PATH PROPERTIES OF CAUCHY’S PRINCIPAL VALUES

RELATED TO LOCAL TIME

Endre Csáki^{1}^{,}^{2} 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^{3} School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada K1S 5B6. E-mail address: mcsorgo@math.carleton.ca

Antónia Földes^{2,}^{4} College of Staten Island, CUNY, 2800 Victory Blvd., Staten Island, New York 10314, U.S.A. E-mail address: afoldes@email.gc.cuny.edu

Zhan Shi^{2} Laboratoire de Probabilités, Université Paris VI, 4 Place Jussieu, F-75252 Paris Cedex 05, France. E-mail address: zhan@proba.jussieu.fr

Abstract: Sample path properties of the Cauchy principal values of Brownian and random walk local times are studied. We establish LIL type results (without exact constants). Large and small increments are discussed. A strong approximation result between the above two processes is also proved. AMS 1991 Subject Classification: Primary 60J15; Secondary 60F15, 60J55. Keywords: Additive functionals, principal values of Brownian and random walk local times, law of the iterated logarithm, large increments, modulus of continuity. Short title: Principal values of local times.

1Research supported by the Hungarian National Foundation for Scientific Research, Grant No. T 019346 and T 029621.

2Research supported by a Fellowship of the Paul Erdős Summer Research Center of Mathematics, Budapest.

3Research 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.

4Research supported by a PSC CUNY Grant, No. 6-67383.

1. Introduction and main results

Recently a collection of papers appeared in Yor (1997) that deal with principal values related to Brownian motion (cf. Part B of Yor 1997).

Let W (·) be a standard Wiener process (Brownian motion) with local time L(x, t), i.e., for any Borel function f ≥ 0 and t ≥ 0,

∫_{0}^{t} f(W (s)) ds = ∫_{R} f(x)L(x, t) dx.

Define the integral ∫_{0}^{t} ds/W^{α}(s) (notation: z^{α} = |z|^{α} sgn(z)) in the sense of Cauchy’s principal value:

Y_{α}(t) =^{def} ∫_{0}^{t} \frac{ds}{W^{α}(s)} = ∫_{0}^{∞} \frac{L(x, t) − L(−x, t)}{x^{α}} dx. (1.1)

It is well-known (McKean 1962, Borodin 1985) that the local time in the space variable is Hölder continuous of order (1/2 − ε) for any ε > 0, which ensures the finiteness (for each t ≥ 0) of the second integral in (1.1) whenever α < 3/2. We shall from now on assume this condition. Strictly speaking, the first integral is defined as Cauchy’s principal value for 1 ≤ α < 3/2 (cf., e.g., Sections 1.1-1.3 in Alili 1997 and references therein), and as Riemann integral for α < 1.

The aim of our paper is to study the almost sure (a.s.) path behaviour of the process Y_{α}(·). For a significant first step along these lines we refer to the paper by Hu and Shi (1997), who proved the local, as well as the global, laws of the iterated logarithm (LIL) for Y_{1}(·). Namely, they established the following results:

(1.2)  lim sup_{t→∞} \frac{Y_{1}(t)}{\sqrt{t log log t}} = 2\sqrt{2} , a.s.

and

(1.3)  lim sup_{h→0} \frac{Y_{1}(h)}{\sqrt{h log log(1/h)}} = 2\sqrt{2} , a.s.

We are interested in studying the modulus of continuity and large increment properties (including the LIL) of Y_{α}(·), as well as appropriate properties of a simple symmetric random walk along these lines. Due however to lack of precise distributional properties of Y_{α}(·), when α ≠ 1, we could not obtain the desirable exact constants, though the rates we establish here are optimal.

We present now our main results. First we prove the upper bounds for the LIL, large increments and modulus of continuity. Concerning (1.4) of Theorem 1.1, we assume that a_{T} is a non-decreasing function of T, such that 0 < a_{T} ≤ T and a_{T} /T is non-increasing.

Theorem 1.1. For 0 < α < 3/2 we have

(1.4)  lim sup_{T →∞} \frac{sup_{0≤t≤T −a_{T}} sup_{0≤s≤a_{T}} |Y_{α}(t + s) − Y_{α}(t)|}{a_{T}^{1−α/2} (log(T/a_{T}) + log log T)^{α/2}} ≤ c_{1}(α),  a.s.
(1.5)  lim sup_{h→0} \frac{|Y_{α}(h)|}{h^{1−α/2}(log log(1/h))^{α/2}} ≤ c_{1}(α),  a.s.
(1.6)  lim sup_{h→0} \frac{sup_{0≤t≤1−h} sup_{0≤s≤h} |Y_{α}(t + s) − Y_{α}(t)|}{h^{1−α/2} (log(1/h))^{α/2}} ≤ c_{1}(α),  a.s.

Here, the constant c_{1}(α) is given by

c_{1}(α) = \frac{3 · 2^{7α/6}}{α^{2α/3}(3 − 2α)^{1−α/3}(2 − α)^{α/3}} . (1.7)

Remark: In the particular case a_{T} = T we get

(1.8)  lim sup_{t→∞} \frac{|Y_{α}(t)|}{t^{1−α/2}(log log t)^{α/2}} ≤ c_{1}(α),  a.s.

Concerning the constant in LIL, we have the following result

Theorem 1.2. For 0 < α < 3/2, there exists a finite positive constant c_{2}(α) such that

(1.9)  lim sup_{t→∞} \frac{|Y_{α}(t)|}{t^{1−α/2}(log log t)^{α/2}} = c_{2}(α) ∈ [ 2^{(3α−2)/2} Γ(3 − α), c_{1}(α) ], a.s.

