Probab. Theory Relat. Fields (2012) 152:407–424 DOI 10.1007/s00440-010-0326-3

Local limit theorem for the maximum of asymptotically stable random walks Vitali Wachtel

Received: 4 May 2010 / Revised: 9 August 2010 / Published online: 9 October 2010 © Springer-Verlag 2010

Abstract Let {Sn ; n ≥ 0} be an asymptotically stable random walk and let Mn denote it’s maximum in the first n steps. We show that the asymptotic behaviour of local probabilities for Mn can be approximated by the density of the maximum of the corresponding stable process if and only if the renewal massfunction based on ascending ladder heights is regularly varying at infinity. We also give some conditions on the random walk, which guarantee the desired regularity of the renewal mass-function. Finally, we give an example of a random walk, for which the local limit theorem for Mn does not hold. Keywords

Limit theorems · Random walks · Renewal theorem

Mathematics Subject Classification (2000)

60G50 · 60G52

1 Introduction and statement of results Let {Sn , n ≥ 0} denote the random walk with increments X i , that is, S0 := 0, Sn :=

n 

X i , n ≥ 1.

i=1

We shall assume that X 1 , X 2 , . . . are independent copies of a random variable X . Moreover, we shall assume that X is taken from the domain of attraction of a stable

V. Wachtel (B) Mathematical Institute, University of Munich, Theresienstrasse 39, 80333 Munich, Germany e-mail: [email protected]

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V. Wachtel

law with characteristic function    t πα α G α,β (t) := exp −|t| 1 − iβ tan |t| 2

(1)

with (α, β) ∈ A := (0, 1) × (−1, 1) ∪ {(1, 1/2)} ∪ (1, 2] × [−1, 1]. If EX exists, we assume that EX = 0. In this case we write X ∈ D (α, β). It is well known, that for every X ∈ D(α, β) there exists a sequence cn regularly varying of index 1/α such that {S[nt] /cn , t ∈ [0, 1]} converges weakly to the stable process {Yα,β (t) t ∈ [0, 1]} characterised by (1), i.e., EeitYα,β (1) = G α,β (t). Denote Mn := max0≤k≤n Sk . It follows from the invariance principle for asymptotically stable random walks that appropriately rescaled Mn converges towards the maximum of the corresponding stable law:   P(Mn ≤ cn x) → P max Yα,β (t) ≤ x as n → ∞. 0≤t≤1

In the present note we pose a question on whether a local version of this convergence is valid. More precisely, we investigate the conditions, under which the convergence cn P(Mn ∈ [cn x, cn x + 1)) → m α,β (x) as n → ∞ is true for every x > 0. (m α,β stands for the density function of max0≤t≤1 Yα,β (t).) Moreover, we study the asymptotic behaviour of P(Mn ∈ [x, x + 1)) for x = o(cn ). It is well known that by investigating local probabilities, one has to distinguish between lattice and non-lattice distributions. We concentrate here on the case of aperiodic lattice distributions, that is, P(X ∈ Z) = 1 and P(X ∈ r Z) < 1 for all r ≥ 2. To formulate our results we need some additional notation. Define ladder epochs τ + := min{k ≥ 1 : Sk > 0} and τ − := min{k ≥ 1 : Sk ≤ 0}. Let χ + and χ − denote the corresponding ladder heights, that is, χ + := Sτ + and χ − := −Sτ − . We finally introduce the following renewal functions: H + (x) =

x 

h + (y), h + (y) := 1{y = 0} +

y=0

∞    P χ1+ + χ2+ + · · · + χk+ = y k=1

and −

H (x) =

x 

∞    h (y), h (y) := 1{y = 0} + P χ1− + χ2− + · · · + χk− = y ,

y=0

k=1

−

−

where χ1+ , χ2+ , . . . and χ1− , χ2− , . . . are independent copies of χ + and χ − respectively.

