PATH PROPERTIES OF CAUCHY S PRINCIPAL VALUES RELATED TO LOCAL TIME

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1 PATH PROPERTIES OF CAUCHY S PRINCIPAL VALUES RELATED TO LOCAL TIME Endre Csáki,2 Mathematical Institute of the Hungarian Academy of Sciences. Budapest, P.O.B. 27, H- 364, Hungary address: csaki@math-inst.hu Miklós Csörgő 3 School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada KS 5B6 address: mcsorgo@math.carleton.ca Antónia Földes 2,4 City University of New York, 28 Victory Blvd., Staten Island, New York 34, U.S.A. address: afoldes@ .gc.cuny.edu Zhan Shi 2 Laboratoire de Probabilités, Université Paris VI, 4 Place Jussieu, F Paris Cedex 5, France address: shi@ccr.jussieu.fr Abstract: Sample path properties of Cauchy s principal values of Brownian and random walk local times are studied. AMS 99 Subject Classification: Primary 6J5; Secondary 6F5, 6J55. 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. Research supported by the Hungarian National Foundation for Scientific Research, Grant No. T 6384 and T Research supported by a Fellowship of the Paul Erdős Summer Research Center of Mathematics, Budapest. 3 Research 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. 4 Research supported by a PSC CUNY Grant, No

2 . Introduction and main results Recently a collection of papers appeared in Yor 997) that deal with principal values related to Brownian motion cf. Part B of Yor 997). Let W ) be a standard Wiener process Brownian motion) with local time Lx, t), i.e. for any Borel function f and t, t fw s)) ds = fx)lx, t). Define the integral t ds/w α s) notation: z α = z α sgnz)) in the sense of Cauchy s principal value:.) Y α t) def = t R ds W α s) = Lx, t) L x, t). It is well-known McKean 962, Borodin 985) that the local time in the space variable is Hölder continuous of order /2 ε) for any ε >, which ensures the finiteness for each t ) of the second integral in.) 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 α < 3/2 cf., e.g., Sections.-.3 in Alili 997 and references therein), and as Riemann integral for α <. 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 997), who proved the local, as well as the global, laws of the iterated logarithm LIL) for Y ). Namely, they established the following results:.2) and.3) t h Y t) t log log t = 2 2, a.s. Y h) h log log/h) = 2 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, 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.6) of Theorem., we assume that a T is a non-decreasing function of T, such that < a T T and a T /T is non-increasing. 2

3 Theorem.. For < α < 3/2 we have.4).5).6).7) t t h h Y α t) t α/2 log log t) c α), a.s. α/2 sup t T at sup s at Y α t + s) Y α t) c a α/2 T logt/a T ) + log log T ) α/2 α), a.s. Y α h) h α/2 log log/h)) c α), a.s. α/2 sup t h sup s h Y α t + s) Y α t) c h α/2 log/h)) α/2 α), a.s. Here, the constant c α) is given by.8) c α) = 3 2 5α/6 α 2α/3 3 2α) α/3 2 α) α/3. Concerning the constant in LIL, we have the following result Theorem.2. For < α < 3/2, there exists a finite positive constant c 2 α) such that.9) t Y α t) t α/2 log log t) α/2 = c 2α) [ 2 3α/2 Γ2 α), c α) ], a.s. The LIL holds true also for random walks via an invariance principle. Let S i, i =, 2,... be a simple symmetric random walk on the line, starting from, and let ξx, n) = #{i : i n, S i = x} be its local time. Define.) G α n) def = n k= [Sk ] S α k = ξi, n) ξ i, n) i α. We prove the following invariance principle. Theorem.3. On a suitable probability space one can define a Wiener process {W t), t } and a simple symmetric random walk {S n, n =, 2,...} such that for any < α < 3/2 and sufficiently small ε > we have.) Y α [t]) G α [t]) = ot α/2 ε ), a.s., as t. 3