The LIL holds true also for random walks via an invariance principle. Let S_{i}, i = 1, 2, … be a simple symmetric random walk on the line, starting from 0, and let ξ(x, n) = #{i : 1 ≤ i ≤ n, S_{i} = x} be its local time. Define

G_{α}(n) =^{def} ∑_{k=1}^{n} \frac{1_{[S_{k}≠0]}}{S_{k}^{α}} = ∑_{i=1}^{∞} \frac{ξ(i, n) − ξ(−i, n)}{i^{α}}. (1.10)

We prove the following invariance principle.

Theorem 1.3. On a suitable probability space one can define a Wiener process {W (t), t ≥ 0} and a simple symmetric random walk {S_{n}, n = 1, 2, …} such that for any 0 < α < 3/2 and sufficiently small ε > 0 we have

|Y_{α}([t]) − G_{α}([t])| = o(t^{1−α/2−ε}), a.s., (1.11)

as t → ∞.

As a consequence of our Theorem 1.3, the LILs (1.2), (1.8) and (1.9) remain true if Y_{α} is replaced by G_{α}.

As it is easily seen, Y_{α} is not defined for α ≥ 3/2. In this case, we consider instead the process

Z_{α}(t) =^{def} ∫_{0}^{t} \frac{1_{[|W (s)|≥1]}}{W^{α}(s)} ds = ∫_{1}^{∞} \frac{L(x, t) − L(−x, t)}{x^{α}} dx. (1.12)

This is a “nice” additive functional, for which the strong approximation of Csáki et al. (1992) holds. The limit process associated with such functionals is V (t) = B(L(t)), where B(·) is a standard Wiener process and L(·) is a Wiener local time at zero, independent of B, a version of iterated Brownian motion (cf., e.g., Csáki et al. 1992).

Theorem 1.4. If α > 3/2, then on a rich enough probability space one can define a Wiener process {W (t), t ≥ 0} and a process {V (t), t ≥ 0} such that for ε > 0 small enough

Z_{α}(t) = σV (t) + o(t^{1/4−ε}), a.s., (1.13)

as t → ∞, where

σ^{2} =^{def} \frac{16}{(α − 1)(2α − 3)}. (1.14)

For the random walk case we have

Theorem 1.5. If α > 3/2, then on a rich enough probability space one can define a simple symmetric random walk {S_{n}, n = 1, 2, …} and a process {V (t), t ≥ 0} such that for ε > 0 small enough G_{α}(t) = σ_{0}V (t) + o(t^{1/4−ε}), a.s., (1.15)

as t → ∞, where

σ_{0}^{2} =^{def} 16 ∑_{ℓ=1}^{∞} ∑_{k=ℓ+1}^{∞} \frac{1}{k^{α}ℓ^{α−1}} + 8 ∑_{k=1}^{∞} \frac{1}{k^{2α−1}} − 2 ∑_{k=1}^{∞} \frac{1}{k^{2α}} . (1.16)

The rest of the paper is organized as follows. In Section 2, we briefly describe our method. Depending on whether we are interested in the process Y_{α} or in its increments, we shall use slightly different approaches, which are discussed in separated subsections for the sake of clarity. Theorems 1.1, 1.2 and 1.3 are proved in Sections 3, 4 and 5 respectively. Finally, Section 6 is devoted to the proof of Theorems 1.4 and 1.5.

Throughout the paper, some universal (finite and positive) constants are denoted by the letter c with subscripts. When they depend on some (possibly multi-dimensional) parameter p, they are denoted by c(p) with subscripts.

2. Preliminaries

We describe the main lines of our approach that will be used in the proofs in the sequel. Throughout, we assume α < 3/2. We shall be using somewhat different approaches for the process Y_{α} and its increments. Our basic tools are: for Y_{α}, a martingale inequality due to Barlow and Yor (1982); and for the increments of Y_{α}, Tanaka’s formula together with some elementary stochastic calculus.

2.1. The process Y_{α}

For any b > 0, consider the decomposition

Y_{α}(t) = ∫_{0}^{b} \frac{L(x, t) − L(−x, t)}{x^{α}} dx + ∫_{b}^{∞} \frac{L(x, t) − L(−x, t)}{x^{α}} dx. (2.1)

It is easy to estimate the second expression on the right hand side. Indeed, by the occupation time formula,

|∫_{b}^{∞} \frac{L(x, t) − L(−x, t)}{x^{α}} dx| ≤ \frac{1}{b^{α}} ∫_{R} L(y, t) dy = \frac{t}{b^{α}} , (2.2)

so that, for any b > 0,

|Y_{α}(t)| ≤ |∫_{0}^{b} \frac{L(x, t) − L(−x, t)}{x^{α}} dx| + \frac{t}{b^{α}} . (2.3)

To treat the integral expression on the right hand side, we first recall the following useful inequality.

Fact 2.1 (Barlow and Yor 1982). For any t > 0, ε ∈ (0, 1/2] and γ ≥ 1,

E ( sup_{0≤s≤t} sup_{x≠y} \frac{|L(x, s) − L(y, s)|}{|x − y|^{1/2−ε}} )^{γ} ≤ c_{3} t^{(1+2ε)γ/4}, (2.4)

where c_{3} = c_{3}(γ, ε).

Fact 2.1 allows us to control the almost sure asymptotics (when t is large) of expressions like sup_{x≠y} |L(x, t) − L(y, t)|/|x − y|^{1/2−ε}. Indeed, let ν ∈ [0, 1/2) and let ε > 0. By Chebyshev’s inequality and (2.4), for any γ ≥ 1 and n ≥ 1,

P ( sup_{0≤s≤n} sup_{x≠y} \frac{|L(x, s) − L(y, s)|}{|x − y|^{ν}} > n^{(1−ν)/2+ε} ) ≤ c_{4}(γ, ν, ε) n^{−γε}.

