On a class of additive functionals of two-dimensional Brownian motion and random walk

Dedicated to Pál Révész on the occasion of his 65th birthday

Endre Csáki^{a} Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, P.O.B. 127, H-1364, Hungary. E-mail address: csaki@math-inst.hu

Miklós Csörgő^{b} School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada K1S 5B6. E-mail address: mcsorgo@math.carleton.ca

Antónia Földes^{c} City University of New York, 2800 Victory Blvd., Staten Island, New York 10314, U.S.A. E-mail address: afoldes@email.gc.cuny.edu

Zhan Shi^{d} Laboratoire de Probabilités UMR 7599, Université Paris VI, 4 Place Jussieu, F-75252 Paris Cedex 05, France. E-mail address: zhan@proba.jussieu.fr

Abstract: Sample path properties of a class of additive functionals of two-dimensional Brownian motion and random walk are studied.

AMS 1991 Subject Classification: Primary 60J15; Secondary 60F15, 60J55.

Keywords: Additive functional, Brownian motion, random walk, strong approximation.

aResearch supported by the Hungarian National Foundation for Scientific Research, Grant No. T 019346 and T 029621, and by a Fellowship of the Paul Erdős Summer Research Center of Mathematics, Budapest.

bResearch supported by an NSERC Canada Grant at Carleton University, Ottawa, and by a Paul Erdős Visiting Professorship of the Paul Erdős Summer Research Center of Mathematics, Budapest.

cResearch supported by a PSC CUNY Grant, No. 6-69413, and by a Fellowship of the Paul Erdős Summer Research Center of Mathematics, Budapest.

dResearch supported by a Fellowship of the Paul Erdős Summer Research Center of Mathematics, Budapest.

1. Introduction and main results

Let {W (t), t ≥ 0} be a standard Wiener process (Brownian motion) with local time L(x, t) and for α < 3/2 define

Y_{α}(t) := ∫_{0}^{t} \frac{ds}{W^{α}(s)} = ∫_{0}^{∞} \frac{L(x, t) − L(−x, t)}{x^{α}} dx.

Here z^{α} = |z|^{α} sgn(z) and the first integral is defined as Cauchy principal value for 1 ≤ α < 3/2. In our previous works [CsCsFS, 99a and 99b], we were interested in the almost sure path properties of Y_{α}(t) and its corresponding counterparts for the case of simple symmetric random walk. The general case 0 < α < 3/2 was studied in [CsCsFS, 99a], and the case α = 1 in [CsCsFS, 99b]. Moreover, in [CsCsFS, 99a], for α > 3/2, we investigated the properties of the modified process

Y_{α}^{∗}(t) := ∫_{0}^{t} \frac{1}{W^{α}(s)}1_{\{|W (s)|≥1\}} ds.

In this note we discuss the corresponding two-dimensional problem.

Let {W(t) := (W_{1}(t), W_{2}(t)), t ≥ 0} be a two-dimensional Wiener process, where W_{1}(t) and W_{2}(t) are two independent one-dimensional Wiener processes, with W_{1}(0) = W_{2}(0) = 0. Put

R(t) := ∥W(t)∥ = \sqrt{W_{1}^{2}(t) + W_{2}^{2}(t)} .

It is well-known that {R(t), t ≥ 0} is a two-dimensional Bessel process. We are interested in the additive functional

Z_{α}(t) := ∫_{0}^{t} \frac{ds}{R^{α}(s)}, (1.1)

the critical case being α = 2 (instead of 3/2). It can be seen that the integral in (1.1) converges for α < 2, but diverges for α ≥ 2 almost surely. In the latter case we define the modified process

Z_{α}^{∗}(t) := ∫_{0}^{t} \frac{1}{R^{α}(s)}1_{\{R(s)≥1\}} ds. (1.2)

Considering the random walk counterpart, let {S_{n}}_{n=1}^{∞} be a simple symmetric random walk on the integer lattice Z^{2}, i.e. S_{n} = ∑_{k=1}^{n} X_{k}, where the random variables X_{i}, i = 1, 2, ... are i.i.d., with

P(X_{1} = (0, 1)) = P(X_{1} = (0, −1)) = P(X_{1} = (1, 0)) = P(X_{1} = (−1, 0)) = \frac{1}{4}.

We also propose to study the discrete process

U_{α}(n) := ∑_{k=1}^{n} \frac{1}{∥S_{k}∥^{α}} 1_{\{S_{k}≠0\}}. (1.3)

Define

ξ(x, n) := #{k; 1 ≤ k ≤ n, S_{k} = x},

for any lattice point x of Z^{2}. This is the local time process of {S_{n}}_{n=1}^{∞}. Let furthermore {ρ_{n}, n ≥ 0} denote the consecutive return times of the walk to zero, that is

ρ_{0} := 0, ρ_{n} := min{k > ρ_{n−1}, S_{k} = 0}.

Our first result concerns the case 0 < α < 2, for which we show that the processes Z_{α}(·) and U_{α}(·) are close to each other.

Theorem 1.1. Let 0 < α < 2. Then on a suitable probability space one can define a two-dimensional Wiener process W(·) with the additive functional Z_{α}(·) as in (1.1) and a two-dimensional simple symmetric random walk S_{·} with the additive functional U_{α}(·) as in (1.3) such that, as n → ∞, for any ε > 0

(1.4) |2^{−α/2}U_{α}(n) − Z_{α}(n)| = \{ \begin{matrix}O(n^{1−α+ε}), & a.s., if 0 < α < 1, \\ O(log^{4+ε} n), & a.s., if α = 1, \\ O(log^{2+ε} n), & a.s., if 1 < α < 2. \end{matrix}

We note that by self-similarity, Z_{α}(t) is of order t^{1−α/2}, so the rates in (1.4) are useful.

In the case when α > 2, the corresponding processes, Z_{α}^{∗} and U_{α} are not close to each other. Instead, as in the one-dimensional case, we show separately, that both processes, suitably centered, are close to certain iterated processes.

Theorem 1.2. Let α > 2. There exists a probability space where one can define

(i) a two-dimensional simple symmetric random walk {S_{n}}_{0}^{∞} with its local time process ξ(x, n), and with the corresponding additive functional process {U_{α}(n), n = 1, 2...} as in (1.3);

(ii) a process {ξ^{(1)}(0, n), n = 1, 2, ...} =^{D} {ξ(0, n), n = 1, 2, ...};

(iii) a standard Wiener process {W (t), t ≥ 0}, independent of {ξ^{(1)}(0, n), n = 1, 2, ...}; such that, for some ε > 0, as n → ∞,

U_{α}(n) − f̄_{α}^{D}ξ(0, n) = σ_{α}^{D}W (ξ^{(1)}(0, n)) + O(log^{1/2−ε} n), a.s.,

where

f̄_{α}^{D} := ∑_{x∈Z^{2}−\{0\}} \frac{1}{∥x∥^{α}} , σ_{α}^{D} := \sqrt{Var(U_{α}(ρ_{1}))} .