4 As a consequence of our Theorem.3, the LILs.2),.4) and.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 t Z α t) def [ W s) ] Lx, t) L x, t).2) = ds =. W α s) This is a nice additive functional, for which the strong approximation of Csáki et al. 992) holds. The limit process associated with such functionals is V t) = BLt)), 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. 992). Theorem.4. If α > 3/2, then on a rich enough probability space one can define a Wiener process {W t), t } and a process {V t), t } such that for ε > small enough.3) as t, where.4) Z α t) = σv t) + ot /4 ε ), σ 2 def = For the random walk case we have 6 α )2α 3). Theorem.5. If α > 3/2, then on a rich enough probability space one can define a simple symmetric random walk {S n, n =, 2,...} and a process {V t), t } such that for ε > small enough.5) as t, where.6) σ 2 def = 6 l= k=l+ k α l α + 8 k= a.s., G α t) = σ V t) + ot /4 ε ), k 2α 2 k= k 2α. 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.,.2 and.3 are proved in Sections 3, 4 and 5 respectively. Finally, Section 6 is devoted to the proof of Theorems.4 and.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 cp) with subscripts. Our use of almost surely is not systematic. 4 a.s.,

5 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 982); and for the increments of Y α, Tanaka s formula together with some elementary stochastic calculus. 2.. The process Y α 2.) For any b >, consider the decomposition Y α t) = Lx, t) L x, t) Lx, t) L x, t) +. b It is easy to estimate the second expression on the right hand side. Indeed, by the occupation time formula, Lx, t) L x, t) b 2.2) Ly, t) dy = t b α R b, α so that, for any b >, 2.3) Y α t) Lx, t) L x, t) + t b. α To treat the integral expression on the right hand side, we first recall the following useful inequality. Fact 2. Barlow and Yor 982). For any t >, ε, /2] and γ, 2.4) where c 3 = c 3 γ, ε). E L x s L y ) γ sup sup s c s t x y x y /2 ε 3 t +2ε)γ/4, Fact 2. 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 /2 ε. Indeed, let ν [, /2) and let ε >. By Chebyshev s inequality and 2.4), for any γ and n, P sup sup s n x y L x s L y ) s > n ν)/2+ε c x y ν 4 γ, ν, ε) n γε. 5

6 Take γ = 2/ε and use the Borel Cantelli lemma to see that L x s L y sup sup s = On ν)/2+ε ), a.s. s n x y x y ν Since t sup 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 [, /2) and ε >, when t, 2.5) L x t L y t sup = ot ν)/2+ε ), a.s. x y x y ν 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 [, /2) to be as close to /2 as possible, such that α ν <. As a consequence, for any fixed b > and ε >, when t, 2.6) Lx, t) L x, t) = ot /4+ε ), a.s The increments of Y α In Section 3, we shall be interested in the increments of Y α. Let us first fix b > and recall 2.) here: Y α t) = Lx, t) L x, t) Lx, t) L x, t) +. b We now use Tanaka s formula for the first term on the right hand side: Lx, t) = W t) x x t sgnw s) x) dw s), with the usual notation in Tanaka s formula: sgn) = ). Hence, 2.7) Lx, t) L x, t) = = Ax, W t)) + Ax, W t)) + 2 t 2 t [ W s) x] dw s) [ W s) b] b α W s) α α 6 dw s),

7 where 2.8) It is easily seen that 2.9) Ax, W t)) def = W t) x W t) + x. Ax, W t)) 2x. 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..6 of Revuz and Yor 994) to conclude that for the stochastic integral t fw s)) dw s), there exists a standard Wiener process B ) such that t 2.) fw s)) dw s) = BUt)), where Ut) def = t Moreover, Ut) is a stopping time for B ), hence also a standard Wiener process for fixed t. To study the increments of Y α ), we have 2.) Here, Y α t + h) Y α t) = f 2 W s)) ds. Ax, W t + h)) Ax, W t)) Bs) def = BUt) + s) BUt)), s, is + BUt + h)) BUt)) Lx, t + h) Lx, t) L x, t + h) L x, t)) +. b 2.2) Hence t b α W s) α ) 2 Ut) = 4 [ W s) b] ds α b b α x α ) 2 = 4 Lx, t). b α Ut + h) Ut) = 4 b b α x α ) 2 Lx, t + h) Lx, t)) α suplx, t + h) Lx, t)) 4 x = suplx, t + h) Lx, t)) c 5 α) b 3 2α, x b b α x α ) 2 α 7