Take γ = 2/ε and use the Borel–Cantelli lemma to see that

sup_{0≤s≤n} sup_{x≠y} \frac{|L(x, s) − L(y, s)|}{|x − y|^{ν}} = O(n^{(1−ν)/2+ε}), a.s.

Since t ↦ sup_{0≤s≤t} sup_{x≠y} |L(x, s) − L(y, s)|/|x − y|^{ν} is non-decreasing, and since ε can be arbitrarily small, we have proved the following result.

Lemma 2.2. For any ν ∈ [0, 1/2) and ε > 0, when t → ∞,

sup_{x≠y} \frac{|L(x, t) − L(y, t)|}{|x − y|^{ν}} = o(t^{(1−ν)/2+ε}), a.s. (2.5)

This lemma will allow us to obtain useful estimates for the integral expression on the right hand side of (2.3). For example, since α < 3/2, we can choose ν ∈ [0, 1/2) to be as close to 1/2 as possible, such that α − ν < 1. As a consequence, for any fixed b > 0 and ε > 0, when t → ∞,

∫_{0}^{b} \frac{L(x, t) − L(−x, t)}{x^{α}} dx = o(t^{1/4+ε}), a.s. (2.6)

2.2. The increments of Y_{α}

In Section 3, we shall be interested in the increments of Y_{α}. Let us first fix b > 0 and recall (2.1) here:

Y_{α}(t) = ∫_{0}^{b} \frac{L(x, t) − L(−x, t)}{x^{α}} dx + ∫_{b}^{∞} \frac{L(x, t) − L(−x, t)}{x^{α}} dx.

We now use Tanaka’s formula for the first term on the right hand side:

L(x, t) = |W (t) − x| − |x| − ∫_{0}^{t} sgn(W (s) − x) dW (s),

(with the usual notation in Tanaka’s formula: sgn(0) = −1). Hence, if α ≠ 1

∫_{0}^{b} \frac{L(x, t) − L(−x, t)}{x^{α}} dx

= ∫_{0}^{b} \frac{A(x, W (t))}{x^{α}} dx + ∫_{0}^{b} \frac{2 ∫_{0}^{t} 1_{[|W (s)|≤x]} dW (s)}{x^{α}} dx

= ∫_{0}^{b} \frac{A(x, W (t))}{x^{α}} dx + 2 ∫_{0}^{t} 1_{[|W (s)|≤b]} \frac{b^{1−α} − |W (s)|^{1−α}}{1 − α} dW (s), (2.7)

where

A(x, W (t)) =^{def} |W (t) − x| − |W (t) + x|. (2.8)

It is easily seen that |A(x, W (t))| ≤ 2x. (2.9)

For the second term on the right hand side of (2.7), we may apply the Dambis–Dubins– Schwarz theorem for continuous local martingales (cf., e.g., Theorem V.1.6 of Revuz and Yor 1999) to conclude that for the stochastic integral ∫_{0}^{t} f(W (s)) dW (s), there exists a standard Wiener process B(·) such that

∫_{0}^{t} f(W (s)) dW (s) = B(U(t)), (2.10)

where

U(t) =^{def} ∫_{0}^{t} f^{2}(W (s)) ds.

Moreover, U(t) is a stopping time for B(·), hence B̃(s) =^{def} B(U(t) + s) − B(U(t)), s ≥ 0, is also a standard Wiener process for fixed t.

To study the increments of Y_{α}(·), we have

Y_{α}(t + h) − Y_{α}(t) = ∫_{0}^{b} \frac{A(x, W (t + h)) − A(x, W (t))}{x^{α}} dx
+ B(U(t + h)) − B(U(t))
(2.11)  + ∫_{b}^{∞} \frac{L(x, t + h) − L(x, t) − (L(−x, t + h) − L(−x, t))}{x^{α}} dx.

Here,

U(t) = 4 ∫_{0}^{t} 1_{[|W (s)|≤b]} (\frac{b^{1−α} − |W (s)|^{1−α}}{1 − α} )^{2} ds
(2.12)  = 4 ∫_{−b}^{b} (\frac{b^{1−α} − |x|^{1−α}}{1 − α} )^{2} L(x, t) dx.

Hence

U(t + h) − U(t) = 4 ∫_{−b}^{b} (\frac{b^{1−α} − |x|^{1−α}}{1 − α} )^{2} (L(x, t + h) − L(x, t)) dx
≤ sup_{x}(L(x, t + h) − L(x, t)) 4 ∫_{−b}^{b} (\frac{b^{1−α} − |x|^{1−α}}{1 − α} )^{2} dx
= sup_{x}(L(x, t + h) − L(x, t)) c_{5}(α) b^{3−2α},

where

c_{5}(α) = 4 ∫_{−1}^{1} (\frac{1 − |u|^{1−α}}{1 − α} )^{2} du = \frac{16}{(2 − α)(3 − 2α)}. (2.13)

Furthermore, denoting

B̃^{∗}(t) =^{def} sup_{0≤s≤t} |B̃(s)|

and

L^{∗}(h, t) =^{def} sup_{x}(L(x, t + h) − L(x, t)),

we obtain |B(U(t + h)) − B(U(t))| ≤ B̃^{∗} (c_{5}(α) b^{3−2α} L^{∗}(h, t)) . (2.14)

For the third term in (2.11) we use the estimate

|∫_{b}^{∞} \frac{L(x, t + h) − L(x, t) − (L(−x, t + h) − L(−x, t))}{x^{α}} dx|

≤ \frac{1}{b^{α}} ∫_{−∞}^{∞}(L(x, t + h) − L(x, t)) dx = \frac{h}{b^{α}} . (2.15)

Putting (2.11), (2.9), (2.14) and (2.15) together, we finally get that, for any b > 0,

|Y_{α}(t + h) − Y_{α}(t)| ≤ (\frac{4b^{2}}{2 − α} + h) \frac{1}{b^{α}} + B̃^{∗} (c_{5}(α) b^{3−2α} L^{∗}(h, t)) . (2.16)

Note that repeating the same procedure in the case α = 1 with obvious modifications, one can easily see that the final conclusion (2.16) holds true in this case as well.