Moreover, the processes ξ(0, n) and ξ^{(1)}(0, n) can be chosen such that for some δ > 0,

ξ(0, n) = ξ^{(1)}(0, n) + O(log^{1−δ} n), a.s.

We would also like to prove a similar result for the corresponding problem of a twodimensional Wiener process. Formulating a claim in this case needs some careful consideration as points are polar for the two-dimensional Wiener process, so that it has no local time. However its norm R(t) does have a local time which could perfectly fit in our desired theorem. For technical reasons we introduce a slightly different definition as our modified local time for R(t). This idea is not new, similar but more general concepts were already discussed e.g. in Burdzy et al. [BPY, 90]. Define two increasing sequences of stopping times {σ_{n}, n ≥ 0} and {θ_{n}, n ≥ 1} by

(1.5)  σ_{0} := inf{t > 0; R(t) = 1},
(1.6)  θ_{n} := inf{t > σ_{n−1}; R(t) = 2},  n ≥ 1,
(1.7)  σ_{n} := inf{t > θ_{n}; R(t) = 1},  n ≥ 1,
(1.8)  τ_{i} := σ_{i} − σ_{i−1},  i = 1, 2, ...

Let furthermore η(t) = max{n; σ_{n} − σ_{0} ≤ t}. (1.9)

Theorem 1.3. Let α > 2. There exists a probability space where one can define

(i) a two-dimensional Wiener process {W(t), t ≥ 0} with its corresponding additive functional process {Z_{α}^{∗}(t), t ≥ 0} as in (1.2), and with {η(t), t > 0} as in (1.9);

(ii) a process {η^{(1)}(t), t > 0} =^{D} {η(t), t > 0};

(iii) a standard Wiener process {W (t), t ≥ 0}, independent of {η^{(1)}(t), t ≥ 0}; such that for some ε > 0,

Z_{α}^{∗}(t) − f̄_{α}^{W} η(t) = σ_{α}^{W} W (η^{(1)}(t)) + O(log^{1/2−ε} t), a.s., (1.10)

where

f̄_{α}^{W} := \frac{2}{(α − 2)^{2}} , (σ_{α}^{W} )^{2} := E (∫_{1}^{∞} \frac{1}{x^{α}} ℓ(x, τ_{1}) dx)^{2} − \binom{}{f̄_{α}^{W}}^{2} ,

and ℓ(x, t) is the local time at time t and position x of {R(t), t ≥ 0}. Furthermore, η and η^{(1)} can be chosen such that for some δ > 0,

η(t) = η^{(1)}(t) + O(log^{1−δ} t), a.s. (1.11)

The rest of the paper is organized as follows. Section 2 is devoted to a collection of known results which will be used later on. Theorems 1.1–1.3 are proved in Sections 3–5 respectively. In Section 6 we present some further weak and strong convergence results, as consequences of our theorems.

2. Preliminary results

In this section we list some results which will be used in the proofs of the theorems.

Let {S_{n}, n ≥ 1} be a simple symmetric random walk on the plane, and let {ρ_{n}, n ≥ 0} denote, as before, the consecutive return times to zero.

For any real valued function f(·) on Z^{2} we define the additve functional

A(f, n) = ∑_{k=1}^{n} f(S_{k}).

We assume that for some δ > 0,

E (|∑_{k=1}^{ρ_{1}} f(S_{k})|^{2+δ}) < ∞. (2.1)

It is easy to compute the expectation of ∑_{k=1}^{ρ_{1}} f(S_{k}). Indeed, by Spitzer [S, 64], E(ξ(x, ρ_{1})) = 1 for any x ∈ Z^{2}, and hence we have

E ( ∑_{k=1}^{ρ_{1}} f(S_{k})) = E ∑_{x∈Z^{2}} ξ(x, ρ_{1})f(x) = ∑_{x∈Z^{2}} f(x) =: f̄.

Now we recall a few known results that will be of use later on.

Theorem A (Csáki and Földes [CsF, 99]). Let (X_{i}, τ_{i})_{i=1}^{∞} be a sequence of i.i.d. vectors, such that τ_{i} ≥ 0 and

P(|X_{i}| > x) < \frac{c}{x^{β}} , P(τ_{i} > x) ≤ \frac{1}{h(x)} (2.2)

for x large enough, where β > 2, c > 0 and h(x) is slowly varying at infinity, increasing with lim_{x→∞} h(x) = +∞. Then on an appropriate probability space one can construct two independent copies (X_{i}^{(1)}, τ_{i}^{(1)})_{i=1}^{∞} and (X_{i}^{(2)}, τ_{i}^{(2)})_{i=1}^{∞}, together with (X_{i}, τ_{i})_{i=1}^{∞} such that

(S_{n}, ρ_{n})_{n=1}^{∞} =^{D} (S_{n}^{(j)}, ρ_{n}^{(j)})_{n=1}^{∞},  j = 1, 2,
sup_{k≤n} |S_{k} − S_{k}^{(2)}| = O(n^{1/β^{∗}}),  a.s.,
sup_{k≤n} |ρ_{k} − ρ_{k}^{(1)}| = O (h^{∗}(n^{γ})) ,  a.s.,

as n → ∞, where S_{k}^{(j)} = ∑_{i=1}^{k} X_{i}^{(j)}, ρ_{k}^{(j)} = ∑_{i=1}^{k} τ_{i}^{(j)}, S_{k} = ∑_{i=1}^{k} X_{i}, ρ_{k} = ∑_{i=1}^{k} τ_{i}, γ < 1, β^{∗} > 2, and h^{∗}(·) is the inverse of h(·).

The next result is an application of Theorem A for simple symmetric random walk in Z^{2}.