123

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409

Theorem 1 Suppose X ∈ D(α, β). Then, as n → ∞, P(Mn = x) ∼ h + (x)P(τ + > n)

(2)

uniformly in x ∈ (0, δn cn ), where δn → 0 arbitrary slowly. Furthermore, cn P(Mn = x) − m α,β (x/cn ) → 0

(3)

uniformly in x ≥ acn , a > 0, if and only if h + is regularly varying at infinity. In the case of finite variance the asymptotic behaviour of P(Mn = x) has been studied by Aleshkyavichene [1] and by Nagaev and Eppel [10]. Moreover, in the case when Eχ + < ∞, (2) has been obtained by Alili and Doney [2]. The restriction x > acn in the second statement of the theorem means that we exclude “small” values of Mn . This seems to be quite natural, because the density function m α,β is not bounded near zero in general. More exactly, it has been proven by Doney and Savov [6] that m α,β (u) ∼ u αρ−1 as u → 0, where ρ is a constant defined in (4). So, m α,β remains bounded if and only if αρ = 1. We expect that in this case one can get the uniform convergence on the set x > xn for any xn → ∞. For random walks with finite variance it has been shown in [10]. Theorem 1 establishes the direct connection between local probabilities of the maximum and the mass-function of the renewal function based on ladder heights. First, to use (2) one has to know the asymptotic behaviour of h + (x) and that of P(τ + > n). The behaviour of the latter probability is well known: If P(Sn > 0) → ρ ∈ (0, 1),

(4)

which is always true for X ∈ D(α, β), then P(τ + > n) = l(n)n −ρ ,

(5)

where l(x) is a regularly varying function. Second, to approximate local probabilities for Mn by the density of a stable law, one needs to know that h + is regularly varying. Thus, we need to understand, under which restrictions on the distribution of X we have the desired property of h + . It is worth mentioning that the regular variation of h + appears as a restriction in some further situations. Caravenna and Chaumont [4], for example, have imposed this restriction in proving an invariance principle for random walks conditioned to stay positive. It is well known that P(χ + ≥ x) is regularly varying of index −αρ, if αρ < 1, and + χ is relative stable if αρ = 1. Then, using the result of Garsia and Lamperti [7], we have that h + is regularly varying for any random walk with αρ ∈ (1/2, 1). It is also known, see Williamson [12], that if the tail of a positive random variable is regularly varying of some index less than 1/2, then the mass-function of the corresponding renewal function is not regularly varying in general. Doney [5] has shown that the mass-function of renewal function is regularly varying under certain assumptions on the local probabilities of underlying random variables. In order to apply his result to ladder heights, we need an information on P(χ + = x).

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Theorem 2 Suppose X ∈ D(α, β) with α < 2 and |β| < 1. If

P(X = x) = O x −1 P(X ≥ x) ,

(6)

then P(χ + = x) ≤ C

P(χ + ≥ x) . x

(7)

If, additionally, α(1 − ρ) > 1/2 or

P(X = −x) = O x −1 P(X ≤ −x) ,

(8)

then, as x → ∞, P(χ + = x) ∼

αρ P(χ + ≥ x). x

(9)

Applying Doney’s result mentioned above, we obtain the following statement. Corollary 3 If X ∈ D(α, β) with α < 2 and |β| < 1 and (6) holds, then, as x → ∞, h + (x) ∼ αρ

H + (x) . x

(10)

Consequently, in view of Theorem 1, cn P(Mn = x) − m α,β (x/cn ) → 0 uniformly in x ≥ acn , a > 0. To the best of our knowledge, the behaviour of P(χ + = x) for oscillating random walks has been studied in the case β = 1 only. Namely, Bertoin and Doney [3] have proven the following result: If Eχ − < ∞, which is a particular case of β = 1, and the right tail of X is long-tailed, then P(χ + = x) ∼

1 P(X ≥ x) as x → ∞. Eχ −

(11)

In this special case the behaviour of P(χ + = x) is resistant against all kinds of irregularity in the local structure of the distribution of X . But this is not true in the case when β < 1. We demonstrate it with the following example. Example 4 Assume that P(X = x) =

123

C2γ n , x ∈ [2n , 2n + 2(1−γ )n /n], n ≥ 1 2n(α+1)

