Abstract:
This paper is concerned with various aspects of the Slepian process $(B_{t+1} - B_t, t \ge 0)$ derived from a one-dimensional Brownian motion $(B_t, t \ge 0 )$. In particular, we offer an analysis of the local structure of the Slepian zero set $\{t : B_{t+1} = B_t \}$, including a path decomposition of the Slepian process for $0 \le t \le 1$. We also establish the existence of a random time $T$ such that $T$ falls in the the Slepian zero set almost surely and the process $(B_{T+u} - B_T, 0 \le u \le 1)$ is standard Brownian bridge.

Abstract:
This paper develops a recursion formula for the conditional moments of the area under the absolute value of Brownian bridge given the local time at 0. The method of power series leads to a Hermite equation for the generating function of the coefficients which is solved in terms of the parabolic cylinder functions. By integrating out the local time variable, this leads to an integral expression for the joint moments of the areas under the positive and negative parts of the Brownian bridge.

Abstract:
Aldous and Pitman (1994) studied asymptotic distributions, as n tends to infinity, of various functionals of a uniform random mapping of a set of n elements, by constructing a mapping-walk and showing these mapping-walks converge weakly to a reflecting Brownian bridge. Two different ways to encode a mapping as a walk lead to two different decompositions of the Brownian bridge, each defined by cutting the path of the bridge at an increasing sequence of recursively defined random times in the zero set of the bridge. The random mapping asymptotics entail some remarkable identities involving the random occupation measures of the bridge fragments defined by these decompositions. We derive various extensions of these identities for Brownian and Bessel bridges, and characterize the distributions of various path fragments involved, using the theory of Poisson processes of excursions for a self-similar Markov process whose zero set is the range of a stable subordinator of index between 0 and 1.

Abstract:
The standard functional central limit theorem for a renewal process with finite mean and variance, results in a Brownian motion limit. This note shows how to obtain a Brownian bridge process by a direct procedure that does not involve conditioning. Several examples are also considered.

Abstract:
We study the problem of stopping an $\alpha$-Brownian bridge as close as possible to its global maximum. This extends earlier results found for the Brownian bridge (the case $\alpha=1$). The exact behavior for $\alpha$ close to $0$ is investigated.

Abstract:
We discuss the distributions of three functionals of the free Brownian bridge: its $\L^2$-norm, the second component of its signature and its L\'evy area. All of these are freely infinitely divisible. We introduce two representations of the free Brownian bridge as series of free semicircular random variables are used, analogous to the Fourier representations of the classical Brownian bridge due to \ts{L\'evy} and \ts{Kac}.

Abstract:
We sharpen a classical result on the spectral asymptotics of the boundary value problems for self-adjoint ordinary differential operator. Using this result we obtain the exact $L_2$-small ball asymptotics for a new class of zero mean Gaussian processes. This class includes, in particular, integrated generalized Slepian process, integrated centered Wiener process and integrated centered Brownian bridge.

Abstract:
We study the law of the minimum of a Brownian bridge, conditioned to take specific values at specific points, and the law of the location of the minimum. They are used to compare some non-adaptive optimisation algorithms for black-box functions for which the Brownian bridge is an appropriate probabilistic model and only a few points can be sampled.

Abstract:
We study the sharp large deviation for the energy of -Brownian bridge. The full expansion of the tail probability for energy is obtained by the change of measure. 1. Introduction We consider the following -Brownian bridge: where is a standard Brownian motion, , , and the constant . Let denote the probability distribution of the solution of (1). The -Brownian bridge is first used to study the arbitrage profit associated with a given future contract in the absence of transaction costs by Brennan and Schwartz [1]. -Brownian bridge is a time inhomogeneous diffusion process which has been studied by Barczy and Pap [2, 3], Jiang and Zhao [4], and Zhao and Liu [5]. They studied the central limit theorem and the large deviations for parameter estimators and hypothesis testing problem of -Brownian bridge. While the large deviation is not so helpful in some statistics problems since it only gives a logarithmic equivalent for the deviation probability, Bahadur and Ranga Rao [6] overcame this difficulty by the sharp large deviation principle for the empirical mean. Recently, the sharp large deviation principle is widely used in the study of Gaussian quadratic forms, Ornstein-Uhlenbeck model, and fractional Ornstein-Uhlenbeck (cf. Bercu and Rouault [7], Bercu et al. [8], and Bercu et al. [9, 10]). In this paper we consider the sharp large deviation principle (SLDP) of energy , where Our main results are the following. Theorem 1. Let be the process given by the stochastic differential equation (1). Then satisfies the large deviation principle with speed and good rate function defined by the following: where . Theorem 2. satisfies SLDP; that is, for any , there exists a sequence such that, for any , when approaches enough, where The coefficients may be explicitly computed as functions of the derivatives of and (defined in Lemma 3) at point . For example, is given by with , and . 2. Large Deviation for Energy Given , we first consider the following logarithmic moment generating function of ; that is, And let be the effective domain of . By the same method as in Zhao and Liu [5], we have the following lemma. Lemma 3. Let be the effective domain of the limit of ; then for all , one has with where and . Furthermore, the remainder satisfies Proof. By It？’s formula and Girsanov’s formula (see Jacob and Shiryaev [11]), for all and , Therefore, If , we can choose such that . Then where , and . Therefore, Proof of Theorem 1. From Lemma 3, we have and is steep; by the G？rtner-Ellis theorem (Dembo and Zeitouni [12]), satisfies the large deviation principle with speed and good

Abstract:
Functionals of Brownian bridge arise as limiting distributions in nonparametric statistics. In this paper we will give a derivation of distributions of extrema of the Brownian bridge based on excursion theory for Brownian motion. Only the Poisson character of the excursion process will be used. Particular cases of calculations include the distributions of the Kolmogorov-Smirnov statistic, the Kuiper statistic, and the ratio of the maximum positive ordinate to the minumum negative ordinate.