3. Proof of Theorem 1.1

The main step in proving Theorem 1.1 is the following upper bound.

Lemma 3.1. For 0 < α < 3/2 and δ > 0 there exist positive constants c_{6} = c_{6}(α, δ) and λ_{0} = λ_{0}(α, δ) such that for any t > 0, h > 0 and λ ≥ λ_{0},

P (|Y_{α}(t + h) − Y_{α}(t)| > λh^{1−α/2}) ≤ c_{6} exp(−c_{7} λ^{2/α}), (3.1)

where

c_{7} = c_{7}(α, δ) = \frac{α^{4/3}(3 − 2α)^{2/α−2/3}(2 − α)^{2/3}}{3^{2/α}2^{7/3}(1 + δ)^{2/α−1}} .

Proof. It follows from (2.16) that, for any b > 0,

P(|Y_{α}(t + h) − Y_{α}(t)| > λh^{1−α/2})

(3.2)  ≤ P (B̃^{∗}(c_{5}(α) b^{3−2α} L^{∗}(h, t)) ≥ λh^{1−α/2} − (\frac{4b^{2}}{2 − α} + h) \frac{1}{b^{α}} ) .

Let δ > 0 be arbitrarily small, and choose

b = h^{1/2} (\frac{3(1 + δ)}{(3 − 2α)λ})^{1/α} , (3.3)

so that

λh^{1−α/2} − (1 + δ)\frac{h}{b^{α}} = \frac{2αλh^{1−α/2}}{3} . (3.4)

Define λ_{0} by

λ_{0} = \frac{3(1 + δ)2^{α}}{(2 − α)^{α/2}(3 − 2α)δ^{α/2}} .

Then for λ ≥ λ_{0},

\frac{4b^{2}}{2 − α} ≤ δh. (3.5)

Collecting (3.2), (3.5) and (3.4) yields that for any A > 0,

P(|Y_{α}(t + h) − Y_{α}(t)| > λh^{1−α/2})

≤ P (B̃^{∗}(c_{5}(α) b^{3−2α} L^{∗}(h, t)) > λh^{1−α/2} − (1 + δ)\frac{h}{b^{α}} )
= P (B̃^{∗}(c_{5}(α) b^{3−2α} L^{∗}(h, t)) > \frac{2αλh^{1−α/2}}{3} )
≤ P(L^{∗}(h, t) > Ah^{1/2}) + P (B̃^{∗}(c_{5}(α) b^{3−2α} Ah^{1/2}) > \frac{2αλh^{1−α/2}}{3} ) .

By the usual estimate for Gaussian tails (cf., e.g. Proposition II.1.8 of Revuz and Yor 1999),

P(B̃^{∗}(t) ≥ x) ≤ 2 exp (−\frac{x^{2}}{2t} ) . (3.6)

On the other hand, for any δ > 0, t > 0 and λ > 0,

(3.7) P ( sup_{x∈R} L(x, t) ≥ λ) ≤ c_{8}(δ) exp (−\frac{λ^{2}}{2(1 + δ)t}) ,

cf. Kesten (1965) and Csáki (1989). For forthcoming applications, we also recall the LIL for the maximum local time:

(3.8)  lim sup_{t→∞} (2t log log t)^{−1/2} sup_{x∈R} L(x, t) = 1,  a.s.

Since L^{∗}(h, t) has the same distribution as sup_{x∈R} L(x, h), we have

P(L^{∗}(h, t) ≥ Ah^{1/2}) ≤ c_{8}(δ) exp (−\frac{A^{2}}{2(1 + δ)}) .

Hence

P(|Y_{α}(t + h) − Y_{α}(t)| > λh^{1−α/2})
≤ c_{9}(δ) (exp (−\frac{A^{2}}{2(1 + δ)}) + exp (−\frac{2α^{2}λ^{2}h^{3/2−α}}{9c_{5}(α) b^{3−2α} A})) .

Using (3.3) and (2.13), and choosing A such that the two exponents above should be equal, we get (3.1). ⊔ ⊓

Proof of Theorem 1.1. (1.4) and (1.6) follows from Csáki and Csörgő (1992, Theorems 3.1 and 3.2) and our Lemma 3.1.

To show (1.5) one can use the following inequality:

Lemma 3.2. For h > 0, λ > 1, ε > 0 we have

P( sup_{0≤s≤h} |Y_{α}(s)| > λh^{1−α/2}) ≤ C exp (−\frac{c_{7}λ^{2/α}}{1 + ε} ) , (3.9)

with some C = C(ε) > 0, and c_{7} as in Lemma 3.1.

Proof. Lemma 3.1 combined with Lemma 2.1 of Csáki and Csörgő (1992), when in the latter we put T = a = h, gives for any integer r that

P ( sup_{0≤s≤h} |Y_{α}(s)| > λ (h + \frac{1}{R})^{1−α/2} + 2 ∑_{j=0}^{∞} x_{j} (\frac{1}{2^{r+j+1}} )^{1−α/2})

≤ Rh(Rh + 1)c_{6} exp(−c_{7}λ^{2/α}) + 4Rhc_{6} ∑_{j=0}^{∞} 2^{j} exp(−c_{7}x_{j}^{2/α}),

where R = 2^{r}. For an arbitrary positive ε > 0 we choose R such that

\frac{1}{ε} < Rh < \frac{2}{ε},

and select

x_{j} = (\frac{j}{c_{7}} + λ^{2/α})^{α/2}

we get our statement by elementary calculations. □

Now (1.5) follows by standard application of the Borel–Cantelli lemma and the monotonicity of h ↦ sup_{0≤s≤h} |Y_{α}(h)|. This completes the proof of Theorem 1.1. □

4. Proof of Theorem 1.2

For brevity, we write throughout the section

h(t) = ( \frac{t}{log log t} )^{1/2}.

