LOCAL LIMIT THEOREMS FOR LADDER EPOCHS 7 0 VLADIMIRA.VATUTINANDVITALIWACHTEL 0 2 Abstract. Let S0 =0,{Sn}n≥1 bea random walkgenerated bya sequence n of i.i.d. random variablesX1,X2,...andletτ− :=min{n≥1: Sn≤0}and a τ+ := min{n≥1: Sn>0}. Assuming that the distribution of X1 belongs J tothedomainofattractionofanα-stablelaw,α6=1,westudytheasymptotic 1 behavior ofP(τ±=n)asn→∞. 3 ] R 1. Introduction and main result P . h Let X,X1,X2,... be a sequence of independent identically distributed random t variables. Denote S =0,S =X +X +...+X . We assume that a 0 n 1 2 n m ∞ 1 ∞ 1 P(S >0)= P(S 0)= . [ n n n n ≤ ∞ n=1 n=1 X X 1 This condition means that S is anoscillating randomwalk,and, in particu- n n≥0 v { } lar, the stopping times 4 1 τ− :=min n 1: S 0 and τ+ :=min n 1: S >0 n n 9 { ≥ ≤ } { ≥ } 1 arewell-definedproperrandomvariables. Furthermore,itfollowsfromtheWiener- 0 Hopf factorization (see, for example, [3, Theorem 8.9.1, p. 376]) that for all z ∈ 7 (0,1), /0 1 Ezτ− =exp ∞ znP(S 0) (1) h n − (− n ≤ ) t n=1 a X and m ∞ zn : 1 Ezτ+ =exp P(Sn >0) . (2) v − (− n ) n=1 i X X Rogozin [15] has shown that the Spitzer condition r n a n−1 P(S >0) ρ (0,1) as n (3) k → ∈ →∞ k=1 X holds if and only if τ+ belongs to the domain of attraction of a spectrally positive stable law with parameter ρ. Since (1) and (2) imply the equality (1 Ezτ+)(1 Ezτ−)=1 z for all z (0,1), − − − ∈ Date:February2,2008. 1991 Mathematics Subject Classification. Primary60G50;Secondary60G40. Key words and phrases. randomwalk,laddermoment,Spitzer condition,stablelaw. ThefirstauthorwassupportedinpartbytheRussianFoundationforBasicResearch(grant05- 01-00035),INTAS(project03-51-5018),andtheprogram”Contemporaryproblemsoftheoretical mathematics”ofRAS. Thesecondauthor wassupportedbytheGIF. 1 2 VATUTINANDWACHTEL one can deduce from the Rogozin result that (3) holds if and only if there exists a function l(n) slowly varying at infinity such that, as n , →∞ l(n) 1 P τ− >n , P τ+ >n . (4) ∼ n1−ρ ∼ Γ(ρ)Γ(1 ρ)nρl(n) − (cid:0) (cid:1) (cid:0) (cid:1) Doney [11] proved that the Spitzer condition is equivalent to P(S >0) ρ (0,1) as n . (5) n → ∈ →∞ Therefore, both relations in (4) are valid under condition (5). To get a more detailed information about the asymptotic properties of l(x) it is necessary to impose additional hypotheses on the distribution of X. Rogozin [15] has shown that l(x) is asymptotically a constant if and only if ∞ 1 P(S >0) ρ < . (6) n n − ∞ nX=1 (cid:16) (cid:17) It follows from the Spitzer-R´osen theorem (see [3, Theorem 8.9.23, p. 382]) that if EX2 is finite, then (6) holds with ρ=1/2, and, consequently, C± P(τ± >n) as n , (7) ∼ n1/2 →∞ where C± are positive constants. If EX2 = much less is known about the form ∞ of l(x). For instance, if the distribution of X is symmetric, then, clearly, 1 1 P(S >0) = P(S =0). (8) n n − 2 2 (cid:12) (cid:12) (cid:12) (cid:12) Furthermore, according t(cid:12)o [14, Theorem(cid:12)III.9, p. 49], there exists C >0 such that (cid:12) (cid:12) for all n 1, ≥ C P(S =0) . n ≤ √n Bythisestimateand(8)weconcludethat(6)holdswithρ=1/2. Thus,(7)isvalid for all symmetric random walks. Assuming that P(X > x) = (xαl (x))−1, x > 0, 0 with 1 < α < 2 and l (x) slowly varying at infinity, Doney [8] established for a 0 numberof casesrelationshipsbetweenthe asymptotic behaviorofl (x) andl(x) at 0 infinity. The aim of the present paper is to study the asymptotic behavior of the proba- bilities P(τ± =n) as n . →∞ We assume throughout that the distribution of X is either non-lattice or arith- metic with span h>0, i.e. the h is the maximal number such that the support of the distribution of X is contained in the set kh, k =0, 1, 2,... . { ± ± } Let := 0<α<1; β <1 1<α<2; β 1 α=2,β =0 A { | | }∪{ | |≤ }∪{ } be a subset in R2. For (α,β) we write X (α,β) if the distribution of X ∈ A ∈ D belongs to the domain of attraction of a stable law with characteristic function t πα Ψ(t):=exp ct α 1 iβ tan ,c>0, (9) − | | − t 2 (cid:26) (cid:18) | | (cid:19)(cid:27) LOCAL LIMIT THEOREMS FOR LADDER EPOCHS 3 and, in addition, EX =0 if 1<α 2. One can show (see, for instance, [16]) that ≤ if X (α,β), then condition (5) holds with ∈D 1 1 πα ρ= + arctan βtan (0,1). (10) 2 πα 2 ∈ (cid:16) (cid:17) Here is our main result. Theorem 1. Assume X (α,β). If α 2 and β <1, then, as n , ∈D ≤ →∞ l(n) P τ− =n =(1 ρ) (1+o(1)). (11) − n2−ρ In the case 1 < α < 2,(cid:0)β = 1 (cid:1)equality (11) remains valid under the additional { } hypothesis ∞ F( x) − dx< . (12) x(1 F(x)) ∞ Z1 − Denote T− :=min n 1: S <0 and set n { ≥ } ∞ zn ∞ Ω(z)=exp P(S =0) =: ω zk. (13) n k n ( ) n=1 k=0 X X Thenextstatementrelatesthe asymptoticbehaviorofP(τ− =n)andP(T− =n). Theorem 2. If (11) holds, then P(T− =n) lim =Ω(1). n→∞ P(τ− =n) Applying Theorems 1 and 2 to the random walk S , one can easily find n n≥0 an asymptotic representation for P(τ+ =n): {− } Theorem 3. Assume X (α,β). If α 2 and β > 1, then, as n , ∈D ≤ − →∞ ρ P τ+ =n = (1+o(1)). (14) Γ(ρ)Γ(1 ρ)n1+ρl(n) − (cid:0) (cid:1) In the case 1 < α < 2, β = 1 equality (14) remains valid under the additional { − } hypothesis ∞ 1 F(x) − dx< . (15) xF( x) ∞ Z1 − InsomespecialcasestheasymptoticbehaviorofP(τ± =n)asn isalready known from the literature. Eppel [12] proved that if EX = 0 and→E∞X2 is finite, then C± P τ± =n . (16) ∼ n3/2 Observe that in this case EX2 <(cid:0) impli(cid:1)es X (2,0). ∞ ∈D Asymptotic representation (16) is valid for all continuous symmetric (implying ρ = 1/2 in (5)) random walks (see [13, Chapter XII, Section 7]). Note that the restriction X (α,β) is superfluous in this situation. ∈D Recently Borovkov[2] has shown that if (3) is valid and n1−ρ P(S >0) ρ const ( , ) as n , (17) n − → ∈ −∞ ∞ →∞ (cid:16) (cid:17) then(11)holdswithℓ(n) const (0, ). ProvingthementionedresultBorovkov ≡ ∈ ∞ does not assume that the distribution of X is taken from the domain of attraction 4 VATUTINANDWACHTEL of a stable law. However, he gives no explanations how one can check the validity of (17) in the general situation. Letχ+ :=S betheascendingladderheight. AliliandDoney[1,Remark1,p.98] τ+ have shown that (14) holds if Eχ+ is finite. By Theorem 3 of [9] the assumption Eχ+ < isequivalentto(15),i.e. forthecase 1<α<2,β = 1 ourTheorem3 ∞ { − } is (implicitly) contained in [1] . Alili and Doney analyzed the distribution of τ+ only. Clearly, one can easily derive the statement of our Theorem 1 for the case 1<α<2,β =1 from their result (for instance, applying Theorem 2). However, { } for these spectrally one-sided cases we present an alternative proof, which clarifies the ”typical” behavior of the random walk on the events τ± = n . See Section { } 3.2 and Section 5 for more details. 2. Auxiliary results 2.1. Notation. InwhatfollowswedenotebyC,C ,C ,...finite positiveconstants 1 2 whichmaybedifferent fromformulatoformulaandbyl(x),l (x),l (x)...functions 1 2 slowly varying at infinity which are, as a rule, fixed. For x 0 let ≥ B (x):=P S (0,x];τ− >n , n n ∈ bn(x):=Bn(cid:0)(x+1) Bn(x)=(cid:1)P Sn (x,x+1];τ− >n . − ∈ Introduce the renewal function (cid:0) (cid:1) ∞ H(x):=1+ P χ++...+χ+ x , x 0, H(x)=0,x<0, 1 k ≤ ≥ k=1 X (cid:0) (cid:1) where χ+ isasequenceofi.i.d. randomvariablesdistributedthesameasχ+. i i≥1 Observe that by the duality principle for random walks for x 0 (cid:8) (cid:9) ≥ ∞ ∞ 1+ B (x)=1+ P S (0,x];τ− >j j j ∈ j=1 j=1 X X (cid:0) (cid:1) ∞ =1+ P(S (0,x];S >S ,S >S ,...,S >S ) j j 0 j 1 j j−1 ∈ j=1 X =H(x). (18) In the sequel we deal rather often with slowly varying functions and, following Doney [9], say that a slowly varying function l∗(x) is an α-conjugate of a slowly varying function l∗∗(x) when the following relations are valid y xαl∗(x) as x if and only if x y1/αl∗∗(y). ∼ →∞ ∼ It is known that if X (α,β) with α (0,2), and F(x):=P(X x), then ∈D ∈ ≤ 1 1 F(x)+F( x) as x , (19) − − ∼ xαl (x) →∞ 0 where l (x) is a function slowly varying at infinity. Besides, for α (0,2), 0 ∈ F( x) 1 F(x) − q, − p as x , (20) 1 F(x)+F( x) → 1 F(x)+F( x) → →∞ − − − − with p+q = 1 and β = p q in (9). Let c be a sequence specified by the − { n}n≥1 relation c :=inf x 0:1 F(x)+F( x) n−1 . (21) n ≥ − − ≤ (cid:8) (cid:9) LOCAL LIMIT THEOREMS FOR LADDER EPOCHS 5 In view of (19) this sequence is regularly varying at infinity with index α−1, i.e. c =n1/αl (n), (22) n 1 where l (x) is a slowly varying function being an α-conjugate of l (x): 1 0 cαl (c ) n as n . (23) n 0 n ∼ →∞ Moreover, S n d Y as n , α c → →∞ n where Y is a random variable obeying an α stable law. α − For the case α = 2 the normalizing sequence c requires a