Counting subsets of the random partition and the ‘Brownian Bridge’ process

Let Ω_m be the set of partitions, ω, of a finite m-element set; induce a uniform probability distribution on Ω_m, and define X_ms(ω) as the number of s-element subsets in ω. We alow the existence of an integer-valued function n=n^(m)(t), t?[0, 1], and centering constants b_ms, 0?s? m, such that

$\text{Z}{}^{\mathrm{(m)}}\text{(t)=}\text{∑}\text{s=0}\text{n(m)(t)}\text{(X}{}_{\mathrm{ms}}\text{?b}{}_{\mathrm{ms}}\text{)}\text{√}\text{∑}\text{s=0}\text{m}\text{b}{}_{\mathrm{ms}}$

converges to the ‘Brownian Bridge’ process in terms of its finite-dimensional distributions.     

Spectral properties of zeroth-order pseudodifferential operators

Let Q be a self-adjoint, classical, zeroth order pseudodifferential operator on a compact manifold X with a fixed smooth measure dx. We use microlocal techniques to study the spectrum and spectral family, {E_S}_{S∈$\text{R}$} as a bounded operator on L²(X, dx).Using theorems of Weyl (Rend. Circ. Mat. Palermo, 27 (1909), 373–392) and Kato (“Perturbation Theory for Linear Operators,” Springer-Verlag, 1976) on spectra of perturbed operators we observe that the essential spectrum and the absolutely continuous spectrum of Q are determined by a finite number of terms in the symbol expansion. In particular Spec_ESSQ = range(q(x, ξ)) where q is the principal symbol of Q. Turning the attention to the spectral family {E_S}_{S∈$\text{R}$}, it is shown that if $\text{dE}\text{ds}$ is considered as a distribution on $\text{R}$×X×X it is in fact a Lagrangian distribution near the set $\text{{σ=0}?}\text{T}{}^1\text{(}\text{R}\text{×X×X)}\text{0}$ where (s, x, y, σ, ξ,η) are coordinates on T¹($\text{R}$×X×X) induced by the coordinates (s, x, y) on $\text{R}$×X×X. This leads to an easy proof that$\text{?(Q)}$ is a pseudodifferential operator if ?∈C^∞($\text{R}$) and to some results on the microlocal character of E_s. Finally, a look at the wavefront set of $\text{dE}\text{ds}$ leads to a conjecture about the existence of absolutely continuous spectrum in terms of a condition on q(x, ξ).     

General theorems on rates of convergence in distribution of random variables I. General limit theorems

Let (X_n)_n?$\text{N}$ be a sequence of real, independent, not necessarily identically distributed random variables (r.v.) with distribution functions F_Xn, and S_n = Σ_i=1ⁿX_i. The authors present limit theorems together with convergence rates for the normalized sums ?(n)S_n, where ?: $\text{N}$ → $\text{R}$⁺, ?(n) → 0, n → ∞, towards appropriate limiting r.v. X, the convergence being taken in the weak (star) sense. Thus higher order estimates are given for the expression ∝_$\text{R}$f(x) d[F_?(n)Sn(x) ? F_X(x)] which depend upon the normalizing function ?, decomposability properties of X and smoothness properties of the function f under consideration. The general theorems of this unified approach subsume O- and o-higher order error estimates based upon assumptions on associated moments. These results are also extended to multi-dimensional random vectors.     

A remark on the tail probability of a distribution

Let {X_n}_n≥1 be a sequence of independent and identically distributed random variables. For each integer n ≥ 1 and positive constants r, t, and ?, let S_n = Σ_j=1ⁿX_j and $\text{E{N}{}_{\mathrm{\infty }}\text{(r, t, ?)} = Σ}{}_{\mathrm{n=1}}{}^{\mathrm{\infty }}\text{n}{}^{\mathrm{r?2}}\text{P{|S}{}_{\mathrm{n}}\text{| > ?n}{}^{\mathrm{<mtext>r</mtext><mtext>t</mtext>}}\text{}}$. In this paper, we prove that (1) lim_?→0+?^α(r?1)E{N_∞(r, t, ?)} = K(r, t) if E(X₁) = 0, Var(X₁) = 1, and E(| X₁ |^t) < ∞, where 2 ≤ t < 2r ≤ 2t, $\text{K(r, t) =}\text{{2}{}^{\mathrm{<mtext>\alpha (r?1)</mtext><mtext>2</mtext>}}\text{Γ(}\text{(1 + α(r ? 1))}\text{2}\text{)}}\text{{(r ? 1) Γ(}\text{1}\text{2}\text{)}}$, and $\text{α =}\text{2t}\text{(2r ? t)}$; (2) $\text{lim}{}_{\mathrm{?\to 0+}}\text{G(t, ?)}\text{H(t, ?)}\text{= 0}$ if 2 < t < 4, E(X₁) = 0, Var(X₁) > 0, and E(|X₁|^t) < ∞, where G(t, ?) = E{N_∞(t, t, ?)} = Σ_n=1^∞n^t?2P{| S_n | > ?n} → ∞ as ? → 0+ and $\text{H(t, ?) = E{N}{}_{\mathrm{\infty }}\text{(t, t, ?)} = Σ}{}_{\mathrm{n=1}}{}^{\mathrm{\infty }}\text{n}{}^{\mathrm{t?2}}\text{P{| S}{}_{\mathrm{n}}\text{| > ?n}{}^{\mathrm{<mtext>2</mtext><mtext>t</mtext>}}\text{} → ∞}\text{as}\text{? → 0+}$, i.e., H(t, ?) goes to infinity much faster than G(t, ?) as ? → 0+ if 2 < t < 4, E(X₁) = 0, Var(X₁) > 0, and E(| X₁ |^t) < ∞. Our results provide us with a much better and deeper understanding of the tail probability of a distribution.     