Local limit theorem for the maximum of random walks

411

for some γ ∈ (0, 1), and that P(X = x) = C x −α−1 for all other values of x > 0. The negative part of X is such that P(X ≤ −x) = O(x −α ) and β < 1. One can easily see, that P(X ≥ x) ∼ C x −α . But lim sup x→∞

xP(χ + = x) = ∞. P(χ + ≥ x)

(12)

xh + (x) = ∞. H + (x)

(13)

If, furthermore, α(1 + ρ) < 1, then lim sup x→∞

We postpone the proof of these relations to the end of the paper. The relation (12) shows that (9) and (7) can not be true if one simply removes the condition (6) from Theorem 2. And (13) shows that (10) can not be valid for all random variables X ∈ D(α, β). Combining (13) with the second statement of Theorem 1, we see that the local limit theorem for Mn takes place not for all asymptotically stable random walks. There is another interesting observation connected to Example 4. Williamson’s counterexample to the local renewal theorem can be seen as a special case of (13) for random walks with positive increments. Since ρ = 1 for positive increments, the condition α(1 + ρ) < 1 changes to α < 1/2. But we know that the local renewal theorem holds for all random walks with α > 1/2. Therefore, the following conjecture seems to be quite plausible: (10) holds for all random walks with α(1 + ρ) > 1. In conclusion we mention that we expect that analogous results are true in the nonlattice case. There one has to replace P(Mn = x) and h + (x) by P(Mn ∈ [x, x + 1)) and H + (x + 1) − H + (x) respectively. 2 Proofs 2.1 Some results from fluctuation theory In this paragraph we collect some known results from the fluctuation theory for random walks. We start with a representation for P(Mn = x), which will be used in the proof of Theorem 1. Lemma 5 For all n, x > 0, P(Mn = x) =

n 

P(Sk = x, τ − > k)P(τ + > n − k).

(14)

k=0

Proof Denote θn := min{k ≥ 0 : Sk = Mn }. By the Markov property, P(Mn = k, θn = k) = P(Sk = x, θk = k)P(θn−k = 0).

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Furthermore, it follows from the duality lemma for random walks that P(Sk = x, θk = k) = P(Sk = x, τ − > k). Noting also that P(θn−k = 0) = P(τ + > n − k), we have P(Mn = x) = =

n  k=0 n 

P(Mn = k, θn = k) P(Sk = x, τ − > k)P(τ + > n − k).

k=0



Thus, the lemma is proved.

Remark 6 Representation (14) has been derived in [2]. The authors have used it by proving an analog of (2) for random walks with Eχ + < ∞. It is worth mentioning that [1] contains another representation for P(Mn = x), which is based on a recursive formula for the characteristic function of Mn . The latter is due to Nagaev [9]. We next note that h + can be written as an infinite sum of P(Sk = x, τ − > k). Indeed, using the duality lemma once again, we have h + (x) = 1{x = 0} +

∞    P χ1+ + χ2+ + · · · + χk+ = x k=1

∞    = 1{x = 0} + P S j = x; S j > S0 , S j > S1 , . . . , S j > S j−1 j=1 ∞    P S j = x; τ − > j . = 1{x = 0} +

(15)

j=1

If X ∈ D(α, β), then the norming sequence cn can be specified by the relation cn := inf

⎧ ⎨ ⎩

u > 0 : u −2

u x 2 P(X ∈ d x) > n

−u

⎫ ⎬ ⎭

, n ≥ 1.

If, furthermore, α < 2 and β > −1, then P(X ≥ x) is regularly varying of index −α and there exists a positive constant C(α, β) such that P(X ≥ cn ) ∼

C(α, β) , as n → ∞. n

As it has been mentioned in the introduction, see (4), P(Sn > 0) → ρ for every asymptotically stable random walk. We have also mentioned, that ρ determines the

123

Local limit theorem for the maximum of random walks

413

asymptotic behaviour of τ + , see (5). Besides this relation one has lim P(τ − > n)P(τ + > n) =

n→∞

sin πρ . π

(16)

We next state a result, which establishes a connection between the tail of τ + and

H + (x).

