Bernard Roynette, Pierre Vallois, Marc Yor
We determine the rate of decay of the expectation Z(t) of some multiplicative functional related to Brownian motion up to time t. This permits to prove that the Wiener measure, penalized by this multiplicative functional, converges as t goes to infinity to a probability measure (p.m.) . We obtain the law of the canonical process under this new p.m.
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The Lévy-Ciesielski Construction of Brownian motion is used to determine non-asymptotic estimates for the maximal deviation of increments of a Brownian motion process $(W_{t})_{t\in \left[ 0,T\right] }$ normalized by the global modulus function, for all positive $\varepsilon $ and $δ$. Additionally, uniform results over $δ$ are obtained. Using the same method, non-asymptotic estimates for the distribution function for the standard Brownian motion normalized by its local modulus of continuity are obtained. Similar results for the truncated Brownian motion are provided and play a crucial role in…
Let B be a Brownian motion and T its first hitting time of the level 1. For U a uniform random variable independent of B, we study in depth the distribution of T^{-1/2}B_{UT}, that is the rescaled Brownian motion sampled at uniform time. In particular, we show that this variable is centered.
G-Brownian motion has a very rich and interesting new structure which nontrivially generalizes the classical one. Its quadratic variation process is also a continuous process with independent and stationary increments. We prove a self-normalized functional central limit theorem for independent and identically distributed random variables under the sub-linear expectation with the limit process being a G-Browian motion self-normalized by its quadratic variation. To prove the self-normalized central limit theorem, we also establish a new Donsker's invariance principle.
Consider $q_n$ a random pointed quadrangulation chosen equally likely among the pointed quadrangulations with $n$ faces. In this paper we show that, when $n$ goes to $+\infty$, $q_n$ suitably normalized converges weakly in a certain sense to a random limit object, which is continuous and compact, and that we name the Brownian map. The same result is shown for a model of rooted quadrangulations and for some models of rooted quadrangulations with random edge lengths. A metric space of rooted (resp. pointed) abstract maps that contains the model of discrete rooted (resp. pointed) quadrangulations…
Let $\{u(t\,, x)\}_{(t, x)\in \mathbb{R}_+\times \mathbb{R}}$ be the density of one-dimensional super-Brownian motion starting from Lebesgue measure. Using the Laplace functional of super-Brownian motion, we prove that as $N\to \infty$, the normalized spatial integral $N^{-1/2}\int_0^{xN}[u(t\,, z)-1 ]\rm{d} z$ converges jointly in $(t, x)$ to Brownian sheet in distribution.
In this paper we prove exact forms of large deviations for local times and intersection local times of fractional Brownian motions and Riemann-Liouville processes. We also show that a fractional Brownian motion and the related Riemann-Liouville process behave like constant multiples of each other with regard to large deviations for their local and intersection local times. As a consequence of our large deviation estimates, we derive laws of iterated logarithm for the corresponding local times. The key points of our methods: (1) logarithmic superadditivity of a normalized sequence of moments of…
Distribution of particles near the front in supercritical branching Brownian motion with compactly supported branching
研究超临界分支布朗运动中前沿附近粒子分布的长期行为，利用抛物型PDE解的渐进行为证明粒子数收敛性及极限分布依赖域位置。
We show that on a manifold whose Riemannian metric evolves under backwards Ricci flow two Brownian motions can be coupled in such a way that the expectation of their normalized L-distance is non-increasing. As an immediate corollary we obtain a new proof of a recent result of Topping (J. reine angew. Math. 636 (2009), 93-122), namely that the normalized L-transportation cost between two solutions of the heat equation is non-increasing as well.
We consider an Ornstein-Uhleneck (OU) process associated to self-normalised sums in i.i.d. symmetric random variables from the domain of attraction of $N(0, 1)$ distribution. We proved the self-normalised sums converge to the OU process (in $C[0, \infty)$). Importance of this is that the OU process is a stationary process as opposed to the Brownian motion, which is a non-stationary distribution (see for example, the invariance principle proved by Csorgo et al (2003, Ann Probab) for self-normalised sums that converges to Brownian motion). The proof uses recursive equations similar to those that…
We consider a random walk $S$ in the domain of attraction of a standard normal law $Z$, \textit{ie} there exists a positive sequence $a_n$ such that $S_n/a_n$ converges in law towards $Z$. The main result of this note is that the rescaled process $(S_{\lfloor nt \rfloor}/a_n, t \geq 0)$ conditioned to stay non-negative, to start and to come back \textit{near the origin} converges in law towards the normalized brownian excursion.