Theorem B (Csáki and Földes [CsF, 99]). For a simple symmetric random walk {S_{n}}_{1}^{∞} in Z^{2} and any real valued function f(·) for which (2.1) holds, there exists a probability space where we can redefine {S_{n}}_{1}^{∞} together with its local time process ξ(0, n) and with the corresponding additive functional A(f, n) in such a way, that on the same probability space there exist

(i) a standard Wiener process {W (t), t ≥ 0},

(ii) and a process

{ξ^{(1)}(0, n), n = 0, 1, 2, ...} =^{D} {ξ(0, n), n = 0, 1, 2, ...}

such that {W (t), t ≥ 0} and {ξ^{(1)}(0, n), n = 0, 1, 2, ...} are independent and, as n → ∞, we have

A(f, n) − f̄ξ(0, n) = σW (ξ^{(1)}(0, n)) + O(log^{κ} n), a.s.,

and

|ξ^{(1)}(0, n) − ξ(0, n)| = O(log^{1−δ} n), a.s.,

where σ = \sqrt{Var(∑_{k=1}^{ρ_{1}} f(S_{k}))}, κ < 1/2, and δ > 0.

Concerning the moments of the local time of ξ(x, ρ_{1}) we quote the following result.

Theorem C (Révész [R, 90]). For any integer m ≥ 1,

|E (ξ(x, ρ_{1}) − 1)^{m} | ≤ q + m!p^{1−m}q,

where p := P(0 ⇝ x) = P({S_{n}} hits x before returning to 0),

and q = 1 − p.

The order of magnitude of p is well-known, namely we have

Theorem D (Spitzer [S, 64], pp. 117, 124, 125, and Révész, [R, 90] p. 219). As ∥x∥ goes to infinity,

P(0 ⇝ x) = \frac{π + o(1)}{4 log ∥x∥}.

The next theorem concerns the asymptotics of ξ(0, n).

Theorem E (Dvoretzky and Erdős [DE, 51], Erdős and Taylor [ET, 60]).

lim_{n→∞} P(ξ(0, n) < x log n) = 1 − e^{−πx}.

We also have

lim sup_{n→∞} \frac{ξ(0, n)}{log n log_{3} n} = \frac{1}{π} , a.s.

lim sup_{n→∞} \frac{sup_{x} ξ(x, n)}{(log n)^{2}} ≤ \frac{1}{π} , a.s.

Furthermore, as n goes to infinity,

P(ρ_{1} > n) = P(ξ(0, n) = 0) = \frac{π}{log n} + O((log n)^{−2}).

Now we collect certain results for the two-dimensional Bessel process {R(t), t ≥ 0} (with R(0) = 0). Let {σ_{n}, n ≥ 0} and {θ_{n}, n ≥ 1} be as in (1.5)–(1.7). It is a consequence of the strong Markov property that {σ_{i} − σ_{i−1}}_{1}^{∞} is an i.i.d. sequence of random variables, for which we quote the following result.

Theorem F (Csáki et al. [CsFRS, 98]). We have,

P(σ_{1} − σ_{0} > x) = \frac{2 log 2}{log x} + O(log^{−2} x),

as x → ∞.

We also need the following iterated logarithm law.

Theorem G (Csáki et al. [CsFRS, 98]). Let {η(t), t > 0} be as in (1.9). Then

(2.3)  lim sup_{t→∞} \frac{η(t)}{(log t) log_{3} t} = \frac{1}{2 log 2},  a.s.

Let us recall now the following theorem concerning the increments of n ↦ ξ(0, n).

Theorem H (Csáki, Földes and Révész [CsFR, 98]). Let a_{n} = exp((log n)^{K}) and b_{n} = exp((log n)^{b}) for some constants K > 0 and b > 0. Then for any ε > 0,

sup_{a≤a_{n}}(ξ(0, a + b_{n}) − ξ(0, a)) = O(log^{b+ε} n), a.s.

The next is a strong approximation theorem for the two-dimensional random walk and the Wiener process.

Theorem J (Révész [R, 90]). On a rich enough probability space one can define a Wiener process W(·) ∈ R^{2} and a simple symmetric random walk S_{n} ∈ Z^{2} such that

∥\sqrt{2} S_{n} − W(n)∥ = O(log n), a.s.

Finally, we will need the following two theorems concerning the two-dimensional Bessel process {R(t), t ≥ 0}. They are both borrowed from Bertoin and Werner [BW, 94] (see their Proposition 4 and Theorem 3, respectively).

Theorem K (Bertoin and Werner [BW, 94]). We have

lim sup_{t→∞} \frac{1}{(log t) log_{3} t} ∫_{0}^{t} 1_{\{R(s)<1\}}ds = \frac{1}{2}, a.s.

Theorem L (Bertoin and Werner [BW, 94]). Let Z_{2}^{∗} be as in (1.2). For any ε > 0, when t → ∞,

Z_{2}^{∗}(t) = o ((log t)^{2}(log_{2} t)^{2+ε}) , a.s.

3. Proof of Theorem 1.1

We begin this section with a few lemmas and facts which will be needed in the proof. Fact 1. For x ≥ 1, y ≥ 1 |x^{α} − y^{α}| ≤ α|x − y|(x^{α−1} + y^{α−1}) if α > 1 (3.1) and |x^{α} − y^{α}| ≤ |x − y| if 0 ≤ α ≤ 1. (3.2)

Fact 1 is a simple consequence of the mean value theorem. Fact 2. For x ≥ 1, y ≥ 1, and α ≥ 0 we have

\frac{1}{xy^{α}} + \frac{1}{yx^{α}} ≤ 2 (\frac{1}{x^{α+1}} + \frac{1}{y^{α+1}} ) . (3.3)

Fact 2 can be checked by elementary computations.

Recall that {R(t), t ≥ 0} is a two-dimensional Bessel process starting from 0, and {ℓ(x, t), x ≥ 0, t ≥ 0} is its local time. First we give a rough estimate for Z_{α}(t) for large t. Lemma 3.1. For 0 ≤ α < 2, and any ε > 0, as t → ∞,

Z_{α}(t) := ∫_{0}^{t} \frac{ds}{R^{α}(s)} = O (t^{1−α/2+ε}) , a.s.