Lemma 7 If X ∈ D(α, β), then there exists a constant C(α, β) such that H + (cn ) ∼

C(α, β) . P(τ + > n)

(17)

This statement is contained implicitly in Lemma 13 of [11]. In our proofs we shall frequently use the following well-known properties of regularly varying sequences. Lemma 8 Let an be regularly varying of index γ . (i) For every ε > 0, ak = (k/n)γ + o(1) as n → ∞ an uniformly in k ∈ [εn, (1 − ε)n]. (ii) If γ > −1, then, for every r > 0, rn  k=1

ak ∼

r 1+γ nan as n → ∞. 1+γ

(iii) If γ < −1, then, for every r > 0, ∞ 

ak ∼ −

k=r n

r 1+γ nan as n → ∞. 1+γ

2.2 Small deviations for the maximum: Proof of (2) Fix any ε ∈ (0, 1). It is easy to see that P(τ + > n) ≤

εn

− + k=0 P(Sk = x, τ > k)P(τ > εn − k=0 P(Sk = x, τ > k)

n − k)

≤ P(τ + > n − εn). (18)

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Using (15), we have εn 

P(Sk = x, τ − > k) =

k=0

∞ 

P(Sk = x, τ − > k) −

∞ 

P(Sk = x, τ − > k)

εn

k=0

= h + (x) −

∞ 

P(Sk = x, τ − > k).

εn

According to Lemma 20 from [11], P(Sk = x, τ − > k) ≤ C

H + (x) , x > 0. kck

(19)

Therefore, ∞ 

P(Sk = x, τ − > k) ≤ C H + (x)

εn

∞  1 H + (x) ≤ Cε−1/α , kck cn εn

where in the last step we have applied Lemma 8(iii) to the sequence (ncn )−1 . Consequently, h + (x) − Cε−1/α

εn

H + (x)  ≤ P(Sk = x, τ − > k) ≤ h + (x). cn

(20)

k=0

Applying Theorem 1.1 from [7] to h + , we have lim inf x→∞

xh + (x) = αρ. H + (x)

This implies that H + (x) ≤ C xh + (x), x > 0. From this bound and (20) we conclude that   εn  1   x   − P(Sk = x, τ > k) − 1 ≤ Cε−1/α .  +  h (x)  cn k=0

Therefore, εn

1  P(Sk = x, τ − > k) → 1 + h (x) k=0

123

(21)

Local limit theorem for the maximum of random walks

415

uniformly in x ≤ δn cn . Applying this to the inequalities in (18), we obtain εn lim inf min

= x, τ − > k)P(τ + > n − k) ≥1 h + (x)P(τ + > n)

(22)

= x, τ − > k)P(τ + > n − k) ≤ (1 − ε)−ρ . h + (x)P(τ + > n)

(23)

k=0 P(Sk

n→∞ x≤δn cn

and εn lim sup max

k=0 P(Sk

n→∞ x≤δn cn

Using (19) once again, we get 

P(Sk = x, τ − > k)P(τ + > n − k) ≤ C H + (x)

εn≤k≤n

 P(τ + > n − k) kck

εn≤k≤n

≤

Cε−1−1/α H + (x) ncn

≤ Cε−1−1/α

n 

P(τ + > j)

j=0 + H (x)P(τ +

cn

> n)

,

where in the last two steps we have used Lemma 8 (i) and (ii) respectively. It follows now from (21) that 

εn≤k≤n

P(Sk = x, τ − > k)P(τ + > n − k) h + (x)P(τ + > n)

→0

(24)

uniformly in x ≤ δn cn . Combining (14) and (22)–(24), we obtain the desired relation. 2.3 Local limit theorem for the maximum: Proof of (3) Let ε be any fixed number from the interval (0, 1/2). Then, using the bound (see Lemma 19 in [11]) P(Sk = x, τ − > k) ≤

C P(τ − > k), x > 0, ck

and applying Lemma 8(ii) to the sequence P(τ + > n), we get n  k=(1−ε)n

P(Sk = x, τ − > k)P(τ + > n − k) ≤

εn

 C P(τ − > n) P(τ + > k) cn k=1

≤ Cε1−ρ

nP(τ −

> n)P(τ + > n) . cn

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V. Wachtel

In view of (16), the sequence nP(τ − > n)P(τ + > n) remains bounded. Therefore, n 

cn

P(Sk = x, τ − > k)P(τ + > n − k) ≤ Cε1−ρ

(25)

k=(1−ε)n

for all x > 0. Applying Theorem 5 of [11], we have, uniformly in x > 0, (1−ε)n 

P(Sk = x, τ − > k)P(τ + > n − k)

k=εn (1−ε)n 

pα,β (x/ck ) P(τ − > k)P(τ + > n − k) ck k=εn ⎛ ⎞ (1−ε)n  P(τ − > k)P(τ + > n − k) ⎠, +o ⎝ ck