In 2005, Nualart and Peccati showed the so-called Fourth Moment Theorem asserting that, for a sequence of normalized multiple Wiener-Itô integrals to converge to the standard Gaussian law, it is necessary and sufficient that its fourth moment tends to 3. A few years later, Kemp et al. extended this theorem to a sequence of normalized multiple Wigner integrals, in the context of the free Brownian motion. The q-Brownian motion, q in (-1,1], introduced by the physicists Frisch and Bourret in 1970 and mathematically studied by Bozejko and Speicher in 1991, interpolates between the classical Browni…
We study some limit theorems for the normalized law of integrated Brownian motion perturbed by several examples of functionals: the first passage time, the nth passage time, the last passage time up to a finite horizon and the supremum. We show that the penalization principle holds in all these cases and give descriptions of the conditioned processes. In particular, it is remarkable that the penalization by the nth passage time is independent of n, and always gives the same conditioned process, i.e. integrated Brownian motion conditioned not to hit 0. Our results rely on some explicit formulae…
In a recent paper by Gorin and Shkolnikov (2016), they have found, as a corollary to their result relevant to random matrix theory, that the area below a normalized Brownian excursion minus one half of the integral of the square of its total local time, is identical in law with a centered Gaussian random variable with variance $1/12$. In this note, we give a pathwise interpretation to their identity; Jeulin's identity connecting normalized Brownian excursion and its local time plays an essential role in the exposition.
It is well known that, under suitable regularity conditions, the normalized fractional process with fractional parameter $d$ converges weakly to fractional Brownian motion for $d>1/2$. We show that, for any non-negative integer $M$, derivatives of order $m=0,1,\dots,M$ of the normalized fractional process with respect to the fractional parameter $d$, jointly converge weakly to the corresponding derivatives of fractional Brownian motion. As an illustration we apply the results to the asymptotic distribution of the score vectors in the multifractional vector autoregressive model.
We extend the Poincaré--Borel lemma to a weak approximation of a Brownian motion via simple functionals of uniform distributions on n-spheres in the Skorokhod space $D([0,1])$. This approach is used to simplify the proof of the self-normalized Donsker theorem in Csörgő et al. (2003). Some notes on spheres with respect to $\ell_p$-norms are given.
We study the transformed path measure arising from the self-interaction of a three-dimensional Brownian motion via an exponential tilt with the Coulomb energy of the occupation measures of the motion by time $t$. The logarithmic asymptotics of the partition function were identified in the 1980s by Donsker and Varadhan [DV83-P] in terms of a variational formula. Recently [MV14] a new technique for studying the path measure itself was introduced, which allows for proving that the normalized occupation measure asymptotically concentrates around the set of all maximizers of the formula. In the pre…
We provide a new construction of Brownian disks in terms of forests of continuous random trees equipped with nonnegative labels corresponding to distances from a distinguished point uniformly distributed on the boundary of the disk. This construction shows in particular that distances from the distinguished point evolve along the boundary as a five-dimensional Bessel bridge. As an important ingredient of our proofs, we show that the uniform measure on the boundary, as defined in the earlier work of Bettinelli and Miermont, is the limit of the suitably normalized volume measure on a small tubul…
Consider the intersection measure $\ell^{\mathrm{IS}}_t$ of $p$ independent Brownian motions on $\mathbb{R}^d$. In this article, we prove the large deviation principle for the normalized intersection measure $t^{-p}\ell^{\mathrm{IS}}_t$ as $t\rightarrow \infty$, before exiting a (possibly unbounded) domain $D\subset\mathbb{R}^d$ with smooth boundary. This is an extension of [W. König and C. Mukherjee: Communications on Pure and Applied Mathematics, 66(2):263--306, 2013] which deals with the case $D$ is bounded. Our essential contribution is to prove the so-called super-exponential estimate for…
Asymptotic Properties of the Derivative of Self-Intersection Local Time of Multidimensional Fractional Brownian Motion
针对d维分数布朗运动，证明自相交局部时一阶导数近似在L²意义下存在的充要条件为H<3/[2(1+d)]，并给出不同H范围下该近似归一化后的中心极限定理。
We present a further analysis of the fragmentation at heights of the normalized Brownian excursion. Specifically we study a representation for the mass of a tagged fragment in terms of a Doob transformation of the 1/2-stable subordinator and use it to study its jumps; this accounts for a description of how a typical fragment falls apart. These results carry over to the height fragmentation of the stable tree. Additionally, the sizes of the fragments in the Brownian fragmentation when it is about to reduce to dust are described in a limit theorem.