8 where 2.3) Furthermore, denoting and we obtain 2.4) 2.5) c 5 α) = 4 ) u α 2 du = α B t) def = sup Bs) s t 6 2 α)3 2α). L h, t) def = suplx, t + h) Lx, t)), x BUt + h)) BUt)) B c 5 α) b 3 2α L h, t) ). For the third term in 2.) we use the estimate Lx, t + h) Lx, t) L x, t + h) L x, t)) b b α Lx, t + h) Lx, t)) = h b α. Putting 2.), 2.9), 2.4) and 2.5) together, we finally get that, for any b >, 2.6) ) 4b 2 Y α t + h) Y α t) 2 α + h b + B c α 5 α) b 3 2α L h, t) ). Note that repeating the same procedure in the case α = with obvious modifications, one can easily see that the final conclusion 2.6) holds true in this case as well. 3. Proof of Theorem. The main step in proving Theorem. is the following upper bound. Lemma 3.. For < α < 3/2 and δ > there exist positive constants c 6 = c 6 α, δ) and λ = λ α, δ) such that for any t >, h > and λ λ, 3.) where P Y α t + h) Y α t) > λh α/2) c 6 exp c 7 λ 2/α ), c 7 = c 7 α, δ) = α4/3 3 2α) 2/α 2/3 2 α) 2/3 3 2/α 2 5/3 + δ) 2/α. 8

9 Proof. It follows from 2.6) that, for any b >, P Y α t + h) Y α t) > λh α/2 ) ) ) 4b P B c 5 α) b 3 2α L h, t)) λh α/ ) 2 α + h. b α Let δ > be arbitrarily small, and choose 3.3) ) /α 3 + δ) b = h /2, 3 2α)λ so that 3.4) Define λ by λh α/2 + δ) h b = 2αλh α/2. α 3 λ = 3 + δ)4α/2 3 2α)δ α/2. Then for λ λ, 4b 2 3.5) 2 α δh. Collecting 3.2), 3.5) and 3.4) yields that for any A >, P Y α t + h) Y α t) > λh α/2 ) P B c 5 α) b 3 2α L h, t)) > λh α/2 + δ) h ) b α ) = P B c 5 α) b 3 2α L h, t)) > 2αλh α/2 3 ) PL h, t) > Ah /2 ) + P B c 5 α) b 3 2α Ah /2 ) > 2αλh α/2. 3 By the usual estimate for Gaussian tails, ) 3.6) P B t) x) 4 exp x2. 2t On the other hand, for any δ >, t > and λ >, ) 3.7) P sup Lx, t) λ x R c 8 δ) exp 9 λ2 2 + δ)t ),

10 cf. Kesten 965) and Csáki 989). For forthcoming applications, we also recall the LIL for the maximum local time: 3.8) t 2t log log t) /2 sup Lx, t) =, x R a.s. Since L h, t) has the same distribution as sup x R Lx, h), we have PL h, t) Ah /2 ) c 8 δ) exp Hence A2 2 + δ) ). P Y α t + h) Y α t) > λh α/2 ) ) c 9 δ) exp A2 2 + δ) + exp 2α2 λ 2 h 3/2 α 9c 5 α) b 3 2α A )). Putting 3.3) and choosing A such that the two exponents above should be equal, we get 3.). Proof of Theorem.. Follows from Csáki and Csörgő 992, Theorems 3. and 3.2) and Lemma Proof of Theorem.2 For brevity, we write throughout the section ht) = t ) /2. log log t The proof of Theorem.2 is based on the following Fact 4. Donsker and Varadhan 977). Let L t x) = t log log t) /2 L x ht), t), x R, and let A denote the space of all uniformly continuous subprobability densities on R, endowed with the topology of uniform convergence on bounded intervals. Then for any continuous functional Φ on A, 4.) Φ L t ) = sup Φf), a.s., t f A, If)