=

(26)

k=εn

where pα,β denotes the density of the corresponding Levy meander. Then, using Lemma 8(i), and taking into account the fact that pα,β is uniformly continuous, we get

(1−ε)n 

pα,β (x/ck ) P(τ − > k)P(τ + > n − k) ck k=εn        (1−ε)n  x k −1/α k ρ−1−1/α k −ρ − + 1− ∼ P(τ > n)P(τ > n) pα,β cn n n n

cn

k=εn

sin πρ ∼ π



1−ε pα,β ε

 x 1/α ρ−1−1/α v v (1 − v)−ρ dv cn

(27)

uniformly in x > 0. Using the same arguments, one can easily get

(1−ε)n  k=εn

123

P(τ − > k)P(τ + > n − k) sin πρ ∼ ck π

1−ε v ρ−1−1/α (1 − v)−ρ dv. ε

(28)

Local limit theorem for the maximum of random walks

417

Combining (26)–(28), we have

cn

(1−ε)n 

P(Sk = x, τ − > k)P(τ + > n − k) =

k=εn

sin πρ (1 + o(1)) π



1−ε pα,β ε

 x −1/α ρ−1−1/α v v (1 − v)−ρ dv + o(1), cn

and the o(1)-terms are uniform in x > 0. Consequently, uniformly in x > acn , ⎛ lim lim ⎝cn

ε→0 n→∞

−

pα,β 0

P(Sk = x, τ − > k)P(τ + > n − k)

k=εn



1

(1−ε)n 

⎞  x −1/α ρ−1−1/α v v (1 − v)−ρ dv ⎠ = 0. cn

(29)

The finiteness of the integral for x > acn follows from the boundedness of pα,β and from the relation pα,β (y) ∼ C y −α−1 , y → ∞. The latter has been proven in [6]. Define N x = max{n : cn ≤ x}. Lemma 9 Assume that X ∈ D(α, β). Then lim lim

x

b→0 x→∞ H + (x)



P(Sk = x, τ − > k) = αρ.

k≥bN x

Proof Using (19) and Lemma 8(ii), we have 

P(Sk = x, τ − > k) ≤ C H + (x)

k≥N x /b

≤ Cb1/α



k≥N x /b + H (x)

c Nx

1 kck

≤ Cb1/α

H + (x) . x

(30)

Applying Theorem 5 of [11] once again, we have  bN x ≤k≤N x /b

P(Sk = x, τ − > k) = (1 + o(1))

 bN x ≤k≤N x /b

P(τ − > k) pα,β (x/ck ). ck

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V. Wachtel

Repeating the arguments, which have been used in deriving (27), and noting that c Nx ∼ x, one can easily see that, as x → ∞, N x /b  k=bN x

P(τ − > k) N x P(τ − > N x ) pα,β (x/ck ) ∼ ck c Nx

1/b v ρ−1−1/α pα,β (v −1/α )dv b

1 sin πρ ∼ + π xP(τ > N x )

1/b v ρ−1−1/α pα,β (v −1/α )dv. b

Taking into account (17), we get N x /b 

P(Sk = x, τ

−

k=bN x

H + (x) > k) ∼ C(α, β) x

1/b v ρ−1−1/α pα,β (v −1/α )dv. (31) b

From (30) and (31) we conclude that lim lim

b→0 x→∞



x H + (x)

P(Sk = x, τ − > k) = C(α, β).

(32)

k≥bN x

The constant on the right hand side is finite, since ∞ v

ρ−1−1/α

pα,β (v

−1/α

∞ )dv = α

0

z −αρ pα,β (z)dz < ∞.