We provide a new construction of the Brownian disks, which have been defined by Bettinelli and Miermont as scaling limits of quadrangulations with a boundary when the boundary size tends to infinity. Our method is very similar to the construction of the Brownian map, but it makes use of the positive excursion measure of the Brownian snake which has been introduced recently. This excursion measure involves a random continuous tree whose vertices are assigned nonnegative labels, which correspond to distances from the boundary in our approach to the Brownian disk. We provide several applications …
The Brownian loop measure is a conformally invariant measure on loops in the plane that arises when studying the Schramm-Loewner evolution (SLE). When an SLE curve in a domain evolves from an interior point, it is natural to consider the loops that hit the curve and leave the domain, but their measure is infinite. We show that there is a related normalized quantity that is finite and invariant under Möbius transformations of the plane. We estimate this quantity when the curve is small and the domain simply connected. We then use this estimate to prove a formula for the Radon-Nikodym derivative…
In this paper we study a random walk on an affine building of type $\tilde{A}_r$, whose radial part, when suitably normalized, converges to the Brownian motion of the Weyl chamber. This gives a new discrete approximation of this process, alternative to the one of Biane \cite{Bia2}. This extends also the link at the probabilistic level between Riemannian symmetric spaces of the noncompact type and their discrete counterpart, which had been previously discovered by Bougerol and Jeulin in rank one \cite{BJ}. The main ingredients of the proof are a combinatorial formula on the building and the est…
Concentration of the truncated variation of fractional Brownian motions of any Hurst index, their $1/H$-variations and local times
本文研究了分数布朗运动任意Hurst指数的截断变化集中性，推导了偏离概率界，并建立了轨迹局部时间的弱收敛性。
In the paper we introduce the truncated variation, upward truncated variation and downward truncated variation. These are closely related to the total variation but are well-defined even if the latter is infinite. Our aim is to explore their feasibility to studies of stochastic processes. We concentrate on a Brownian motion with drift for which we prove the convergence of the above- mentioned quantities. For example, we study the truncated variation when the truncation parameter c tends to 0. We prove in this case that for "small" c's it is well-approximated by a deterministic process. Moreove…
Rough volatility models are becoming increasingly popular in quantitative finance. In this framework, one considers that the behavior of the log-volatility process of a financial asset is close to that of a fractional Brownian motion with Hurst parameter around 0.1. Motivated by this, we wish to define a natural and relevant limit for the fractional Brownian motion when $H$ goes to zero. We show that once properly normalized, the fractional Brownian motion converges to a Gaussian random distribution which is very close to a log-correlated random field.
We consider a (one-dimensional) branching Brownian motion process with a general offspring distribution having at least two moments, and in which all particles have a drift towards the origin where they are immediately absorbed. It is well-known that if and only if the branching rate is sufficiently large, then the population survives forever with positive probability. We show that throughout this super-critical regime, the number of particles inside any fixed set normalized by the mean population size converges to an explicit limit, almost surely and in $L^1$. As a consequence, we get that al…
The convergence of properly time-scaled and normalized maxima of independent standard Brownian motions to the Brown-Resnick process is well-known in the literature. In this paper, we study the extremal functional behavior of non-Gaussian processes, namely squared Bessel processes and scalar products of Brownian motions. It is shown that maxima of independent samples of those processes converge weakly on the space of continuous functions to the Brown-Resnick process.