The proof of Theorem 1.2 is based on the following

Fact 4.1 (Donsker and Varadhan 1977). Let

L̂_{t}(x) = (t log log t)^{−1/2}L (x h(t), t) , x ∈ R,

and let A denote the space of all absolutely continuous subprobability densities on R, endowed with the topology of uniform convergence on bounded intervals. Then for any continuous functional Φ on A,

(4.1)  lim sup_{t→∞} Φ(L̂_{t}) = sup_{f∈A, I(f)≤1} Φ(f),  a.s.,

where

I(f) = \frac{1}{8} ∫_{R} \frac{(f′(y))^{2}}{f(y)} dy.

Proof of Theorem 1.2. That the expression on the left hand side of (1.9) should be equal to a constant (possibly zero or infinite) follows from Kolmogorov’s 0–1 law. In view of (1.8), it only remains to check that c_{2}(α) ≥ 2^{(3α−2)/2}Γ(3 − α).

Let δ ∈ (0, 1/2). Observe that

Y_{α}(t) = ( ∫_{0}^{δh(t)} + ∫_{δh(t)}^{h(t)/δ} + ∫_{h(t)/δ}^{∞} )\frac{L(x, t) − L(−x, t)}{x^{α}} dx
= ( ∫_{0}^{δh(t)} + ∫_{h(t)/δ}^{∞} )\frac{L(x, t) − L(−x, t)}{x^{α}} dx

+t^{1−α/2}(log log t)^{α/2} ∫_{δ}^{1/δ} \frac{L̂_{t}(x) − L̂_{t}(−x)}{x^{α}} dx.

For any fixed δ ∈ (0, 1/2), we can apply (4.1) to see that, almost surely,

lim sup_{t→∞} ∫_{δ}^{1/δ} \frac{L̂_{t}(x) − L̂_{t}(−x)}{x^{α}} dx = sup_{f∈A, I(f)≤1} ∫_{δ}^{1/δ} \frac{f(x) − f(−x)}{x^{α}} dx.

Let q > 1 and take the function

f(x) = \{ \begin{matrix}p x^{q} exp(−rx), & if x ≥ 0, \\ 0, & if x < 0, \end{matrix}

for some positive constants p and r. It is easily checked that if p Γ(q + 1) ≤ r^{q+1}, (4.2) then f is an absolutely continuous subprobability density function. Furthermore, whenever pq Γ(q − 1) ≤ 8r^{q−1}, (4.3) we have I(f) ≤ 1. Therefore, under (4.2) and (4.3),

lim sup_{t→∞} ∫_{δ}^{1/δ} \frac{L̂_{t}(x) − L̂_{t}(−x)}{x^{α}} dx ≥ ∫_{δ}^{1/δ} p x^{q−α}e^{−rx} dx, a.s.

Since δ can be as small as possible, this leads to: for q > 1, p Γ(q+1) < r^{q+1} and pq Γ(q−1) < 8r^{q−1},

lim inf_{δ→0} lim sup_{t→∞} ∫_{δ}^{1/δ} \frac{L̂_{t}(x) − L̂_{t}(−x)}{x^{α}} dx ≥ \frac{p Γ(q − α + 1)}{r^{q−α+1}}, a.s.

Take q = 2, r = 2^{3/2}, and send p to 2^{7/2} from the left, to arrive at:

lim inf_{δ→0} lim sup_{t→∞} ∫_{δ}^{1/δ} \frac{L̂_{t}(x) − L̂_{t}(−x)}{x^{α}} dx ≥ 2^{(3α−2)/2} Γ(3 − α), a.s.

Write for brevity ϕ_{α}(t) = t^{1−α/2}(log log t)^{α/2}. (4.4)

If we could furthermore show that almost surely,

(4.5)  lim_{δ→0} lim sup_{t→∞} \frac{1}{ϕ_{α}(t)} ∫_{0}^{δh(t)} \frac{L(x, t) − L(−x, t)}{x^{α}} dx = 0,
(4.6)  lim_{δ→0} lim sup_{t→∞} \frac{1}{ϕ_{α}(t)} ∫_{h(t)/δ}^{∞} \frac{L(x, t) − L(−x, t)}{x^{α}} dx = 0,

then the proof of Theorem 1.2 would be complete. The rest of the section is devoted to the verification of (4.5) and (4.6). ⊔ ⊓

Proof of (4.6). Follows immediately from (2.2) (taking b = h(t)/δ there) and the definition of the function h(·). ⊔ ⊓

Proof of (4.5). Assume α ≠ 1 for the moment. Recall (2.7):

∫_{0}^{b} \frac{L(x, t) − L(−x, t)}{x^{α}} dx

(4.7)  = ∫_{0}^{b} \frac{A(x, W (t))}{x^{α}} dx + 2 ∫_{0}^{t} 1_{[|W (s)|≤b]} \frac{b^{1−α} − |W (s)|^{1−α}}{1 − α} dW (s),

with |A(x, W (t))| ≤ 2x for any t > 0 and x > 0. The first term on the right hand side is easily controlled, since its modulus is smaller than or equal to a constant multiple of b^{2−α}. To study the second term, let us write

Λ(t, b) =^{def} ∫_{0}^{t} 1_{[|W (s)|≤b]} \frac{b^{1−α} − |W (s)|^{1−α}}{1 − α} dW (s).

The proof of (4.5) will be complete as soon as we show

(4.8)  lim_{δ→0} lim sup_{t→∞} \frac{Λ(t, δh(t))}{ϕ_{α}(t)} = 0,  a.s.