Proof: By the scaling property, for any fixed t > 0,

\{R(ut), u ≥ 0\} =^{D} \{\sqrt{t} R(u), u ≥ 0\},

which implies that

Z_{α}(t) =^{D} t^{1−α/2} ∫_{0}^{1} \frac{du}{R^{α}(u)}. (3.4)

Observe furthermore that R^{2}(u)/u has the exponential distribution with mean 2, thus we have

E (∫_{0}^{1} \frac{du}{R^{α}(u)}) = c_{1} ∫_{0}^{1} \frac{du}{u^{α/2}} = c_{2},

where c_{1} and c_{2} are finite constants. Hence by Markov’s inequality, we get for any A > 0 that

P (Z_{α}(t) > t^{1−α/2} A) ≤ \frac{c_{2}}{A} . (3.5)

For any given ε > 0, let β > 1/ε and t_{k} = k^{β}. From (3.5) we get that

P (Z_{α}(t_{k+1}) > t_{k}^{1−α/2+ε}) ≤ \frac{2c_{2}}{k^{βε}} ,

for k large enough, as t_{k+1}/t_{k} → 1. Using the Borel–Cantelli lemma and the monotonicity of t ↦ Z_{α}(t), we obtain that

Z_{α}(t) = O (t^{1−α/2+ε}) , a.s.,

completing the proof of Lemma 3.1. □

Lemma 3.1 implies

Lemma 3.2. For 0 ≤ α < 2, and any ε > 0

Z_{α}^{∗}(t) := ∫_{0}^{t} \frac{1_{\{R(s)≥1\}}}{R^{α}(s)} ds = O (t^{1−α/2+ε}) , a.s.

Lemma 3.3. For any b > 2, and any 0 ≤ α < 2, as t → ∞,

∫_{0}^{t} \frac{1_{\{R(s)<1\}}}{R^{α}(s)} ds = O(log^{b} t), a.s.

Proof: Using the notation introduced in (1.5)–(1.9) we can decompose the local time ℓ(x, t) of R(t) as follows: for any δ > 0 and all large t,

ℓ(x, t) = ℓ(x, σ_{0}) + ∑_{i=1}^{η(t)}(ℓ(x, σ_{i}) − ℓ(x, σ_{i−1})) + ℓ(x, t) − ℓ(x, σ_{η(t)})

≤ ℓ(x, σ_{0}) + ∑_{i=1}^{η(t)+1}(ℓ(x, σ_{i}) − ℓ(x, σ_{i−1}))

(3.6)  ≤ ℓ(x, σ_{0}) + ∑_{i=1}^{log^{1+δ} t}(ℓ(x, σ_{i}) − ℓ(x, σ_{i−1})),

the last inequality following from Theorem G. Since b > 2, we can choose δ > 0 such that 2 + δ < b.

By writing ∫_{0}^{t} \frac{1_{\{R(s)<1\}}}{R^{α}(s)} ds = ∫_{0}^{1} \frac{ℓ(x,t)}{x^{α}} dx, and in view of (3.6) (noting that ∫_{0}^{1} \frac{ℓ(x,σ_{0})}{x^{α}} dx = Z_{α}^{∗}(σ_{0}) < ∞ almost surely), it only remains to check that

(3.7) Q(t) := ∑_{i=1}^{log^{1+δ} t} ∫_{0}^{1} \frac{ℓ(x, σ_{i}) − ℓ(x, σ_{i−1})}{x^{α}} dx = O(log^{b} t), a.s.

To this end, we look at the Laplace transforms of ℓ(x, θ_{1}) − ℓ(x, σ_{0}) and ℓ(x, σ_{1}) − ℓ(x, θ_{1}) which can be found in Borodin and Salminen ([BS, 96] p. 297, Formula 2.3.1): for any γ > 0,

E (e^{−γ(ℓ(x,θ_{1})−ℓ(x,σ_{0}))}) = \{ \begin{matrix}\frac{1+2γx| log x|}{1+2γx| log(x/2)|} & if 0 < x ≤ 1 \\ \frac{1}{1+2γx| log(x/2)|} & if 1 ≤ x ≤ 2 \\ 1 & if x ≥ 2 \end{matrix} ,

E (e^{−γ(ℓ(x,σ_{1})−ℓ(x,θ_{1}))}) = \{ \begin{matrix}1 & if 0 < x ≤ 1 \\ \frac{1}{1+2γx| log x|} & if 1 ≤ x ≤ 2 \\ \frac{1+2γx| log(x/2)|}{1+2γx| log x|} & if x ≥ 2 \end{matrix} .

Since ℓ(x, θ_{1}) − ℓ(x, σ_{0}) and ℓ(x, σ_{1}) − ℓ(x, θ_{1}) are independent, we get the Laplace transform of ℓ(x, σ_{1}) − ℓ(x, σ_{0}):

(3.8) E (e^{−γ(ℓ(x,σ_{1})−ℓ(x,σ_{0}))}) = \{ \begin{matrix}\frac{1+2γx| log x|}{1+2γx| log(x/2)|} & if 0 < x ≤ 1 \\ \frac{1}{(1+2γx| log x|)(1+2γx| log(x/2)|)} & if 1 ≤ x ≤ 2 \\ \frac{1+2γx| log(x/2)|}{1+2γx| log x|} & if x ≥ 2. \end{matrix} .

Hence for 0 < x ≤ 1,

E (ℓ(x, σ_{i}) − ℓ(x, σ_{i−1})) = E (ℓ(x, σ_{1}) − ℓ(x, σ_{0})) = 2x log 2.

As a consequence, when t goes to ∞,

(3.9) E(Q(t)) ≤ (log^{1+δ} t) ∫_{0}^{1} \frac{x log 2}{x^{α}} dx = O(log^{1+δ} t)

for all 0 ≤ α < 2. Based on (3.9), we can finish the proof in the same way as that of Lemma 3.1, using the monotonicity of t ↦ Q(t), the subsequence t_{k} = e^{k} and the Borel–Cantelli lemma. Indeed, by (3.9) and Markov’s inequality,

P (Q(t_{k+1}) > (log t_{k})^{b}) ≤ c_{3} \frac{(log t_{k+1})^{1+δ}}{(log t_{k})^{b}} ≤ c_{4} \frac{(k + 1)^{1+δ}}{k^{b}} ,

which sums for k (recalling that b > 2 + δ). This yields (3.7), and completes the proof of Lemma 3.3. □

Lemma 3.4. Let α ≥ 0. As n → ∞, we have almost surely for any ε > 0,

U_{2α}(n) := ∑_{k=1}^{n} \frac{1_{\{∥S_{k}∥≠0\}}}{∥S_{k}∥^{2α}} = \{ \begin{matrix}O (n^{1−α+ε}) & if α < 1 \\ O(log^{3+ε} n) & if α = 1 \\ O(log^{2} n) & if α > 1. \end{matrix}

Proof: Let M_{n} = max_{k≤n} ∥S_{k}∥. Then by the LIL, (3.10) M_{n} = O ((n log log n)^{1/2}) , a.s. By Theorem E,

(3.11) ∑_{k=1}^{n} \frac{1_{\{∥S_{k}∥≠0\}}}{∥S_{k}∥^{2α}} = ∑_{x∈Z^{2}−\{0\}} \frac{ξ(x, n)}{∥x∥^{2α}} = O(log^{2} n) ∑_{x∈Z^{2}−\{0\},∥x∥≤M_{n}} \frac{1}{∥x∥^{2α}} .