In this note - starting from $d$-dimensional (with $d>1$) fuzzy vectors - we prove Donsker's classical invariance principle. We consider a fuzzy random walk ${S^*_n}=X^*_1+\cdots+X^*_n,$ where $\{X^*_i\}_1^{\infty}$ is a sequence of mutually independent and identically distributed $d$-dimensional fuzzy random variables whose $α$-cuts are assumed to be compact and convex. Our reasoning and technique are based on the well known conjugacy correspondence between convex sets and support functions, which allows for the association of an appropriately normalized and interpolated time-continuous fuzzy…
We propose a new algorithm to generate a fractional Brownian motion, with a given Hurst parameter, 1/2<H<1 using the correlated Bernoulli random variables with parameter p; having a certain density. This density is constructed using the link between the correlation of multivariate Gaussian random variables and the correlation of their dichotomized binary variables and the relation between the correlation coefficient and the persistence parameter. We prove that the normalized sum of trajectories of this proposed random walk yields a Gaussian process whose scaling limit is the desired fractional…
This paper establishes a functional law of large numbers and a functional central limit theorem for marked Hawkes point measures and their corresponding shot noise processes. We prove that the normalized random measure can be approximated in distribution by the sum of a Gaussian white noise process plus an appropriate lifting map of a correlated one-dimensional Brownian motion. The Brownian results from the self-exiting arrivals of events. We apply our limit theorems for Hawkes point measures to analyze the population dynamics of budding microbes in a host.
We study the transition density of a standard two-dimensional Brownian motion killed when hitting a bounded Borel set $A$. We derive the asymptotic form of the density, say $p^A_t({\bf x},{\bf y})$, for large times $t$ and for ${\bf x}$ and ${\bf y}$ in the exterior of $A$ valid uniformly under the constraint $|{\bf x}|\vee |{\bf y}| =O(t)$. Within the parabolic regime $|{\bf x}|\vee |{\bf y}| = O(\sqrt t)$ in particular $p^A_t({\bf x},{\bf y})$ is shown to behave like $4e_A({\bf x})e_A({\bf y}) (\lg t)^{-2} p_t({\bf y}-{\bf x})$ for large $t$, where $p_t({\bf y}-{\bf x})$ is the transition ke…
In this article, we consider the so-called modified Euler scheme for stochastic differential equations (SDEs) driven by fractional Brownian motions (fBm) with Hurst parameter $\frac13<H<\frac12$. This is a first-order time-discrete numerical approximation scheme, and has been recently introduced by Hu, Liu and Nualart in order to generalize the classical Euler scheme for Itô SDEs to the case $H>\frac12$. The current contribution generalizes the modified Euler scheme to the rough case $\frac13<H<\frac12$. Namely, we show a convergence rate of order $n^{\frac12-2H}$ for the scheme, and we argue …
Local times of deterministic paths and self-similar processes with stationary increments as normalized numbers of interval crossings
本文建立了càdlàg路径区间穿越规范化次数极限与占据测度的一般关系，并据此将分数布朗运动局部时定义为规范化穿越次数的弱极限，同时讨论了半鞅及Rosenblatt过程的情形。
In this paper we show that the continuous version of the self normalised process $Y_{n,p}(t)= S_n(t)/V_{n,p}+(nt-[nt])X_{[nt]+1}/V_{n,p}$ where $S_n(t)=\sum_{i=1}^{[nt]} X_i$ and $V_{(n,p)}= \sum_{i=1}^{n}|X_i|^p)^{\frac{1}{p}}$ and $X_i$ i.i.d. random variables belong to $DA(α)$, has a non trivial distribution iff $p=α=2$. The case for $2 > p > α$ and $p \le α< 2$ is systematically eliminated by showing that either of tightness or finite dimensional convergence to a non-degenerate limiting distribution does not hold. This work is an extension of the work by Csörgö et al. who showed Donsker's …
In this paper we derive weak limits for the discretization errors of sampling barrier-hitting and extreme events of Brownian motion by using the Euler discretization simulation method. Specifically, we consider the Euler discretization approximation of Brownian motion to sample barrier-hitting events, i.e. hitting for the first time a deterministic "barrier" function; and to sample extreme events, i.e. attaining a minimum on a given compact time interval or unbounded closed time interval. For each case we study the discretization error between the actual time the event occurs versus the time t…
Central limit theorem and Cramér-type moderate deviations for Milstein scheme
本文针对乘性布朗运动驱动的随机微分方程的Milstein数值格式，证明其不变测度经验估计量的中心极限定理，并推导两类Cramér型中偏差结果。
Parameter estimation of integrated fractional Brownian motion
本文提出归一化积分分数布朗运动（nifBm）以刻画平均化长记忆数据，推导其协方差与遍历性质，并为单 nifBm 及双 nifBm 线性组合模型构建了强一致且渐近正态的参数估计量。
A fixed-time stable limit theorem for non-normalized functionals on irregular time grids
本文证明了在非正则时间网格上观测的Itô半鞅的非规范化泛函的固定时间稳定收敛定理，给出了极限的跳跃贡献形式，并指出仅靠正性和右连左极正则性不足以替代对倒数采样强度的局部控制。
We present some limit theorems for the normalized laws (with respect to functionals involving last passage times at a given level up to time t) of a large class of null recurrent diffusions. Our results rely on hypotheses on the Lévy measure of the diffusion inverse local time at 0. As a special case, we recover some of the penalization results obtained by Najnudel, Roynette and Yor in the (reflected) Brownian setting.