For each b > 0, t ↦ Λ(t, b) is a Brownian time change as we briefly described in Section 2. However, to check (4.8), we have to consider the situation when b depends on t. Obviously, the process t ↦ Λ(t, δh(t)) is no longer a local martingale (it is not even clear whether it is a semimartingale). So we have to handle it with care.

Fix b > 0 for the moment. Then by (2.10) and (2.12), there exists a Brownian motion B such that Λ(t, b) = B(U(t)) for all t ≥ 0, where

U(t) = 4 ∫_{−b}^{b} (\frac{b^{1−α} − |x|^{1−α}}{1 − α} )^{2} L(x, t) dx
(4.9)  ≤ 4c_{5}(α) b^{3−2α} sup_{x∈R} L(x, t),

with L being the local time of B. For the value of the constant c_{5}(α), cf. (2.13). Note that both the Brownian motion B and its clock U depend on b. Also, (4.9) holds even in the case α = 1 (so from now on we only make the general assumption α < 3/2).

We have, for any y > 0 and z > 0,

P ( sup_{0≤s≤t} |Λ(s, b)| > y) ≤ P ( sup_{0≤s≤t} |B(U(s))| > y)
≤ P ( sup_{x∈R} L(x, t) > z) + ( sup_{0≤u≤4c_{5}(α) b^{3−2α} z} |B(u)| > y)
≤ c_{10} exp (−\frac{z^{2}}{3t} ) + 2 exp (−\frac{y^{2}}{8c_{5}(α) b^{3−2α} z} ) ,

where we used (3.7) and (3.6) in the last inequality. Take z = (ty^{2}/b^{3−2α})^{1/3} to see that

P ( sup_{0≤s≤t} |Λ(s, b)| > y) ≤ (c_{10} + 2) exp (−c_{11}(α) \frac{y^{4/3}}{b^{2−4α/3}t^{1/3}} ) .

Now define t_{n} = (1 + δ)^{n}. Applying the Borel–Cantelli lemma gives that almost surely for all large n,

sup_{0≤s≤t_{n}} |Λ(s, δh(t_{n}))| ≤ c_{12}(α) δ^{3/2−α}ϕ_{α}(t_{n}), (4.10)

where ϕ_{α}(·) is as in (4.4), and we choose c_{12}(α) such that c_{12}^{4/3}(α) c_{11}(α) > 1 should hold.

On the other hand, by (4.7) and the inequality |A(x, W (t))| ≤ 2x, we have, for r ∈ [t_{n−1}, t_{n}],

|Λ(r, δh(t_{n})) − Λ(r, δh(r))|

≤ 2 ∫_{0}^{δh(t_{n})} x^{1−α} dx + \frac{1}{2} ∫_{δh(r)}^{δh(t_{n})} \frac{|L(x, r) − L(−x, r)|}{x^{α}} dx

≤ \frac{2}{2 − α}(δh(t_{n}))^{2−α} + sup_{x∈R} L(x, t_{n}) ∫_{δh(t_{n−1})}^{δh(t_{n})} \frac{dx}{x^{α}}.

For all large n and all x ∈ [δh(t_{n−1}), δh(t_{n})], we have x^{−α} ≤ 2(δh(t_{n}))^{−α}. Moreover, since t ↦ h′(t) is decreasing (for large t), we have, by the mean value theorem, when n is sufficiently large,

δh(t_{n}) − δh(t_{n−1}) ≤ δ(t_{n} − t_{n−1})h′(t_{n−1})
≤ \frac{δ(t_{n} − t_{n−1})}{2\sqrt{t_{n−1} log log t_{n−1}}} = \frac{δ^{2}}{2\sqrt{1 + δ}} \begin{matrix}√ & \overline{t_{n}} \\ √ & \overline{log log t_{n−1}} \end{matrix}
≤ \frac{δ^{2}}{\sqrt{1 + δ}} \begin{matrix}√ & \overline{t_{n}} \\ √ & \overline{log log t_{n}} \end{matrix} .

Thus

∫_{δh(t_{n−1})}^{δh(t_{n})} \frac{dx}{x^{α}} ≤ \frac{2δ^{2−α}}{\sqrt{1 + δ}} t_{n}^{1/2−α/2} (log log t_{n})^{α/2−1/2}.

In view of (3.8), we obtain, almost surely when n → ∞,

sup_{r∈[t_{n−1},t_{n}]} |Λ(r, δh(t_{n})) − Λ(r, δh(r))|

≤ \frac{2}{2 − α}(δh(t_{n}))^{2−α} + \frac{2δ^{2−α}}{\sqrt{1 + δ}} (2^{1/2} + o(1)) ϕ_{α}(t_{n}).

Taking into account of (4.10), we have, with probability one,

lim sup_{n→∞} \frac{1}{ϕ_{α}(t_{n})} sup_{r∈[t_{n−1},t_{n}]} |Λ(r, δh(r))| ≤ c_{12}(α) δ^{3/2−α} + \frac{8^{1/2} δ^{2−α}}{\sqrt{1 + δ}} .

Consequently,

lim sup_{t→∞} \frac{1}{ϕ_{α}(t)} |Λ(t, δh(t))| ≤ c_{12}(α) δ^{3/2−α} + \frac{8^{1/2} δ^{2−α}}{\sqrt{1 + δ}} , a.s.

Since α < 3/2, this clearly yields (4.8), hence completes the proof of (4.5). ⊔ ⊓

5. Proof of Theorem 1.3

Throughout this section, {S_{k}; k = 1, 2, ⋯} denotes a simple symmetric random walk on Z starting from 0, with local time ξ(x, n), and {W (t); t ≥ 0} denotes a Wiener process with local time L(x, t). The proof of Theorem 1.3 is based on a strong invariance principle for local time due to Révész (1981).