11 where If) = 8 R f y)) 2 fy) dy. Proof of Theorem.2. Let δ, /2). Observe that Y α t) = δht) ht)/δ + + ) Lx, t) L x, t) δht) ht)/δ = δht) + ) Lx, t) L x, t) ht)/δ +t α/2 log log t) α/2 /δ δ L t x) L t x). For any fixed δ, /2), we can apply 4.) to see that, almost surely, /δ t δ L t x) L t x) /δ fx) f x) = sup. f A, If) δ Let q > and take the function p x q exp rx), if x, fx) = p x q exp r x ), if x <, for some positive constants p, r and r. It is easily checked that if 4.2) p Γq + ) p Γq + ) +, r q+ r q+ then f is a continuous subprobability density function. Furthermore, whenenever 4.3) p Γq) r q + p Γq) r q 8, we have If). Therefore, under 4.2) and 4.3), /δ t δ L t x) L t x) /δ δ p x q α e rx e r x ). By the dominated convergence theorem, we can send r to infinity to see that, for all δ, /2) and all positive p and r and q > satisfying p Γq + ) < r q+ and p Γq) < 8r q, /δ t δ L t x) L t x) /δ δ p x q α e rx, a.s.

12 Since δ can be as small as possible, this leads to: for q >, p Γq + ) < r q+ and p Γq) < 8r q, /δ L t x) L t x) p Γq α + ) lim inf, a.s. δ t δ r q α+ Let first q +, and then send p and r respectively to 8 and 8, to arrive at: lim inf δ /δ t δ L t x) L t x) 2 3α/2 Γ2 α), a.s. Write for brevity 4.4) φ α t) = t α/2 log log t) α/2. If we could furthermore show that almost surely, 4.5) 4.6) lim δ t lim δ t φ α t) φ α t) δht) ht)/δ Lx, t) L x, t) =, Lx, t) L x, t) =, then the proof of Theorem.2 would be completed. 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 = ht)/δ there) and the definition of the function h ). Proof of 4.5). Assume α for the moment. Recall 2.7): 4.7) = Lx, t) L x, t) Ax, W t)) + 2 t [ W s) b] b α W s) α α dw s), with Ax, W t)) 2x for any t > and x >. 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 Λt, b) def = t [ W s) b] b α W s) α α dw s). 2

13 The proof of 4.5) will be completed as soon as we show 4.8) lim δ t Λt, δht)) φ α t) =, a.s. For each b >, t Λt, 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 Λt, δht)) is no longer a local martingale it is even not clear whether it is a semimartingale). So we have to handle it with care. Fix b > for the moment. Then by 2.) and 2.2), there exists a Brownian motion B such that Λt, b) = BUt)) for all t, where 4.9) Ut) = 4 b b α x α ) 2 Lx, t) α 4c 5 α) b 3 2α sup Lx, t), x R L being the local time of B. For the value of the constant c 5 α), cf. 2.3)). Note that both the Brownian motion B and its clock U depend on b. Also, 4.9) holds even in the case α = so from now on we only make the general assumption α < 3/2). We have, for any y > and z >, ) ) P sup Λs, b) > y s t P P sup s t BUs)) > y sup Lx, t) > z x R c exp z2 3t ) ) exp sup u 4c 5 α) b 3 2α z y 2 ) 8c 5 α) b 3 2α z Bu) > y we have used 3.7) and 3.6) in the last inequality). Take z = ty 2 /b 3 2α ) /3 to see that P sup Λs, b) > y s t ) c + 4) exp c α) y 4/3 ). b 2 4α/3 t /3 Now define t n = + δ) n. Applying the Borel Cantelli lemma gives that almost surely for all large n, 4.) sup Λs, δht n )) c 2 α) δ 3/2 α φ α t n ), s t n where φ α ) is as in 4.4). 3, )