special de- { n}n≥1 scription. Let V(x) = x y2dF(x) be the truncated variance of X. Clearly, −x liminf V(x) > 0 for every nondegenerate random variable X. Furthermore, x→∞ R it is known ([13], Chapter XVII, Section 5) that X (2,0) if and only if V(x) ∈ D varies slowly at infinity. In this case the normalizing sequence c satisfies n V(c ) C n as n . (24) c2 ∼ n →∞ n The last relation means that (22) holds with α = 2 and l (x) is a 2-conjugate of 1 1/V(x). Besides, x2(1 F(x)+F( x)) lim − − =0. (25) x→∞ V(x) 2.2. Basic lemmas. Now we formulate a number of results concerning the distri- butions of the random variables τ−,τ+ and χ+. Recall that a random variable ζ is called relatively stable if there exists a nonrandom sequence d as n n →∞ →∞ such that n 1 p ζ 1 as n , k d → →∞ n k=1 X d where ζ =ζ, k=1,2,... and are independent. k Lemma 4. (see[15]and [10,Theorem9]) AssumeX (α,β). Then, as x , ∈D →∞ 1 P χ+ >x if αρ<1, (26) ∼ xαρl (x) 2 (cid:0) (cid:1) and χ+ is relatively stable if αρ=1. Lemma 5. Suppose X (α,β). If αρ<1, then, as x , ∈D →∞ xαρl (x) 2 H(x) . (27) ∼ Γ(1 αρ)Γ(1+αρ) − If αρ=1, then, as x , →∞ H(x) xl (x), (28) 3 ∼ where x −1 l (x):= P χ+ >y dy , x>0. 3 (cid:18)Z0 (cid:19) In addition, there exists a constant C(cid:0) >0 su(cid:1)ch that in both cases H(c ) Cnρl(n) for all n 1. (29) n ≤ ≥ 6 VATUTINANDWACHTEL Proof. If αρ<1, then by [13, Chapter XIV, formula (3.4)] 1 1 H(x) as x . ∼ Γ(1 αρ)Γ(1+αρ)P(χ+ >x) →∞ − Hence, recalling (26), we obtain (27). If αρ=1, then (28) follows from Theorem 2 in [15]. Let us demonstrate the validity of (29). We know from [15] (see also [7]) that τ+ (ρ,1) under the conditions of the lemma and, in addition, χ+ (αρ,1) ∈ D ∈ D if αρ < 1. This means, in particular, that for sequences a and b n n≥1 n n≥1 { } { } specified by 1 1 P(τ+ >a ) and P(χ+ >b ) as n , (30) n n ∼ n ∼ n →∞ and vectors (τ+,χ+) , being independent copies of (τ+,χ+), we have { k k }k≥1 n n 1 1 τ+ d Y and χ+ d Y as n . (31) a k → ρ b k → αρ →∞ n n k=1 k=1 X X Moreover,it was established by Doney (see Lemma in [10], p. 358) that b Cc as n , (32) n ∼ [an] →∞ where [x] stands for the integer part of x. Therefore, cn ∼ Cb[a−1(n)], where, with a slight abuse of notation, a−1(n) is the inverse function to a . Hence, on account n of (30), C P(χ+ >cn)∼C1P(χ+ >b[a−1(n)])∼ a−1(1n) C ∼C2P(τ+ >a[a−1(n)])∼C3P(τ+ >n)∼ nρl(4n). (33) This proves (29) for αρ<1. If αρ = 1, then, instead of the second equivalence in (30), one should define b n by 1 bn 1 P(χ+ >y)dy as n b ∼ n →∞ n Z0 (see [15, p. 595]). In this case the second convergence in (31) transforms to n 1 χ+ p 1 as n , b k → →∞ n k=1 X while (33) should be changed to 1 cnP(χ+ >y)dy C1 b[a−1(n)]P(χ+ >y)dy C1 cn Z0 ∼ b[a−1(n)] Z0 ∼ a−1(n) C ∼C1P(τ+ >a[a−1(n)])∼C2P(τ+ >n)∼ nρl(3n). The lemma is proved. (cid:3) The next result is a part of Corollary 3 in [9]. LOCAL LIMIT THEOREMS FOR LADDER EPOCHS 7 Lemma 6. Assume X (α,1) with 1<α<2 (implying ρ=1 α−1). Then ∈D − C C P(τ− >n) as n ∼ c ∼ n1/αl (n) →∞ n 1 if and only if ∞ F( x) − dx< . x(1 F(x)) ∞ Z1 − Now we prove a useful result which may be viewed as a statement concerning ”small” deviations of S on the set τ− >n . n { } Let h be the span and g (x) be the density of a stable distribution with pa- α,β rameters α and β in (9) (we agree to consider h=0 for non-lattice distributions). For a set A taken from the Borel σ-algebra on (0, ) denote ∞ µ(A)=g (0) H(x h)ν(dx), α,β − ZA where ν is the counting measure on h,2h,3h,... in the arithmetic case and the { } Lebesgue measure on (0, ) in the non-lattice case. ∞ Lemma 7. Suppose X (α,β). Then ∈D lim nc P(S A; τ− >n)=µ(A) (34) n n n→∞ ∈ for any A taken from the Borel σ-algebra on (0, ). ∞ Proof. Assume firstthat the distributionofX isnon-lattice. Using the Stone local limit theorem (see, for instance, [3, Section 8.4, p. 351]) it is not difficult to show that for λ>0, ∞ g (0) lim c E e−λSn; S >0 =g (0) e−λydy = α,β . (35) n n α,β n→∞ λ Z0 (cid:0) (cid:1) Set ∞ E e−λSn; S >0 n G(λ):= (36) n n=1 (cid:0) (cid:1) X and specify a sequence of measures µ (dx):=nc P(S dx; τ− >n), n 1. n n n ∈ ≥ Since c varies regularly and (35) is valid, applying Theorem 2 from [6] to { n}n≥1 the equality ∞ ∞ zn znE e−λSn; τ− >n =exp E e−λSn; S >0 (37) n n ( ) n=0 n=1 X (cid:0) (cid:1) X (cid:0) (cid:1) shows that for all λ>0, ∞ lim nc E e−λSn; τ− >n = lim e−λxµ (dx) n n n→∞ n→∞ Z0 (cid:0) (cid:1) g (0) α,β = exp G(λ) . (38) λ { } 8 VATUTINANDWACHTEL It follows from (37) that ∞ g (0) g (0) α,β exp G(λ) = α,β 1+ E e−λSk; τ− >k λ { } λ (cid:16) Xk=1 (cid:0) (cid:1)(cid:17) g (0) g (0) ∞ ∞ = α,β + α,β e−λx P S dx; τ− >k k λ λ Z0 k=1 ∈ ! X (cid:0) (cid:1) g (0) g (0) ∞ ∞ = α,β + α,β e−λx P(χ++...+χ+ dx) , λ λ  1 j ∈  Z0 j=1 X   where at the last step we have used the duality principle. Integrating by parts and recalling the definition of H(x), we get g (0) g (0) ∞ α,β exp G(λ) = α,β +g (0) e−λx(H(x) 1)dx α,β λ { } λ − Z0 ∞ =g (0) e−λxH(x)dx. (39) α,β Z0 Combining(38)and (39) andusing the continuity theoremforLaplace transforms, we obtain (34) for non-lattice distributions. In the arithmetic case we have by the Gnedenko local limit theorem ∞ g (0)e−λh lim c E e−λSn; S >0 =g (0) e−λhk = α,β . (40) n→∞ n n α,β 1 e−λh k=1 − (cid:0) (cid:1) X Proceeding as by the derivation of (39), we obtain g (0)e−λh g (0)e−λh ∞ α,β exp G(λ) = α,β 1+ E e−λSk; τ− >k 1 e−λh { } 1 e−λh − − (cid:16) Xk=1 (cid:0) (cid:1)(cid:17) g (0)e−λh g (0)e−λh ∞ = α,β + α,β e−λhj(H(hj) H(hj h)) 1 e−λh 1 e−λh − − − − j=1 X ∞ ∞ =g (0)e−λh e−λhjH(hj)=g (0) e−λhkH(hk h). α,β α,β − j=0 k=1 X X This, together with (40), finishes the proof of the lemma. (cid:3) Lemma8. UndertheconditionsofTheorem1foranyα (0,2)thereexistsC >0 ∈ such that for all y >0 and all n 1, ≥ C l(n) b (y) (41) n ≤ c n1−ρ n and C(y+1) l(n) B (y) . (42) n ≤ c n1−ρ n Proof. Forn=1the statementofthe lemmaisobvious. Let S∗ be arandom { n}n≥0 walk distributed as S and independent of it. One can easily check that for { n}n≥0 LOCAL LIMIT THEOREMS FOR LADDER EPOCHS 9 each n 2, ≥ b (y)=P y <S y+1;τ− >n n n ≤ ∞ = (cid:0) P y S <S S(cid:1) y+1 S ;S dz;τ− >n [n/2] n [n/2] [n/2] [n/2] − − ≤ − ∈ Z0∞ (cid:16) (cid:17) P y z <S∗ y+1 z;S dz;τ− >[n/2] ≤ − n−[n/2] ≤ − [n/2] ∈ Z0 (cid:16) (cid:17) P τ− >[n/2] supP z <S∗ z+1 . (43) ≤ n−[n/2] ≤ z (cid:16) (cid:17) (cid:16) (cid:17) Since the density of any α-stable law is bounded, it follows from the Gnedenko and Stone local limit theorems that if the distribution of X is either arithmetic or non-lattice,thenthereexistsaconstantC >0suchthatforalln 1andallz 0, ≥ ≥ C∆ P(S (z,z+∆]) . (44) n ∈ ≤ c n Hence it follows, in particular, that, for any z >0, C(z+1) P(S (0,z]) . (45) n ∈ ≤ c n Substituting (44) into (43), and recalling (22) and properties of regularly varying functions, we get (41). Estimate (42) follows from (41) by summation. (cid:3) Lemma9. Undertheconditions ofTheorem 1foranyα (1,2]thereexistsC >0 ∈ such that for all n 1 and all x>0, ≥ H(x+1) l(n) x+1 b (x) C + (46) n ≤ nc n1−ρ c2 (cid:18) n n (cid:19) and (x+1)H(x+1) l(n) (x+1)2 B (x) C + . (47) n ≤ ncn n1−ρ c2n ! Proof. According to formula (5) in [12], n−1 x nB (x)=P(S (0,x])+ B (x y)P(S dy). (48) n n n−k k ∈ − ∈ k=1Z0 X Hence we get n−1 x nb (x)=P(S (x,x+1])+ b (x y)P(S dy) n n n−k k ∈ − ∈ k=1Z0 X n−1 x+1 + B (x+1 y)P(S dy). (49) n−k k − ∈ k=1Zx X 10 VATUTINANDWACHTEL Using (41), (45), (22), the inequality 1/α < 1 and properties of slowly varying functions, we deduce [n/2] x [n/2] l(n k) kX=1 Z0 bn−k(x−y)P(Sk ∈dy)≤C kX=1 cn−k(n−−k)1−ρP(Sk ∈[0,x]) [n/2] 1 l(n k) C (x+1) − ≤ 1 kX=1 ck cn−k(n−k)1−ρ [n/2] x+1 l(n) 1 C ≤ 2 c n1−ρ c n k k=1 X nρl(n) C (x+1) . (50) ≤ 3 c2 n On the other hand, in view of (44) and monotonicity of B (x) in x we conclude k (assuming that x is integer without loss of generality and letting B ( 1)= 0 and k − H( 1)=0) that − n x b (x y)P(S dy) n−k k − ∈ k=[Xn/2]+1Z0 n x (B (x j+1) B (x j 1))P(S (j,j+1]) n−k n−k k ≤ − − − − ∈ k=[Xn/2]+1Xj=0 n x C (B (x j+1) B (x j 1)) n−k n−k ≤ − − − − c k k=[Xn/2]+1Xj=0 x ∞ C (B (x j+1) B (x j 1)) k k ≤ c − − − − n j=0k=0 XX x C = (H(x j+1) H(x j 1)) c − − − − n j=0 X C 2C (H(x)+H(x+1)) H(x+1), ≤ c ≤ c n n where for the intermediate equality we have used (18). This gives n x C b (x y)P(S dy) H(x+1). (51) n−k k − ∈ ≤ c k=[n/2]+1Z0 n X Since x B (x) increases for every n, n 7→ n−1 x+1 n−1 B (x+1 y)P(S dy) B (1)P(S (x,x+1]). (52) n−k k n−k k − ∈ ≤ ∈ k=1Zx k=1 X X Further, in view of (42) and (44) we have [n/2] C l(n) [n/2] 1 C nρl(n) B (1)P(S (x,x+1]) 1 2 . (53) n−k k ∈ ≤ c n1−ρ c ≤ c2 k=1 n k=1 k n X X