0

To finish the proof it is sufficient to show that C(α, β) = αρ for some special random walk. We consider any X ∈ D(α, β) with the following property: P(X = x) is regularly varying of index −α − 1. From Lemma 7 of Jones [8] we have P(Sk = x, τ − > k) ≤ CkP(X = x)P(τ − > k). Thus, using Lemma 8(ii), we get bN x 

P(Sk = x, τ − > k) ≤ CP(X = x)

k=1

≤ ≤

kP(τ − > k)

k=1 2 N x P(τ − > Cb Cb1+ρ N x P(τ − > + 1+ρ H (x)

≤ Cb

123

bN x 

1+ρ

x

.

N x )P(X = x) N x )N x P(X ≥ x)/x

Local limit theorem for the maximum of random walks

419

Combining this bound with (32), we get h + (x) =

∞ 

P(Sk = x, τ − > k) ∼ C(α, β)

k=1

H + (x) . x

x + + Recalling that H + (x) = y=0 h (y) is regularly varying of index αρ, h (x) ∼ + αρ H (x)/x. Thus, the proof of the lemma is finished.

We now continue the proof of the local limit theorem. Assume first that h + (x) is regularly varying. Then h + (x) ∼ αρ H + (x)/x. From this relation and Lemma 9 we infer that x x lim lim P(Sk = x, τ − > k) = 0. b→0 x→∞ H + (x) bN

k=1

The latter yields εn

 cn P(Sk = x, τ − > k) = 0 H + (cn )

lim lim

ε→0 n→∞

k=1

uniformly in x ≥ acn . Using (17) once again, we have lim lim cn

ε→0 n→∞

εn 

P(Sk = x, τ − > k)P(τ + > n − k) = 0.

(33)

k=1

Combining (25), (28), (29) and (33), we obtain 

1 cn P(Mn = x) −

pα,β 0

 x −1/α ρ−1−1/α v v (1 − v)−ρ dv = o(1) cn

uniformly in x ≥ acn . It was shown in [6] that the integral in the latter formula is equal to m α,β (x/cn ). Thus, we have proven that the regular variation of h + is sufficient for the local limit theorem. To get the reversed statement, we note that the convergence cn P(Mn = [cn ]) → m α,β (1) implies, in view of (25), (28) and (29), that lim lim cn

ε→0 n→∞

εn 

P(Sk = [cn ], τ − > k)P(τ + > n − k) = 0.

k=1

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V. Wachtel

(Here [x] denotes the integer part of x.) The latter is equivalent to εn

 cn P(Sk = [cn ], τ − > k) = 0. lim lim ε→0 n→∞ H + (cn ) k=1

Substituting x = [cn ], we get εN

lim lim

ε→0 x→∞

x x P(Sk = x, τ − > k) = 0. H + (x) k=1

This, together with Lemma 9, implies that h + (x) ∼ αρ

H + (x) . x

Thus, the proof of the second part of Theorem 1 is completed. 2.4 Proof of Theorem 2 Using Wiener–Hopf factorisation one can get, see formula (5) in [3], P(χ + = x) =

∞ 

P(X = x + y)h − (y).

(34)

y=0

It follows from (6) that max P(X = j x + y) ≤ C

0≤y≤x

P(X ≥ x) P(X ≥ j x) ≤ C j −α−1 , jx x

j ≥ 1.

(35)

Applying this bound to the summands on the right hand side of (34), we get P(χ + = x) =

∞  x−1 

P(X = j x + y)h − (( j − 1)x + y)

j=1 y=0 ∞

≤C

 P(X ≥ x)  −α−1  − H ( j x − 1) − H − (( j − 1)x − 1) . j x j=1

Since H − is regularly varying of index α(1 − ρ), lim

x→∞

123

H − ( j x − 1) − H − (( j − 1)x − 1) α(1−ρ) α(1−ρ) = j . (36) − ( j − 1) H − (x)

Local limit theorem for the maximum of random walks

421

As a result we have P(χ + = x) ≤ C H − (x)

P(X ≥ x) . x

(37)