The purpose of this paper is to construct a Brownian motion $X := (X_t)_{t\geq 0}$ taking values in a Riemannian manifold $M$, together with a compact valued process $D:= (D_t)_{t\geq 0}$ such that, at least for small enough ${\mathscr F}^D$-stopping time $τ> 0$ and conditioned by ${\mathscr F}_τ^D$, the law of $X_τ$ is the normalized Lebesgue measure on $D_τ$. This intertwining result is a generalization of Pitman theorem. We first construct regular intertwined processes related to Stokes' theorem. Then using several limiting procedures we construct synchronous intertwined, free intertwined, …
Rates of convergence to the local time of Oscillating and Skew Brownian Motions
本文基于高频观测研究振荡与偏斜布朗运动统计量向局部时间的收敛速率，并将其应用于偏斜参数估计和扩散系数估计，首次给出了偏斜参数估计量的收敛速率。
For a given normalized Gaussian symmetric matrix-valued process $Y^{(n)}$, we consider the process of its eigenvalues $\{(λ_{1}^{(n)}(t),\dots, λ_{n}^{(n)}(t)); t\ge 0\}$ as well as its corresponding process of empirical spectral measures $μ^{(n)}=(μ_{t}^{(n)}; t\geq0)$. Under some mild conditions on the covariance function associated to $Y^{(n)}$, we prove that the process $μ^{(n)}$ converges in probability to a deterministic limit $μ$, in the topology of uniform convergence over compact sets. We show that the process $μ$ is characterized by its Cauchy transform, which is a rescaling of the s…
Fluctuations of the additive martingales related to super-Brownian motion
本文研究超临界超布朗运动相关加性鞅收敛到极限的波动速率，证明规范化尾鞅的极限律在小参数时由分支机制在0附近的行为决定，在大参数时由极限极值过程决定。
Bounds for expected supremum of fractional Brownian motion with drift
本文针对带漂移的分数布朗运动，给出了其期望上确界的上下界，区分Hurst参数区间，并提出数值程序降低上界，同时为区间[0,1]上的上确界均值提供紧上界。
Consider the supercritical branching random walk on the real line in the boundary case and the associated Gibbs measure $ν_{n,β}$ on the $n^\text{th}$ generation, which is also the polymer measure on a disordered tree with inverse temperature $β$. The convergence of the partition function $W_{n,β}$, after rescaling, towards a nontrivial limit has been proved by A\"ıdékon and Shi in the critical case $β= 1$ and by Madaule when $β>1$. We study here the near-critical case, where $β_n \to 1$, and prove the convergence of $W_{n,β_n}$, after rescaling, towards a constant multiple of the limit of the…
First-passage time of a Brownian motion: two unexpected journeys
本文发现并研究了两类新的扩散过程，它们在负漂移条件下与带漂移布朗运动具有相同的首达时间分布，揭示了首达时间分布无法唯一确定底层扩散过程。
We study the asymptotic behaviour of a properly normalized time changed Wiener processes. The time change reflects the fact that we consider the Laplace operator (which generates a Wiener process) multiplied by a possibly degenerate state-space dependent intensity $λ(x)$. Applying a functional limit theorem for the superposition of stochastic processes, we prove functional limit theorems for the normalized time changed Wiener process. The normalization depends on the asymptotic behaviour of the intensity function $λ$. One of the possible limits is a skew Brownian motion.