Due to the following equiconvergence relation

∑_{x∈Z^{2}−\{0\},∥x∥≤N} \frac{1}{∥x∥^{2α}} ∼ ∫_{1}^{N} r^{1−2α} dr, (3.12)

and the fact that

(3.13) ∫_{1}^{N} r^{1−2α} dr = \{ \begin{matrix}O (N^{2−2α}) & if α < 1 \\ O(log N) & if α = 1 \\ O(1) & if α > 1. \end{matrix}

we have our statement combining (3.10)–(3.13). □ Proof of Theorem 1.1: In view of Lemma 3.3, we only have to check that almost surely,

(3.14) |2^{−α/2}U_{α}(n) − Z_{α}^{∗}(n)| = \{ \begin{matrix}O(n^{1−α+ε}), & if 0 < α < 1, \\ O(log^{4+ε} n), & if α = 1, \\ O(log^{2+ε} n), & if 1 < α < 2. \end{matrix}

We start with the following decomposition:

|2^{−α/2}U_{α}(n) − Z_{α}^{∗}(n)| = |∑_{k=1}^{n} ∫_{k−1}^{k} (\frac{1_{\{S_{k}≠0\}}}{∥S_{k}^{∗}∥^{α}} − \frac{1_{\{R(s)≥1\}}}{R^{α}(s)} ) ds|
≤ |∑_{k=1}^{n} ∫_{k−1}^{k} \frac{R^{α}(s) − ∥S_{k}^{∗}∥^{α}}{∥S_{k}^{∗}∥^{α}R^{α}(s)} 1_{\{S_{k}≠0,R(s)≥1\}} ds|
+ |∑_{k=1}^{n} ∫_{k−1}^{k} \frac{1_{\{S_{k}=0,R(s)≥1\}}}{R^{α}(s)} ds| + |∑_{k=1}^{n} ∫_{k−1}^{k} \frac{1_{\{S_{k}≠0,R(s)<1\}}}{∥S_{k}^{∗}∥^{α}} ds|
(3.15)  = I_{1} + I_{2} + I_{3},

where S_{k}^{∗} = \sqrt{2} S_{k}.

First observe that

I_{2} ≤ ∑_{k=1}^{n} 1_{{S_{k}=0}} = ξ(0, n) = O(log^{1+ε} n), a.s. (3.16)

for any ε > 0, by Theorem E. Similarly, by Theorem K, we have that for any ε > 0

I_{3} ≤ ∫_{0}^{n} 1_{{R(s)<1}} ds = O(log^{1+ε} n), a.s. (3.17)

The estimation of I_{1} is more delicate and we have to consider two cases. First we consider the case 0 < α ≤ 1. According to (3.2) we have

I_{1} ≤ ∑_{k=1}^{n} ∫_{k−1}^{k} \frac{|R^{α}(s) − ∥S_{k}^{∗}∥^{α}|}{∥S_{k}^{∗}∥^{α}R^{α}(s)} 1_{\{S_{k}≠0,R(s)≥1\}} ds
≤ ∑_{k=1}^{n} ∫_{k−1}^{k} \frac{|R(s) − ∥S_{k}^{∗}∥ |}{∥S_{k}^{∗}∥^{α}R^{α}(s)}1_{\{S_{k}≠0,R(s)≥1\}} ds
≤ O(log n) ∑_{k=1}^{n} ∫_{k−1}^{k} \frac{1_{\{S_{k}≠0,R(s)≥1\}}}{∥S_{k}^{∗}∥^{α}R^{α}(s)} ds,

where in the last line we used Theorem J. Since xy ≤ (x^{2} + y^{2})/2, this yields

I_{1} ≤ O(log n) ∑_{k=1}^{n} ∫_{k−1}^{k} 1_{\{S_{k}≠0,R(s)≥1\}} (\frac{1}{∥S_{k}^{∗}∥^{2α}} + \frac{1}{R^{2α}(s)}) ds
≤ O(log n) ( ∑_{k=1}^{n} \frac{1_{\{S_{k}≠0\}}}{∥S_{k}^{∗}∥^{2α}} + ∫_{0}^{n} \frac{1_{\{R(s)≥1\}}}{R^{2α}(s)} ds)
(3.18)  ≤ O(log n) (U_{2α}(n) + Z_{2α}^{∗}(n)) .

Applying Lemmas 3.2 and 3.4 yields that for 0 < α < 1,

I_{1} = O (n^{1−α+ε}) , a.s.

For α = 1, we can use (3.18), Lemma 3.4 and Theorem L to see that

I_{1} = O (log^{4+ε} n) , a.s.

These estimates, together with (3.15)–(3.17), yield (3.14) when 0 < α ≤ 1.

Consider now the case 1 < α < 2. We estimate I_{1} differently. By (3.1) and Theorem J,

I_{1} ≤ ∑_{k=1}^{n} ∫_{k−1}^{k} \frac{|R^{α}(s) − ∥S_{k}^{∗}∥^{α}|}{∥S_{k}^{∗}∥^{α}R^{α}(s)}1_{\{S_{k}≠0,R(s)≥1\}} ds
≤ ∑_{k=1}^{n} ∫_{k−1}^{k} \frac{2|R(s) − ∥S_{k}^{∗}∥ | (R^{α−1}(s) + ∥S_{k}^{∗}∥^{α−1})}{∥S_{k}^{∗}∥^{α}R^{α}(s)}1_{\{S_{k}≠0,R(s)≥1\}} ds
≤ O(log n) ∑_{k=1}^{n} ∫_{k−1}^{k} (\frac{1}{∥S_{k}^{∗}∥^{α}R(s)} + \frac{1}{∥S_{k}^{∗}∥R^{α}(s)}) 1_{\{S_{k}≠0,R(s)≥1\}} ds.

Using (3.3) gives

I_{1} ≤ O(log n) ∑_{k=1}^{n} \frac{1}{∥S_{k}^{∗}∥^{α+1}} 1_{\{S_{k}≠0\}} + O(log n) ∫_{0}^{n} \frac{1}{R^{α+1}(s)}1_{\{R(s)≥1\}} ds
≤ O(log n) (U_{α+1}(n) + Z_{α+1}^{∗}(n)) .