We next note that (34) yields

P(χ

+

≥ x) =

∞ 

−

P(X ≥ x + y)h (y) =

y=0

∞  x−1 

P(X ≥ j x + y)h − (( j − 1)x + y)

j=1 y=0

Using (36) and the fact that P(X ≥ x) is regularly varying, one can easily obtain C1 P(X ≥ x)H − (x) ≤ P(χ + ≥ x) ≤ C2 P(X ≥ x)H − (x). Combining the lower bound with (37), we have P(χ + = x) ≤ C

P(χ + ≥ x) , x

i.e., the first statement is proven. We now turn to the proof of (9). First we note that h − (x) is regularly varying of index −1 − α(1 − ρ). Indeed, if α(1 − ρ) > 1/2, then it follows from the renewal theorem of Garsia and Lamperti. And if (8) holds, then, in view of (7), it is a consequence of Theorem 3 in [5]. Fix any ε ∈ (0, 1). In view of (6), 

P(X = x + y)h − (y) ≤ C

0≤y<εx

P(X ≥ x) − H (εx) x

≤ Cεα(1−ρ)

P(X ≥ x) − H (x). x

(38)

(In the last step we used the fact that H is regularly varying of index α(1 − ρ).) It is easy to see that P(X ≥ ux|X ≥ x) → u −α , i.e. given X > x, X/x converges weakly to the Pareto distribution. Moreover, as we have already proven, h − is regularly varying. This implies that h − (ux − x)/ h − (x) is bounded on (1 + ε, ∞) and, moreover, converges to (u − 1)−1−α(1−ρ) pointwise. Therefore, 

P(X = x + y)h − (y)

y≥εx

  h − (X − x) X ≥ x 1{X ≥ (1 + ε)x} h − (x) ∞ = (1 + o(1))h − (x)P(X ≥ x) (u − 1)−1−α(1−ρ) αu −α−1 du.

= h − (x)P(X ≥ x)E



(39)

1+ε

123

422

V. Wachtel

Applying (38) and (39) to the right hand side of (34), and letting ε → 0, we obtain

P(χ

+

∞ = x) ∼ h (x)P(X ≥ x) (u − 1)−1−α(1−ρ) αu −α−1 du. −

1

In particular, P(χ + = x) is regularly varying, as a product of two regularly varying functions. Since P(χ + ≥ x) is also regularly varying, we conclude that P(χ + = x) ∼ αρP(χ + ≥ x)/x. Thus, the proof of the theorem is completed. 2.5 Calculations related to Example 4 It follows from (34) that P(χ + = 2n + z) ≥

r n −z

P(X = 2n + z + y)h − (y),

y=0

where rn = 2(1−γ )n /n. And according to our choice of the law of X , P(χ + = 2n + z) ≥

C2γ n − H (rn − z), z < rn . 2(α+1)n

Recalling that H − is regularly varying of index α(1 − ρ), we conclude that P(χ

+

C2γ n = 2 + z) ≥ (α+1)n 2



n

2(1−γ )n n

≥ C2−n(1+αρ)

α(1−ρ) L − (rn )

2γ n(1−α(1−ρ)) − L (rn ), z < rn /2. n α(1−ρ)

Using that P(χ + ≥ x) is also regularly varying, we get finally the bound P(χ + = x) ≥ C2nγ



P(χ + ≥ x) , x ∈ [2n , 2n +rn /2], γ  < γ (1−α(1−ρ)). (40) x

This implies (12). We obtain (13) as a particular case of a more general observation, which can be seen as a generalisation of the well-known example of Williamson, see [12]. Assume that X is positive and that there exists a sequence Rn < 2n such that P(X = x) ≥

123

2γ n 2(α+1)n

(2n ), x ∈ [2n , 2n + Rn ]

Local limit theorem for the maximum of random walks

423

and P(X = x) =

(x) , x ∈ (2n + Rn , 2n+1 ) x α+1

for some γ , α ∈ (0, 1) and for some slowly varying function . One can easily verify that the additional restriction Rn 2(1−γ )n yields that we can choose X in a such way that P(X ≥ x) ∼

(x) as x → ∞, xα

i.e., X ∈ D(α, 1). We next derive a lower bound for P(Sk = 2n + Rn ). It is clear that P(Sk = 2n + Rn ) ≥ kP(X 1 ≥ 2n , Sk = 2n + Rn ) ≥ k min P(X = 2n + y)P(Sk−1 ≤ Rn ) y≤Rn

= kP(Sk−1 < Rn ) Since h + (x) = the inequality

∞

k=1 P(Sk

2γ n 2(α+1)n

(2n ).