Fact 5.1 (Révész 1981). On a suitable probability space we have, for any ε > 0, when n → ∞,

sup_{x∈Z} |ξ(x, n) − L(x, n)| = o(n^{1/4+ε}), a.s.

Proof of Theorem 1.3. Without loss of generality, we may and will suppose that t is an integer (the [·] sign for integer part is thus omitted).

Let ε > 0. For convenience, we write

γ = \frac{(1 − α)^{+}}{2} .

Note that γ < 3/4 − α/2 whenever α < 3/2. The reason for introducing γ is that we shall several times use the following relations:

∫_{1}^{t^{1/2+ε}} \frac{dx}{x^{α}} = o(t^{γ+ε}),  ∑_{i=1}^{t^{1/2+ε}} \frac{1}{i^{α}} = o(t^{γ+ε}). (5.1)

The usual LIL for random walk (cf. e.g. Révész 1990, p. 35) says that

lim sup_{n→∞} (2n log log n)^{−1/2} max_{1≤k≤n} |S_{k}| = 1, a.s.

Thus, almost surely for all ε > 0 and large t, ξ(i, t) = 0 whenever |i| > t^{1/2+ε}. Accordingly,

∑_{k=1}^{t} \frac{1}{S_{k}^{α}} 1_{[S_{k}≠0]} = ∑_{i=1}^{∞} \frac{ξ(i, t) − ξ(−i, t)}{i^{α}}

= ∑_{i=1}^{t^{1/2+ε}} \frac{ξ(i, t) − ξ(−i, t)}{i^{α}}

= ∑_{i=1}^{t^{1/2+ε}} \frac{(ξ(i, t) − L(i, t)) − (ξ(−i, t)) − L(−i, t))}{i^{α}}

+ ∑_{i=1}^{t^{1/2+ε}} \frac{L(i, t) − L(−i, t)}{i^{α}}. (5.2)

By Fact 5.1 and (5.1), as t → ∞, the first summation term on the right hand side of (5.2) is

= ∑_{i=1}^{t^{1/2+ε}} \frac{o(t^{1/4+ε})}{i^{α}} = o(t^{1/4+γ+2ε}), a.s.

Since γ < 3/4 − α/2, we can choose ε sufficiently small so that 1/4 + γ + 2ε < 1 − α/2 − ε. Therefore, the proof of Theorem 1.3 is reduced to showing the following estimate: almost surely, when t goes to infinity,

∫_{0}^{∞} \frac{L(x, t) − L(−x, t)}{x^{α}} dx − ∑_{i=1}^{t^{1/2+ε}} \frac{L(i, t) − L(−i, t)}{i^{α}} = o (t^{1−α/2−ε}) ,

or equivalently (since L(x, t) = 0 for all large t and |x| > t^{1/2+ε}),

I(t) = o (t^{1−α/2−ε}) , a.s., (5.3)

where

I(t) = ∫_{0}^{1+t^{1/2+ε}} \frac{L(x, t) − L(−x, t)}{x^{α}} dx − ∑_{i=1}^{t^{1/2+ε}} \frac{L(i, t) − L(−i, t)}{i^{α}},

and we wrote 1 + t^{1/2+ε} instead of t^{1/2+ε} in order to simplify writings later on. Observe that

I(t) = ∑_{i=1}^{t^{1/2+ε}} ∫_{i}^{i+1} \frac{L(x, t) − L(−x, t) − L(i, t) + L(−i, t)}{x^{α}}dx

+ ∫_{0}^{1} \frac{L(x, t) − L(−x, t)}{x^{α}}dx

+ ∑_{i=1}^{t^{1/2+ε}}(L(i, t) − L(−i, t)) ∫_{i}^{i+1} (\frac{1}{x^{α}} − \frac{1}{i^{α}} ) dx

=^{def} I_{1}(t) + I_{2}(t) + I_{3}(t).

According to Bass and Griffin (1985) (or simply by (2.5)),

sup_{|x−y|≤1} |L(x, t) − L(y, t)| = o(t^{1/4+ε}), a.s.,

as t → ∞. Therefore,

|I_{1}(t)| ≤ o (t^{1/4+ε}) ∑_{i=1}^{t^{1/2+ε}} ∫_{i}^{i+1} \frac{dx}{x^{α}}
= o (t^{1/4+ε}) ∫_{1}^{t^{1/2+ε}} \frac{dx}{x^{α}}
= o (t^{1/4+γ+2ε}) ,  a.s.,

where we have used (5.1) in the last identity.

The expression I_{2}(t) was already estimated in Section 2. Indeed, according to (2.6),

I_{2}(t) = o (t^{1/4+ε}) , a.s.

Since γ < 3/4−α/2 and min(1/2, α) > α−1, the proof of (5.3) (and thus also that of Theorem 1.3) will be complete as soon as we prove the following estimate: for any 0 < ν < min(1/2, α),

I_{3}(t) = o(t^{(1−ν)/2+ε}), a.s. (5.4)

To check (5.4), let us note that

∫_{i}^{i+1} (\frac{1}{i^{α}} − \frac{1}{x^{α}} ) dx ∈ (0, \frac{c_{13}(α)}{i^{α+1}}),

for some constant c_{13}(α) < ∞. By (2.5), for any 0 < ν < min(1/2, α),

sup_{i≥1} \frac{|L(i, t) − L(−i, t)|}{i^{ν}} = o(t^{(1−ν)/2+ε}), a.s.