When α > 1, we have α + 1 > 2, so that we can use Theorems 1.2 and 1.3 (or rather their consequences as stated in (6.4) and (6.5) in Section 6) to see that U_{α+1}(n) = O(log^{1+ε} n) and Z_{α+1}^{∗}(n) = O(log^{1+ε} n) almost surely. This yields that for 1 < α < 2,

I_{1} = O (log^{2+ε} n) , a.s.

Combining this with (3.15)–(3.17) yields (3.14) when 1 < α < 2. Consequently, Theorem 1.1 is now proved. □

4. Proof of Theorem 1.2

In view of Theorem B in Section 2, to prove Theorem 1.2, it suffices to show that, when α > 2, the function x ↦ f(x) = 1_{{x≠0}}/∥x∥^{α} satisfies (2.1) with an appropriately chosen δ > 0.

Write ∥Y ∥_{p} := (E(|Y |^{p}))^{1/p} for any random variable Y and p ≥ 1. Fix an integer m ≥ 1. It follows from Theorems C and D that for any x ∈ Z^{2} − {0},

∥ξ(x, ρ_{1})∥_{m} ≤ 1 + ∥ξ(x, ρ_{1}) − 1∥_{m}
≤ 1 + C_{m}p^{(1−m)/m}
≤ C_{m} (log ∥x∥ + 1)^{(m−1)/m}
(4.1)  ≤ C_{m} (log ∥x∥ + 1) ,

where C_{m} denotes a (finite and positive) constant depending only on m, whose value varies from line to line.

Let f(x) = 1_{{x≠0}}/∥x∥^{α}. To check (2.1), observe that

∑_{k=1}^{ρ_{1}} f(S_{k}) = ∑_{x∈Z^{2}−\{0\}} \frac{ξ(x, ρ_{1})}{∥x∥^{α}}.

Hence for any integer m ≥ 1, by the triangle inequality and (4.1),

∥ ∑_{k=1}^{ρ_{1}} f(S_{k})∥_{m} ≤ ∑_{x∈Z^{2}−\{0\}} \frac{∥ξ(x, ρ_{1})∥_{m}}{∥x∥^{α}} ≤ C_{m} ∑_{x∈Z^{2}−\{0\}} \frac{1 + log ∥x∥}{∥x∥^{α}},

which is finite whenever α > 2. Thus any moments of order m ≥ 1 of ∑_{k=1}^{ρ_{1}} f(S_{k}) exist, proving Theorem 1.2. □

5. Proof of Theorem 1.3

The proof of this theorem is based on Theorem A. Denote by ℓ(x, t) the local time (at time t and position x) of the Bessel process {R(t), t ≥ 0}. Let {σ_{n}, n ≥ 0} and {θ_{n}, n ≥ 1} be as in (1.5)–(1.7). Both {θ_{i} −σ_{i−1}}_{1}^{∞} and {σ_{i} −θ_{i}}_{1}^{∞} are i.i.d. sequences. The Laplace transform of ℓ(x, σ_{1}) − ℓ(x, σ_{0}) is given in (3.8), from which we can obtain: for x ≥ 1,

(5.1)  E(ℓ(x, σ_{1}) − ℓ(x, σ_{0})) = 2x log x,
(5.2)  E ((ℓ(x, σ_{1}) − ℓ(x, σ_{0}))^{3}) ≤ Cx^{3}(log x)^{2},

where C > 0 is a constant.

Define, for i ≥ 1,

(5.3)  τ_{i} := σ_{i} − σ_{i−1},
(5.4)  X_{i} := ∫_{σ_{i−1}}^{σ_{i}} \frac{1}{R^{α}(s)}1_{\{R(s)≥1\}} ds.

We show that the sequence (X_{i}, τ_{i})_{i=1}^{∞} of i.i.d. vectors satisfies the conditions of Theorem A. Indeed, by (5.2),

∥X_{i}∥_{3} = (E_{1} (∫_{σ_{0}}^{σ_{1}} \frac{1}{R^{α}(s)}1_{\{R(s)≥1\}} ds)^{3})^{1/3}

= (E (∫_{1}^{∞} \frac{ℓ(x, τ_{1})}{x^{α}} dx)^{3})^{1/3}
≤ ∫_{1}^{∞} \frac{∥ℓ(x, τ_{1})∥_{3}}{x^{α}} dx
≤ C ∫_{1}^{∞} \frac{(x^{3}(log x)^{2})^{1/3}}{x^{α}} dx,

which is finite, whenever α > 2. Using Chebyshev’s inequality, this leads to

P (|X_{i}| > x) ≤ \frac{c}{x^{3}} , (5.5)

for some constant c > 0 and all x > 0. Thus, the first condition of (2.2) is satisfied by X_{i} with β = 3. It is clear that the second condition in (2.2) is satisfied by τ_{i} with h(x) = C log x, since by Theorem F, P(τ_{i} > x) ≤ C/ log x.

Consequently, the sequence of vectors (X_{i}, τ_{i})_{i=1}^{∞} defined in (5.3)–(5.4) satisfies the conditions of Theorem A. Let

S_{i} := ∫_{σ_{0}}^{σ_{i}} \frac{1}{R^{α}(s)}1_{\{R(s)≥1\}} ds = Z_{α}^{∗}(σ_{i}),  i ≥ 1,
ρ_{n} := σ_{n} − σ_{0},  n ≥ 0.

According to Theorem A, on an appropriate probability space one can construct two independent copies, say (X_{i}^{(1)}, τ_{i}^{(1)})_{i=1}^{∞} and (X_{i}^{(2)}, τ_{i}^{(2)})_{i=1}^{∞}, of (X_{i}, τ_{i})_{i=1}^{∞}, such that

(5.6)  (S_{n}, ρ_{n})_{n=1}^{∞} =^{D} (S_{n}^{(1)}, ρ_{n}^{(j)})_{n=1}^{∞},  j = 1, 2,
(5.7)  sup_{k≤n} |S_{k} − S_{k}^{(2)}| = O (n^{1/β^{∗}}) ,  a.s.,
(5.8)  sup_{k≤n} |ρ_{k} − ρ_{k}^{(1)}| = O (e^{cn^{γ}} ) ,  a.s.

where S_{k}^{(j)} = ∑_{i=1}^{k} X_{i}^{(j)}, ρ_{k}^{(j)} = ∑_{i=1}^{k} τ_{i}^{(j)}, γ < 1 and 2 < β^{∗} < 3,

Apply now the Komlós–Major–Tusnády theorem [KMT, 75] (see also Csörgő and Révész [CsR, 81], p. 108, Theorem 2.6.6) to get that, there exists a Wiener process {W (t), t ≥ 0} such that for any 2 < β^{∗} < 3, when N goes to infinity,

S_{N}^{(2)} − NE(X_{1}) − σ_{α}^{W} W (N) = O(N^{1/β^{∗}}), a.s., (5.9)

where σ_{α}^{W} = \sqrt{Var(X_{1})}. Since W is constructed from \{S_{n}^{(2)}, n ≥ 1\}, it can be chosen to be independent of (X_{i}^{(1)}, τ_{i}^{(1)})_{i=1}^{∞}.