14 On the other hand, by 4.7) and the inequality Ax, W t)) 2x, we have, for r [t n, t n ], Λr, δht n )) Λr, δhr)) δhtn ) 2 x α + δhtn ) Lx, r) L x, r) 2 δhr) 2 2 α δht n)) 2 α δhtn ) + sup Lx, t n ) δht n ) x. α For all large n and all x [δht n ), δht n )], we have x α 2δht n )) α. Moreover, since t h t) is decreasing for large t), we have, by the mean value theorem, when n is sufficiently large, Thus x R δht n ) δht n ) δt n t n )h t n ) δt n t n ) 2 = t n log log t n tn. log log tn δhtn) δht n ) δ 2 + δ δ δ 2δ2 α + δ t /2 α/2 n log log t n ) α/2 /2. In view of 3.8), we obtain, almost surely when n, sup Λr, δht n )) Λr, δhr)) r [t n,t n ] tn log log tn 2 2 α δht n)) 2 α + 2δ2 α + δ 2 /2 + o)) φ α t n ). Taking into account of 4.), we have, with probability one, Consequently, n φ α t n ) sup Λr, δhr)) c 2 α) δ 3/2 α + 8/2 δ 2 α. r [t n,t n ] + δ t φ α t) Λt, δht)) c 2α) δ 3/2 α + 8/2 δ 2 α + δ, a.s. Since α < 3/2, this clearly yields 4.8), hence completes the proof of 4.5). 4

15 5. Proof of Theorem.3 Throughout the section, {S k ; k =, 2, } denotes a simple symmetric random walk on Z starting from, with local time ξx, n), and {W t); t } denotes a Wiener process with local time Lx, t). The proof of Theorem.3 is based on a strong invariance principle for local time due to Révész 98). Fact 5. Révész 98). On a suitable probability space we have, for any ε >, when n, sup ξx, n) Lx, n) = on /4+ε ), a.s. x Proof of Theorem.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 >. For convenience, we write γ = α)+. 2 Note that γ < 3/4 α/2 whenever α < 3/2. The reason for which we have introduced γ is that, for several times we shall use the following relations: 5.) t /2+ε = otγ+ε ), t /2+ε i α = otγ+ε ). The usual LIL for random walk cf. e.g. Révész 99, p. 35) says that 2n log log n) /2 max S k =, n k n a.s. Thus, almost surely for all ε > and large t, ξi, t) = whenever i > t /2+ε. Accordingly, 5.2) t k= Sk α [Sk ] = = = t /2+ε t /2+ε ξi, t) ξ i, t) i α ξi, t) ξ i, t) i α ξi, t) Li, t)) ξ i, t)) L i, t)) i α t /2+ε Li, t) L i, t) +. i α 5

16 By Fact 5. and 5.), as t, the first summation term on the right hand side of 5.2) is = t /2+ε ot /4+ε ) i α = ot /4+γ+2ε ), a.s. Since γ < 3/4 α/2, we can choose ε sufficiently small so that /4 + γ + 2ε < α/2 ε. Therefore, the proof of Theorem.3 is reduced to showing the following estimate: almost surely, when t goes to infinity, Lx, t) L x, t) t /2+ε Li, t) L i, t) i α = o t α/2 ε), or equivalently since Lx, t) = for all large t and x > t /2+ε ), 5.3) It) = o t α/2 ε), a.s., where It) = +t /2+ε Lx, t) L x, t) t /2+ε Li, t) L i, t) i α. We have written + t /2+ε instead of t /2+ε in order to simply writings later). Observe that It) = t /2+ε i+ + i t /2+ε + Lx, t) L x, t) Li, t) + L i, t) Lx, t) L x, t) def = I t) + I 2 t) + I 3 t). According to Bass and Griffin 985), Li, t) L i, t)) i+ i x ) α i α sup x y Lx, t) Ly, t) = ot /4+ε ), a.s., 6