Therefore,

|I_{3}(t)| ≤ c_{13}(α) sup_{i≥1} \frac{|L(i, t) − L(−i, t)|}{i^{ν}} ∑_{i=1}^{∞} \frac{1}{i^{1+α−ν}}
= c_{14}(α, ν) sup_{i≥1} \frac{|L(i, t) − L(−i, t)|}{i^{ν}},

proving (5.4). ⊔ ⊓

6. Proof of Theorems 1.4 and 1.5

In the case α > 2, the theorems follow from Csáki et al. (1992), where for the additive functionals ∑_{i=1}^{n} f(S_{i}) and ∫_{0}^{t} g(W (s)) ds, respectively, it was assumed that

∑_{x=−∞}^{∞} |x|^{1+δ} |f(x)| < ∞ (6.1)

and ∫_{−∞}^{∞} |x|^{1+δ} |g(x)| dx < ∞, (6.2) respectively, for some δ > 0.

It is easy to see that these conditions are satisfied for Z_{α} and G_{α} when α > 2. We however claim that the conditions (6.1) and (6.2), respectively, can be replaced by the weaker conditions

∑_{x=−∞}^{∞} |x|^{1/2+δ} |f(x)| < ∞ (6.3)

and ∫_{−∞}^{∞} |x|^{1/2+δ} |g(x)| dx < ∞, (6.4)

respectively, for some δ > 0.

In order to see this, we have to show that (6.3) implies

E| ∑_{i=1}^{ρ} f(S_{i})|^{2+δ} < ∞, (6.5)

where ρ is the time of the first return to zero of the random walk, since it follows from the proof of the strong approximation in Csáki et al. (1992) that (6.5) is sufficient for the conclusion to hold. To prove (6.5), we use

E(ξ(x, ρ))^{m} ≤ c_{15}(m) (1 + |x|)^{m−1},

(cf. (2.23) in Csáki et al. 1992) and the triangle inequality, to obtain (writing c_{16} for c_{15}(2 + δ)):

(E| ∑_{i=1}^{ρ} f(S_{i})|^{2+δ})^{1/(2+δ)} ≤ ∑_{x∈Z} |f(x)| (E(ξ(x, ρ))^{2+δ})^{1/(2+δ)}
≤ c_{16} ∑_{x∈Z} |f(x)|(1 + |x|^{(1+δ)/(2+δ)})
≤ c_{16} ∑_{x∈Z} |f(x)|(1 + |x|^{1/2+δ}) < ∞.

Now one can see that (6.3) is satisfied whenever α > 3/2, hence the conclusion of Theorem 1 in Csáki et al. (1992) holds, which gives Theorem 1.5.

Similar proof holds for the Wiener case, i.e., for Theorem 1.4. Let T_{.} be the inverse of the local time L(0, t). Then (cf. Csáki et al. 1992)

E (L(x, T_{1}))^{m} ≤ c_{17}(m)(1 + |x|)^{m−1},

and similarly to the above estimation, we get from the triangle inequality

(E| ∫_{0}^{T_{1}} g(W (s)) ds|^{2+δ})^{1/(2+δ)} ≤ c_{18} ∫_{−∞}^{∞} |g(x)|(1 + |x|^{1/2+δ}) dx < ∞.

Hence Theorem 2 in Csáki et al. (1992) applies whenever (6.4) is satisfied. This proves our Theorem 1.4.

Acknowledgements

We are grateful to Marc Yor for helpful discussions. Cooperation between the authors was supported by the joint French-Hungarian Intergovernmental Grant ‘Balaton’ (grant no. F25/97).

References

Alili, L. (1997). On some hyperbolic principal values of Brownian local times. In: Exponential Functionals and Principal Values Related to Brownian Motion (M. Yor, ed.), pp. 131–154. Biblioteca de la Revista Matemática Iberoamericana, Madrid.

Barlow, M.T. and Yor, M. (1982). Semi-martingale inequalities via the Garsia–Rodemich– Rumsey lemma, and applications to local times. J. Funct. Anal. \underline{49} 198–229.

Bass, R.F. and Griffin, P.S. (1985). The most visited site of Brownian motion and simple random walk. Z. Wahrsch. Verw. Gebiete \underline{70} 417–436.

Borodin, A.N. (1985). Distribution of the supremum of increments of Brownian local time. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI ) \underline{142} 6–24.

Csáki, E. (1989). An integral test for the supremum of Wiener local time. Probab. Th. Rel. Fields \underline{83} 207–217.

Csáki, E. and Csörgő, M. (1992). Inequalities for increments of stochastic processes and moduli of continuity. Ann. Probab. \underline{20} 1031–1052.

Csáki, E., Csörgő, M., Földes, A. and Révész, P. (1992). Strong approximation of additive functionals. J. Theoretical Probab. \underline{5} 679–706.

Donsker, M.D. and Varadhan, S.R.S. (1977). On laws of the iterated logarithm for local times. Comm. Pure Appl. Math. \underline{30} 707–753.

Hu, Y. and Shi, Z. (1997). An iterated logarithm law for Cauchy’s principal value of Brownian local times. In: Exponential Functionals and Principal Values Related to Brownian Motion (M. Yor, ed.), pp. 211–223. Biblioteca de la Revista Matemática Iberoamericana, Madrid.

Kesten, H. (1965). An iterated logarithm law for the local time. Duke Math. J. \underline{32} 447–456.

McKean, H.P. (1962). A Hölder condition for Brownian local time. J. Math. Kyoto Univ. \underline{1} 195–201.

Révész, P. (1981). Local time and invariance. In: Analytical Methods in Probability Theory (D. Dugué et al., eds.). Lecture Notes in Mathematics \underline{861}, pp. 128–145. Springer, Berlin.

Révész, P. (1990). Random Walk in Random and Non–Random Environments. World Scientific, Singapore.

Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion. Springer, Berlin, 3rd edition.

Yor, M., editor (1997). Exponential Functionals and Principal Values Related to Brownian Motion. Biblioteca de la Revista Matemática Iberoamericana, Madrid.