By (5.1),

E(X_{1}) = E ∫_{1}^{∞} \frac{ℓ(x, σ_{1} − σ_{0})}{x^{α}} dx = ∫_{1}^{∞} \frac{2 log x}{x^{α−1}} dx = \frac{2}{(α − 2)^{2}} =: f̄_{α}^{W} .

It follows from (5.9) and (5.7) that

S_{N} − Nf̄_{α}^{W} − σ_{α}^{W} W (N) = O(N^{1/β^{∗}}), a.s.

Now let η(t) be as in (1.9). We get

S_{η(t)} − f̄_{α}^{W} η(t) − σ_{α}^{W} W (η(t)) = O ((η(t))^{1/β^{∗}}) , a.s.,

which, in view of (2.3), yields that, for any small enough ε > 0,

S_{η(t)} − f̄_{α}^{W} η(t) − σ_{α}^{W} W (η(t)) = O(log^{1/2−ε} t), a.s. (5.10)

Now we want to get an almost sure upper bound for |Z_{α}^{∗}(t)−S_{η(t)}|. Since η(t) < t ≤ η(t)+1, we have

|Z_{α}^{∗}(t) − S_{η(t)}| ≤ X_{η(t)+1}.

On the other hand, a routine Borel–Cantelli argument, using (5.5), yields that for a small enough δ > 0, X_{k} = O(k^{1/2−δ}) almost surely (when k goes to infinity). Therefore, by (2.3), X_{η(t)+1} = O(log^{1/2−δ^{∗}} t), for any 0 < δ^{∗} < δ. Accordingly,

Z_{α}^{∗}(t) − S_{η(t)} = O(log^{1/2−δ^{∗}} t), a.s.

Going back to (5.10), we arrive at

Z_{α}^{∗}(t) − f̄_{α}^{W} η(t) = σ_{α}^{W} W (η(t)) + O(log^{1/2−γ^{∗}} t), a.s., (5.11)

with a small enough γ^{∗} > 0.

What remains to show is that on the right hand side of (5.11) we can replace W (η(t)) by W (η^{(1)}(t)), at the price of a possible small increase in the order of the error term in the approximation. For this purpose we need two preliminary estimates. The first one (Lemma 5.1) is analogous to Theorem H. Its proof, which goes along the same lines as that of Theorem H, is omitted.

Lemma 5.1. Let a_{t} = exp((log t)^{K}) and b_{t} = exp((log t)^{b}), for K > 0 and b > 0. Then for any ε > 0,

sup_{a≤a_{t}}(η(a + b_{t}) − η(a)) = O(log^{b+ε} t), a.s.

Lemma 5.2. There exists ν < 1 such that, as t → ∞,

|η^{(1)}(t) − η(t)| = O(log^{ν} t), a.s.

Proof: Since η(t) = η^{(1)}(ρ_{η(t)}^{(1)}), η^{(1)}(t) = η(ρ_{η^{(1)}(t)}), t ≥ ρ_{η(t)} and t ≥ ρ_{η^{(1)}(t)}^{(1)}, we have

η^{(1)}(ρ_{η(t)}) − η^{(1)}(ρ_{η(t)}^{(1)}) ≤ η^{(1)}(t) − η(t) ≤ η(ρ_{η^{(1)}(t)}) − η(ρ_{η^{(1)}(t)}^{(1)}). (5.12)

By (2.3), for any ε > 0 and all large t, η(t) < (log t)^{1+ε}. It follows from (5.8) that for some γ < 1 and all large t,

|ρ_{η(t)} − ρ_{η(t)}^{(1)}| ≤ sup_{i≤(log t)^{1+ε}} |ρ_{i} − ρ_{i}^{(1)}| ≤ exp (c (log t)^{(1+ε)γ}) , a.s.

Now apply Lemma 5.1 to b_{t} = exp(c (log t)^{(1+ε)γ}) and a_{t} = t (noting that ρ_{η^{(1)}(t)}^{(1)} ≤ t) to see that

η^{(1)}(ρ_{η(t)}) − η^{(1)}(ρ_{η(t)}^{(1)}) = O(log^{(1+2ε)γ} t), a.s.

A similar argument shows that

η(ρ_{η^{(1)}(t)}) − η(ρ_{η^{(1)}(t)}^{(1)}) = O(log^{(1+2ε)γ} t), a.s.

In view of (5.12), we have

η^{(1)}(t) − η(t) = O(log^{(1+2ε)γ} t), a.s.

Choosing ε > 0 so small that (1 + 2ε)γ < 1, gives the lemma. □

We now go back to the proof of Theorem 1.3. Note that Lemma 5.2 yields (1.11). To check (1.10), we can control the the increments of W by using Theorem 1.2.1 of Csörgő and Révész [CsR, 81]: for any ε > 0 and any ν > 0,

sup_{0≤u,v≤(log t)^{1+ε}, |u−v|≤(log t)^{ν}} |W (u) − W (v)| = O ((log t)^{ν/2}(log log t)^{1/2}) , a.s.

Since η(t) = O(log^{1+ε} t) (cf. Theorem G), it follows from Lemma 5.2 and (5.11) that for any 1 > ν^{∗} > ν,

Z_{α}^{∗}(t) − f̄_{α}^{W} η(t) = σ_{α}^{W} W (η^{(1)}(t)) + O(log^{ν^{∗}/2} t) + O(log^{1/2−γ^{∗}} t), a.s.