17 as t. Therefore, I t) o t /4+ε) t /2+ε i+ i = o t /4+ε) t /2+ε = o t /4+γ+2ε), a.s., where we have used 5.) 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 /4+ε), a.s. Since γ < 3/4 α/2 and min/2, α) > α, the proof of 5.3) thus of Theorem.3) will be completed as soon as we prove the following estimate: for any < ν < min/2, α), 5.4) I 3 t) = ot ν)/2+ε ), a.s. To check 5.4), let us note that i+ i i ) α, c 3α) ), iα+ for some constant c 3 α) <. By 2.5), for any < ν < min/2, α), sup i Li, t) L i, t) i ν = ot ν)/2+ε ), a.s. Therefore, I 3 t) c 3 α) sup i = c 4 α, ν) sup i Li, t) L i, t) i ν Li, t) L i, t) i ν, i +α ν proving 5.4). 7

18 6. Proof of Theorems.4 and.5 In the case α > 2, the theorems follow from Csáki et al. 992), where for the additive functionals n fs i ) and t gw s)) ds, resp. it was assumed that 6.) x +δ fx) < and 6.2) x= x +δ gx) <, resp., for some δ >. 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.) and 6.2), resp. can be replaced by the weaker conditions 6.3) and 6.4) x= x /2+δ fx) < x /2+δ gx) <, resp. for some δ >. In order to see this we have to show that 6.3) implies 6.5) ρ E fs i ) 2+δ <, 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. 992), that 6.5) is sufficient for the conclusion to hold. To prove 6.5), we use Eξx, ρ)) m c 5 m) + x ) m, cf. 2.23) in Csáki et al. 992) and the triangle inequality, to obtain writing c 6 for c δ)): ρ E fs i ) 2+δ)/2+δ) fx) Eξx, ρ)) 2+δ) /2+δ) x Z c 6 fx) + x +δ)/2+δ) ) x Z c 6 8 x Z fx) + x /2+δ ) <.

19 Now one can see that 6.3) is satisfied whenever α > 3/2, hence the conclusion of Theorem in Csáki et al. 992) holds, which gives Theorem.4. Similar proof holds for the Wiener case, i.e. for Theorem.5. Acknowledgements We are grateful to Marc Yor for helpful discussions. Cooperation between E. Cs. and Z. S. was supported by the joint French-Hungarian Intergovernmental Grant Balaton grant no. F25/97). References Alili, L. 997). On some hyperbolic principal values of Brownian local times. In: Exponential Functionals and Principal Values Related to Brownian Motion M. Yor, ed.), pp Biblioteca de la Revista Matemática Iberoamericana, Madrid. Barlow, M.T. and Yor, M. 982). Semi-martingale inequalities via the Garsia Rodemich Rumsey lemma, and applications to local times. J. Funct. Anal Bass, R.F. and Griffin, P.S. 985). The most visited site of Brownian motion and simple random walk. Z. Wahrsch. Verw. Gebiete Borodin, A.N. 985). Distribution of the supremum of increments of Brownian local time. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. LOMI ) Csáki, E. 989). An integral test for the supremum of Wiener local time. Probab. Th. Rel. Fields Csáki, E. and Csörgő, M. 992). Inequalities for increments of stochastic processes and moduli of continuity. Ann. Probab Csáki, E., Csörgő, M., Földes, A. and Révész, P. 992). Strong approximation of additive functionals. J. Theoretical Probab Donsker, M.D. and Varadhan, S.R.S. 977). On laws of the iterated logarithm for local times. Comm. Pure Appl. Math

20 Hu, Y. and Shi, Z. 997). 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 Biblioteca de la Revista Matemática Iberoamericana, Madrid. Kesten, H. 965). An iterated logarithm law for the local time. Duke Math. J McKean, H.P. 962). A Hölder condition for Brownian local time. J. Math. Kyoto Univ Révész, P. 98). Local time and invariance. In: Analytical Methods in Probability Theory D. Dugué et al., eds.). Lecture Notes in Mathematics 86, pp Springer, Berlin. Révész, P. 99). Random Walk in Random and Non Random Environments. World Scientific, Singapore. Revuz, D. and Yor, M. 994). Continuous Martingales and Brownian Motion. Springer, Berlin, 2nd edition. Yor, M., editor 997). Exponential Functionals and Principal Values Related to Brownian Motion. Biblioteca de la Revista Matemática Iberoamericana, Madrid. 2