scholarly journals Bridges and random truncations of random matrices

2014 ◽  
Vol 03 (02) ◽  
pp. 1450006 ◽  
Author(s):  
Vincent Beffara ◽  
Catherine Donati-Martin ◽  
Alain Rouault
Keyword(s):  
Gaussian Process ◽  
Random Matrices ◽  
Brownian Bridge ◽  
Two Parameter ◽  
The Way

Let U be a Haar distributed matrix in 𝕌(n) or 𝕆(n). In a previous paper, we proved that after centering, the two-parameter process [Formula: see text] converges in distribution to the bivariate tied-down Brownian bridge. In the present paper, we replace the deterministic truncation of U by a random one, in which each row (respectively, column) is chosen with probability s (respectively, t) independently. We prove that the corresponding two-parameter process, after centering and normalization by n-1/2 converges to a Gaussian process. On the way we meet other interesting convergences.

scholarly journals Random permutations and their discrepancy process

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Guillaume Chapuy
Keyword(s):  
Computational Biology ◽  
Symmetric Group ◽  
Gaussian Process ◽  
Brownian Bridge ◽  
Random Permutation ◽  
Partial Sums ◽  
Random Permutations ◽  
New Process ◽  
International Audience

International audience Let $\sigma$ be a random permutation chosen uniformly over the symmetric group $\mathfrak{S}_n$. We study a new "process-valued" statistic of $\sigma$, which appears in the domain of computational biology to construct tests of similarity between ordered lists of genes. More precisely, we consider the following "partial sums": $Y^{(n)}_{p,q} = \mathrm{card} \{1 \leq i \leq p : \sigma_i \leq q \}$ for $0 \leq p,q \leq n$. We show that a suitable normalization of $Y^{(n)}$ converges weakly to a bivariate tied down brownian bridge on $[0,1]^2$, i.e. a continuous centered gaussian process $X^{\infty}_{s,t}$ of covariance: $\mathbb{E}[X^{\infty}_{s,t}X^{\infty}_{s',t'}] = (min(s,s')-ss')(min(t,t')-tt')$.

Some lower bounds for the distribution of the supremum of the Yeh-Wiener process over a rectangular region

1975 ◽  
Vol 12 (04) ◽  
pp. 824-830
Author(s):  
Arthur H. C. Chan
Keyword(s):  
Lower Bound ◽  
Gaussian Process ◽  
Wiener Process ◽  
Lower Bounds ◽  
Upper Bound ◽  
Exact Distribution ◽  
Rectangular Region ◽  
Process With Independent Increments ◽  
Independent Increments ◽  
Two Parameter

Let W (s, t), s, t ≧ 0, be the two-parameter Yeh–Wiener process defined on the first quadrant of the plane, that is, a Gaussian process with independent increments in both directions. In this paper, a lower bound for the distribution of the supremum of W (s, t) over a rectangular region [0, S]×[0, T], for S, T > 0, is given. An upper bound for the same was known earlier, while its exact distribution is still unknown.

A gaussian process with parabolic covariances

1991 ◽  
Vol 28 (04) ◽  
pp. 898-902 ◽  
Author(s):  
Enrique M. Cabaña
Keyword(s):  
Gaussian Process ◽  
Brownian Bridge ◽  
Stationary Gaussian Process ◽  
String Equation ◽  
Distribution Of The Maximum ◽  
Close Relationship ◽  
Vibrating String

The centred, periodic, stationary Gaussian process X(z), ≧ z ≧ 1 with covariances , appears when one studies the solutions of the vibrating string equation forced by noise, corresponding to the case of a finite string with the extremes tied together. The close relationship between this process and a Brownian bridge permits us to compute the distribution of the maximum excursion of the string at particular times.

scholarly journals TRUNCATIONS OF HAAR DISTRIBUTED MATRICES, TRACES AND BIVARIATE BROWNIAN BRIDGES

2012 ◽  
Vol 01 (01) ◽  
pp. 1150007 ◽  
Author(s):  
CATHERINE DONATI-MARTIN ◽  
ALAIN ROUAULT
Keyword(s):  
Brownian Bridge ◽  
Brownian Bridges ◽  
Two Parameter

Let U be a Haar distributed matrix in 𝕌(n) or 𝕆(n). We show that after centering the two-parameter process [Formula: see text] converges in distribution to the bivariate tied-down Brownian bridge.

Some lower bounds for the distribution of the supremum of the Yeh-Wiener process over a rectangular region

1975 ◽  
Vol 12 (4) ◽  
pp. 824-830 ◽  
Author(s):  
Arthur H. C. Chan
Keyword(s):  
Lower Bound ◽  
Gaussian Process ◽  
Wiener Process ◽  
Lower Bounds ◽  
Upper Bound ◽  
Exact Distribution ◽  
Rectangular Region ◽  
Process With Independent Increments ◽  
Independent Increments ◽  
Two Parameter

Let W (s, t), s, t ≧ 0, be the two-parameter Yeh–Wiener process defined on the first quadrant of the plane, that is, a Gaussian process with independent increments in both directions. In this paper, a lower bound for the distribution of the supremum of W (s, t) over a rectangular region [0, S]×[0, T], for S, T > 0, is given. An upper bound for the same was known earlier, while its exact distribution is still unknown.

The maximum of a Gaussian process whose mean path has a maximum, with an application to the strength of bundles of fibres

1989 ◽  
Vol 21 (02) ◽  
pp. 315-333 ◽  
Author(s):  
H. E. Daniels
Keyword(s):  
Brownian Motion ◽  
Gaussian Process ◽  
Gaussian Processes ◽  
Joint Distribution ◽  
Covariance Function ◽  
Brownian Bridge ◽  
Breaking Load ◽  
Plastic Yield ◽  
Analogous Process ◽  
The Mean

Daniels and Skyrme (1985) derived the joint distribution of the maximum, and the time at which it is attained, of a Brownian path superimposed on a parabolic curve near its maximum. In the present paper the results are extended to include Gaussian processes which behave locally like Brownian motion, or a process transformable to it, near the maximum of the mean path. This enables a wider class of practical problems to be dealt with. The results are used to obtain the asymptotic distribution of breaking load and extension of a bundle of fibres which can admit random slack or plastic yield, as suggested by Phoenix and Taylor (1973). Simulations confirm the approximations reasonably well. The method requires consideration not only of a Brownian bridge but also of an analogous process with covariance function t 1(1 + t 2), .

The maximum of a Gaussian process whose mean path has a maximum, with an application to the strength of bundles of fibres

1989 ◽  
Vol 21 (2) ◽  
pp. 315-333 ◽  
Author(s):  
H. E. Daniels
Keyword(s):  
Brownian Motion ◽  
Gaussian Process ◽  
Gaussian Processes ◽  
Joint Distribution ◽  
Covariance Function ◽  
Brownian Bridge ◽  
Breaking Load ◽  
Plastic Yield ◽  
Analogous Process ◽  
The Mean

Daniels and Skyrme (1985) derived the joint distribution of the maximum, and the time at which it is attained, of a Brownian path superimposed on a parabolic curve near its maximum. In the present paper the results are extended to include Gaussian processes which behave locally like Brownian motion, or a process transformable to it, near the maximum of the mean path. This enables a wider class of practical problems to be dealt with. The results are used to obtain the asymptotic distribution of breaking load and extension of a bundle of fibres which can admit random slack or plastic yield, as suggested by Phoenix and Taylor (1973). Simulations confirm the approximations reasonably well. The method requires consideration not only of a Brownian bridge but also of an analogous process with covariance function t1(1 + t2), .

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