Take

κ := max (\frac{1}{2} − γ^{∗}, \frac{ν^{∗}}{2} ) < \frac{1}{2}

to get

Z_{α}(t) − f̄_{α}^{W} η(t) = σ_{α}^{W} W (η^{(1)}(t)) + O(log^{κ} t), a.s.,

as t → ∞. Since W was chosen to be independent of (X_{i}^{(1)}, τ_{i}^{(1)})_{i=1}^{∞}, it is a fortiori independent of {η^{(1)}(t), t > 0}. This completes the proof of Theorem 1.2. □

6. Further results

Our theorems can be used to deduce weak and strong laws for the additive functionals studied in this paper. Considering Z_{α}(t), defined by (1.1), by scaling we have (cf. (3.4)), for any t > 0,

Z_{α}(t) =^{D} t^{1−α/2}Z_{α}(1). (6.1)

Hence, Theorem 1.1 implies for 0 < α < 2,

2^{−α/2}n^{α/2−1}U_{α}(n) →^{D} Z_{α}(1), n → ∞,

where →^{D} denotes convergence in distribution. It follows also that the LIL for the process Z_{α}(·) is inherited by U_{α}(·).

Now we study limsup and liminf properties of Z_{α}. It is obvious from (6.1) that first we have to study the upper and lower tail behaviour of Z_{α}(1). A direct approach however does not seem available, instead, we apply Corollary XI.1.12 of Revuz and Yor [RY, 99] telling us that

Z_{α}(1) := ∫_{0}^{1} \frac{ds}{R^{α}(s)} =^{D} (\frac{2}{2 − α})^{α} (∫_{0}^{1} R^{β}(t) dt)^{−(1−α/2)} , (6.2)

where β = 2α/(2 − α). For β ≥ 1, Azencott [A, 80] shows that there exists a finite constant c(β) such that

log P (∫_{0}^{1} R^{β}(t) dt > u) ∼ −c(β)u^{2/β}, u → ∞,

which combined with (6.2) yields

log P(Z_{α}(1) < x) ∼ −c_{1}(α)x^{−2/α}, x → 0,

where

c_{1}(α) = \frac{4c(β)}{(2 − α)^{2}} .

Now the usual argument gives the following liminf result for 2/3 ≤ α < 2:

lim inf_{t→∞} t^{−(1−α/2)}(log log t)^{α/2}Z_{α}(t) = K(α), a.s.,

where

K(α) = (c_{1}(α))^{α/2}.

Hence, by Theorem 1.1, we have also for 2/3 ≤ α < 2

lim inf_{n→∞} n^{−(1−α/2)}(log log n)^{α/2}2^{−α/2}U_{α}(n) = K(α), a.s.

Concerning the limsup results, we can use the following small deviation theorem of Borovkov and Mogulskii [BM, 91]: for β ≥ 1

log P (∫_{0}^{1} R^{β}(t) dt < u) ∼ −c̃(β)u^{−2/β}, u → 0, (6.3)

where c̃(β) is a finite positive constant. (We note that in [BM, 91] the analogue result is proved for one-dimensional Wiener process, but as they remark, their result holds also true for higher dimensional case.)

Based on (6.3) and (6.2), similarly to the liminf case, one can easily obtain

lim sup_{t→∞} \frac{Z_{α}(t)}{t^{1−α/2}(log log t)^{α/2}} = K̃(α), a.s.,

with

K̃(α) = (c̃(β))^{−α/2} \frac{2^{α}}{(2 − α)^{α}}

and consequently

lim sup_{n→∞} \frac{2^{−α/2}U_{α}(n)}{n^{1−α/2}(log log n)^{α/2}} = K̃(α), a.s.

Turning now to the case α > 2, we first consider some limiting distribution results. Let E be a random variable with density function e^{−|x|}/2, x ∈ R. Then E is a bilateral exponential random variable and |E| has the exponential distribution of parameter 1, i.e., it has the density function e^{−x}, x ∈ [0, ∞). By Theorem E,

\frac{πξ(0, n)}{log n} →^{D} |E|, n → ∞,

and similarly, from Theorem F one can see that

\frac{(2 log 2)η(t)}{log t} →^{D} |E|, t → ∞.

Moreover, as easily seen,

\frac{\sqrt{2π} W (ξ^{(1)}(0, n))}{\sqrt{log n}} →^{D} E, n → ∞

and

\frac{2\sqrt{log 2} W (η^{(1)}(t))}{\sqrt{log t}} →^{D} E, t → ∞,

where W (ξ^{(1)}(0, ·)) and W (η^{(1)}(·)), resp. are the processes in Theorems 1.2 and 1.3, resp. Now our Theorems 1.2 and 1.3 imply the following weak convergence results:

\frac{πU_{α}(n)}{f̄_{α}^{D} log n} →^{D} |E|, n → ∞,
\frac{U_{α}(n) − f̄_{α}^{D}ξ(0, n)}{σ_{α}^{D}\sqrt{log n}}\sqrt{2π} →^{D} E, n → ∞,
\frac{(2 log 2)Z_{α}^{∗}(t)}{f̄_{α}^{W} log t} →^{D} |E|, t → ∞,
\frac{Z_{α}^{∗}(t) − f̄_{α}^{W} η(t)}{σ_{α}^{W} \sqrt{log t}}2\sqrt{log 2} →^{D} E, t → ∞.

Concerning strong limit results, our Theorems 1.2 and 1.3 combined with Theorems E and G yield

(6.4)  lim sup_{n→∞} \frac{U_{α}(n)}{log n log_{3} n} = \frac{f̄_{α}^{D}}{π} ,  a.s.,

and

(6.5)  lim sup_{t→∞} \frac{Z_{α}^{∗}(t)}{log t log_{3} t} = \frac{f̄_{α}^{W}}{2 log 2},  a.s.

Moreover, the following LIL is true (cf. [MR, 94], [CsFR, 98], [CsF, 99])

lim sup_{n→∞} \frac{W (ξ^{(1)}(0, n))}{\sqrt{log n} log_{3} n} = \frac{1}{\sqrt{2π}} , a.s.

Similarly, one can see that

lim sup_{t→∞} \frac{W (η^{(1)}(t))}{\sqrt{log t} log_{3} t} = \frac{1}{2\sqrt{log 2}}, a.s.

Consequently, we have the following LIL’s:

lim sup_{n→∞} \frac{U_{α}(n) − f̄_{α}^{D}ξ(0, n)}{\sqrt{log n} log_{3} n} = \frac{σ_{α}^{D}}{\sqrt{2π}} , a.s.

and

lim sup_{t→∞} \frac{Z_{α}^{∗}(t) − f̄_{α}^{W} η(t)}{\sqrt{log t} log_{3} t} = \frac{σ_{α}^{W}}{2\sqrt{log 2}}, a.s.

Acknowledgements

Cooperation between the authors was supported by the joint French-Hungarian Intergovernmental Grant “Balaton” (grant no. F25/97).

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