= x) in the case of positive random variables, we have 

h + (2n + Rn ) ≥

∞ 

 kP(Sk−1 < Rn )

k=1

2γ n 2(α+1)n

(2n ).

It follows from the convergence to a stable law, that P(Sk−1 < Rn ) ≥ C > 0 for all k such that ck ≤ Rn , say k ≤ Tn . Then ∞ 

kP(Sk−1 < Rn ) ≥ C Tn2 ≥

k=1

C , (P(X ≥ Rn ))2

in the last step we have used the relation Tn ∼ 1/P(X ≥ Rn ), which follows from the properties of the norming sequence cn . As a result we have h + (2n + Rn ) ≥ C

2γ n

2

R 2α (α+1)n n

(2n ) 2(α−1)n 2γ n Rn2α ( (2n ))2 = C . ( (Rn ))2 (n) 22αn ( (Rn ))2

from this bound we conclude that lim sup (x)x 1−α h + (x) = ∞

(41)

x→∞

123

424

V. Wachtel

provided that Rn satisfies the condition 2(γ /2)n Rnα (2n ) → ∞. 2αn (Rn )

(42)

If α > 1/2 it is not possible. But if α < 1/2, then we can choose Rn = 2(1−γ )n−δn with some 0 < δ < γ (1 − 2α). We now come back to random variables which take values of both signs. It follows from (40) that (42) holds with αρ instead of α and Rn = rn /2 if 1 − α − αρ > 0. Then, (41) yields (13). We finish the paper with the following remark. The additional restriction α(1+ρ) < 1 in (13) reflects the fact that the local behaviour of χ + is much smoother in the case when P(X < 0) > 0 than in the case of positive summands [note that (41) holds without any additional assumption]. This effect appears due to convolution of P(X = x) with h − , see formula (34). This gives rise to the following question: Is it possible, at least for some α and β with αρ ≤ 1/2, to infer the regular behaviour of h + from that of h − ? References 1. Aleshkyavichene, A.K.: Local theorems for the maximum of sums of independent identically distributed random variables. Lit. Matem. Sb. 13, 163–174 (1973) 2. Alili, L., Doney, R.A.: Wiener–Hopf factorization revisited and some applications. Stoc. Stoc. Rep. 66, 87–102 (1999) 3. Bertoin, J., Doney, R.A.: On the local behaviour of ladder height distributions. J. Appl. Prob. 31, 816–821 (1994) 4. Caravenna, F., Chaumont, L.: Invariance principles for random walks conditioned to stay positive. Ann. Inst. H. Poincare Probab. Statist. 44, 170–190 (2008) 5. Doney, R.A.: One-sided local large deviation and renewal theorems in the case of infinite mean. Probab. Theory Relat. Fields 107, 451–465 (1997) 6. Doney, R.A., Savov, M.: The asymptotic behaviour of densities related to the supremum of a stable process. Ann. Probab. 38, 316–326 (2010) 7. Garsia, A., Lamperti, J.: A discrete renewal theorem with infinite mean. Comm. Math. Helv. 37, 221–234 (1963) 8. Jones, E.M.: Large deviations for random walks and Levy processes. PhD thesis, Manchester (2008) 9. Nagaev, S.V.: An estimate of the rate of convergence of the distribution of the maximum of the sums of independent random variables. Sib. Math. J. 10, 443–458 (1969) 10. Nagaev, S.V., Eppel, M.S.: On a local limit theorem for the maximum of sums of independent random variables. Theory Probab. Appl. 21, 384–385 (1976) 11. Vatutin, V.A., Wachtel, V.: Local probabilities for random walks conditioned to stay positive. Probab. Theory Relat. Fields 143, 177–217 (2009) 12. Williamson, J.A.: Random walks and Riesz kernels. Pacific J. Math. 25, 393–415 (1968)

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