Functional limit theorems for a time-changed multidimensional Wiener process
本文研究强度函数在各卦限无穷远有不同极限的时间变换多维Wiener过程，利用随机过程叠加的泛函极限定理证明其归一化后的泛函极限定理，极限包含斜Brown运动的多维类似物。
Interval-Constrained Brownian Paths: Exact Interpolation and Extrapolation
本文研究区间约束布朗运动与布朗桥的精确插值和外推，提出统一采样套件，含边界端点并自动切换密度表示，提升接受率，并扩展至多时刻骨架路径模拟。
Matrix Random Walks and the Lima Bean Law
本文研究矩阵随机游走在维数与步数趋于无穷时的特征值极限分布，证明其经验特征值分布收敛于自由概率极限的布朗测度，并显式计算了支撑于“利马豆”形区域的极限测度及其相变，最终证明布朗运动是该过程的普适极限。
Limit theorems for the normalized laws with respect to two kinds of weight functionals are studied for any symmetric stable Lévy process of index $ 1 < α\le 2 $. The first kind is a function of the local time at the origin, and the second kind is the exponential of an occupation time integral. Special emphasis is put on the role played by a stable Lévy counterpart of the universal $ σ$-finite measure, found in [9] and [10], which unifies the corresponding limit theorems in the Brownian setup for which $ α=2 $.
Motivated by nanoscale growth of ultra-thin films, we study a model of deposition, on an interval substrate, of particles that perform Brownian motions until any two meet, when they nucleate to form a static island, which acts as an absorbing barrier to subsequent particles. This is a continuum version of a lattice model studied in the applied literature. We show that the associated interval-splitting process converges in the sparse deposition limit to a Markovian process (in the vein of Brennan and Durrett) governed by a splitting density with a compact Fourier series expansion but, apparentl…
Stochastic Completeness for Landmark Shape Spaces with Induced Metrics
本文证明了带诱导度量的地标形状空间在一般度量（含退化度量）下是随机完备的，即黎曼布朗运动对所有时间存在，从而排除了初始不同地标碰撞的可能性，为形状随机建模提供了理论基础。
Onsager-Machlup functional for $\text{SLE}_κ$ loop measures
本文建立 Brown 环路测度两种重整化的联系，并据此将 Loewner 能量表述为 $\text{SLE}_κ$ 及同中心荷 MKS 环路测度的 Onsager–Machlup 泛函。
Limit error distributions of Milstein scheme for stochastic Volterra equations with singular kernels
本文证明奇异核随机Volterra方程的Milstein格式收敛速率$n^{-2H}$最优，并揭示归一化误差的极限分布为随机系数线性Volterra方程的非零解。
Consider an intersection measure $\ell_t ^{\mathrm{IS}}$ of $p$ independent (possibly different) $m$-symmetric Hunt processes up to time $t$ in a metric measure space $E$ with a Radon measure $m$. We derive a Donsker-Varadhan type large deviation principle for the normalized intersection measure $t^{-p}\ell_t ^{\mathrm{IS}}$ on the set of finite measures on $E$ as $t \rightarrow \infty$, under the condition that $t$ is smaller than life times of all processes. This extends earlier work by W. König and C. Mukherjee (2013), in which the large deviation principle was established for the intersect…
The Properties of Fractional Gaussian Process and Their Applications
本文研究分数高斯过程的性质，量化其再生核希尔伯特空间与分数布朗运动的关系，证明 Gladyshev 定理成立，并给出由其驱动的遍历分数 Ornstein-Uhlenbeck 过程统计估计的 Berry-Esséen 上界。
The integrated super-Brownian excursion (ISE) is the occupation measure of the spatial component of the head of the Brownian snake with lifetime process the normalized Brownian excursion. It is a random probability measure on $\mathbb{R}$, and it is known to describe the continuum limit of the distribution of labels in various models of random discrete labelled trees. We show that $f_{ISE}$, its (random) density has a.s. a derivative $f'_{ISE}$ which is continuous and $\left(\frac{1}{2}-a\right)$-Hölder for any $a >0$ but for no $a<0$ (proving a conjecture of Bousquet-Mélou and Janson). We con…
Statistically self-similar measures on $[0,1]$ are limit of multiplicative cascades of random weights distributed on the $b$-adic subintervals of $[0,1]$. These weights are i.i.d, positive, and of expectation $1/b$. We extend these cascades naturally by allowing the random weights to take negative values. This yields martingales taking values in the space of continuous functions on $[0,1]$. Specifically, we consider for each $H\in (0,1)$ the martingale $(B_{n})_{n\geq1}$ obtained when the weights take the values $-b^{-H}$ and $b^{-H}$, in order to get $B_n$ converging almost surely uniformly t…
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