ∣∣∣∣∫ R (e−2 cos(y √ t) − C)ς(yn−α √ t)e− (y−x)2 2 dy ∣∣∣∣ → 0, n, t→ +∞. (23) Denote wς−2(z) = sup\|x−y\|≤z \|ς−2(x)− ς−2(y)\|. We have that wς−2(z) → 0, z → 0+. Let ε > 0 be fixed; consider D, δ > 0 such that∫ D −D 1√ 2π e− y2 2 dy ≥ 1 − ε, sup \|x−y\|≤δ \|e− x2 2 − e− y2 2 \| ≤ ε 2D + 1 . For a given x ∈ R, t ∈ R+, put k−(x, t) = max { k ∈ Z : 2πk√ t ≤ x−D } , k+(x, t) = min { k ∈ Z : 2πk√ t ≥ x+D } , A LIMIT THEOREM FOR THE NUMBER OF SIGN CHANGES 91 then k+(x, t) − k−(x, t) ≤ 2D + 4π√ t . Now we have 1√ 2π ∣∣∣∣∫ R (e−2 cos(y √ t) − C)ς−2(yn−α √ t)e− (y−x)2 2 dy ∣∣∣∣ ≤ ε(e2 + C) sup z∈R ς−2(z) + 1√ 2π ∣∣∣∣∣∣∣∣ 2k+(x,t)π√ t∫ 2k−(x,t)π√ t (e−2 cos(y √ t) − C)ς−2(yn−α √ t)e− (y−x)2 2 dy ∣∣∣∣∣∣∣∣ ≤ ε(e2 + C)(sup z∈R ς−2(z) + wς−2 (( D + 2π√ t ) n−α ) + ς−2(x)√ 2π ∣∣∣∣∣∣∣∣ 2k+(x,t)π√ t∫ 2k−(x,t)π√ t (e−2 cos(y √ t) − C)e− (y−x)2 2 dy ∣∣∣∣∣∣∣∣ ≤ ε(e2 + C)(sup z∈R ς−2(z) + wς−2 (( D + 2π√ t ) n−α ) + ς−2(x)√ 2π k+(x,t)−1∑ k=k−(x,t) ∣∣∣∣∣∣∣∣ 2(k+1)π√ t∫ 2kπ√ t (e−2 cos(y √ t) − C)(e− (y−x)2 2 − e− ( 2kπ√ t −x)2 2 ) dy ∣∣∣∣∣∣∣∣ . Here, in the last inequality, we have used the fact that ∫ 2(k+1)π√ t 2kπ√ t (e−2 cos(y √ t) − C) dy = 0, k ∈ Z. If t is such that 2π√ t < δ and 4π√ t < 1, then sup x 1√ 2π ∣∣∣∣∫ R (e−2 cos(y √ t) − C)ς(yn−α √ t)e− (y−x)2 2 dy ∣∣∣∣ ≤ ≤ ε(e2 + C) ( 1 + 1√ 2π )( sup z∈R ς−2(z) + ε(e2 + C)wς−2 (( D + 1 2 ) n−α )) . (24) Now we obtain (23) by passing to the limit in (24) first as n→ ∞ and then as ε→ 0+. Condition (E) holds true with �(x) = ς(0) exp cosx. Condition (A) holds true obvi- ously. At last, condition (B) holds true provided that the SDE dX(t) = C− 1 2 ς(X(t)) dW (t) (25) possesses the path-wise uniqueness property. Thus, the number of sign changes (3) converges weakly to the local time of the process X , defined by SDE (25), at the point 0, multiplied by ς−2(0) ∫ R exp[−2 cos z]P ( Z(1) · z < 0 ∣∣∣Z(0) = z ) dz, where the process Z is defined by the SDE dZ(t) = ς(0)ecos Z(t) dW (t). Bibliography 1. I.I.Gikhman, Some limit theorems for the number of intersections of the boundary of a domain by a random function, Sci. Notes of Kiev Univ. 16 (1957), no. 10, 149 – 164. (Ukrainian) 2. I.I.Gikhman, Asymptotic distributions for the number of intersections of the boundary of a domain by a random function, Visnyk Kiev Univ., Ser. Astronomy, Mathematics and Mechanics 1 (1958), no. 1, 25 – 46. (Ukrainian) 92 ALEXEY M. KULIK 3. N.I.Portenko, The development of I.I.Gikhman’s idea concerning the methods for investigating local behavior of diffusion processes and their weakly convergent sequences, Theor. Probability and Math. Statist. 50 (1994), 7 – 22. 4. H.Al Farah, M.I.Portenko, Limit theorem for the number of intersections of the fixed level by weakly convergent sequence of diffusion processes, Preprint 2007.6 (2007), Institute of Math., Kiev, 24. (Ukrainian) 5. Yu.N.Kartashov, A.M.Kulik, Invariance principle for additive functionals of Markov chain, submitted (2006). (Russian; preprint, in English translation, available at arXiv:0704.0508v1) 6. I.I.Gikhman, A.V.Skorokhod, Stochastic Differential Equations and Their Applications, 2-nd ed, Kiev, Nauk. Dumka, 1982. (Russian) 7. E.B.Dynkin, Markov Processes, Moscow, Fizmatgiz, 1963. (Russian) 8. A.M.Kulik, Markov approximation of stable processes by random walks, Theory of Stochastic Processes 12(28) (2006), no. 1-2, 87 – 93. 9. J.L.Doob, Stochastic Processes, NY, Wiley, 1953. 10. A.M.Kulik, The optimal coupling property and its applications: limit theorems for non-elliptic diffusions and construction of canonical stochastic flow, Theory of Stochastic Processes 9(25) (2003), no. 1-2, 82 – 98. 11. N.Ikeda, S.Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Hol- land, Amsterdam, 1981. 12. O.F.Porper, S.D.Eidelman, Asymptotic behavior of the classical and generalized one-dimen- sional parabolic equations of second order, Trudy Mosk. Mat. Ob., vol. 36, Mosc. Univ. publ., 1978, pp. 85 – 130. (Russian) 13. S.Nakao, On the pathwise uniqueness of solutions of one-dimensional stochastic differential equations, Osaka J. Math 9 (1972), 513 – 518. 14. J.M.Harrison, L.A.Shepp, On skew Brownian motion, Annals of Probability 9 (1981), no. 2, 309 – 313. !��" �#��# &��"�� %�� E-mail : kulik@imath.kiev.ua

A limit theorem for the number of sign changes for a sequence of one-dimensional diffusions

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