Scaling limits of Gaussian vector fields

For Gaussian vector fields {X(t) ∈ Rⁿ:t ∈ R^d} we describe the covariance functions of all scaling limits Y(t) = $\text{L}$lim_α↓0 B^?1(α) X(αt) which can occur when B(α) is a d × d matrix function with B(α) → 0. These matrix covariance functions $\text{r(t, s) = EY(t) Y}{}^1\text{(s)}$ are found to be homogeneous in the sense that for some matrix L and each α > 0, $\text{(}{}^1\text{) r(αt, αs) = α}{}^{\mathrm{L1}}\text{r(t, s) α}{}^{\mathrm{L}}$. Processes with stationary increments satisfying (¹) are further analysed and are found to be natural generalizations of Lévy's multiparameter Brownian motion.     

Limiting distributions of two random sequences

For fixed p (0 ≤ p ≤ 1), let {L₀, R₀} = {0, 1} and X₁ be a uniform random variable over {L₀, R₀}. With probability p let {L₁, R₁} = {L₀, X₁} or = {X₁, R₀} according as $\text{X}{}_1\text{≥}\text{1}\text{2}\text{(L}{}_0\text{+ R}{}_0\text{)}\text{or}\text{<}\text{1}\text{2}\text{(L}{}_0\text{+ R}{}_0\text{)}$; with probability 1 ? p let {L₁, R₁} = {X₁, R₀} or = {L₀, X₁} according as $\text{X}{}_1\text{≥}\text{1}\text{2}\text{(L}{}_0\text{+ R}{}_0\text{)}\text{or}\text{<}\text{1}\text{2}\text{(L}{}_0\text{+ R}{}_0\text{)}$, and let X₂ be a uniform random variable over {L₁, R₁}. For n ≥ 2, with probability p let {L_n, R_n} = {L_{n ? 1}, X_n} or = {X_n, R_{n ? 1}} according as $\text{X}{}_{\mathrm{n}}\text{≥}\text{1}\text{2}\text{(L}{}_{\mathrm{n\; ?\; 1}}\text{+ R}{}_{\mathrm{n\; ?\; 1}}\text{)}\text{or}\text{<}\text{1}\text{2}\text{(L}{}_{\mathrm{n\; ?\; 1}}\text{+ R}{}_{\mathrm{n\; ?\; 1}}\text{)}$, with probability 1 ? p let {L_n, R_n} = {X_n, R_{n ? 1}} or = {L_{n ? 1}, X_n} according as $\text{X}{}_{\mathrm{n}}\text{≥}\text{1}\text{2}\text{(L}{}_{\mathrm{n\; ?\; 1}}\text{+ R}{}_{\mathrm{n\; ?\; 1}}\text{)}\text{or}\text{<}\text{1}\text{2}\text{(L}{}_{\mathrm{n\; ?\; 1}}\text{+ R}{}_{\mathrm{n\; ?\; 1}}\text{)}$, and let X_{n + 1} be a uniform random variable over {L_n, R_n}. By this iterated procedure, a random sequence {X_n}_{n ≥ 1} is constructed, and it is easy to see that X_n converges to a random variable Y_p (say) almost surely as n → ∞. Then what is the distribution of Y_p? It is shown that the Beta, (2, 2) distribution is the distribution of Y₁; that is, the probability density function of Y₁ is g(y) = 6y(1 ? y) I_0,1(y). It is also shown that the distribution of Y₀ is not a known distribution but has some interesting properties (convexity and differentiability).     

Limit distributions for the maxima of stationary Gaussian processes

Let {X_n} be a stationary Gaussian sequence with E{X₀} = 0, {X²₀} = 1 and E{X₀X_n} = rⁿ_n Let c_n = (2ln n)^{$\text{built1}\text{2}$}, b_n = c_n? $\text{1}\text{2}$c^-1_n ln(4π ln n), and set M_n = max_{0 ?k?n}X_k. A classical result for independent normal random variables is that

$\text{P[c}{}_{\mathrm{n}}\text{(M}{}_{\mathrm{n}}\text{?b}{}_{\mathrm{n}}\text{)?x]→exp[-e}{}^{\mathrm{-x}}\text{] as n → ∞ for all x.}$

Berman has shown that (1) applies as well to dependent sequences provided r_nlnn = o(1). Suppose now that {r_n} is a convex correlation sequence satisfying r_n = o(1), (r_nlnn)^-1 is monotone for large n and o(1). Then

$\text{P[r}{}_{\mathrm{n}}{}^{\mathrm{<mtext>-1</mtext><mtext>2</mtext>}}\text{(M}{}_{\mathrm{n}}\text{? (1?r}{}_{\mathrm{n}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{b}{}_{\mathrm{n}}\text{)?x] → Ф(x)}$

for all x, where Ф is the normal distribution function. While the normal can thus be viewed as a second natural limit distribution for {M_n}, there are others. In particular, the limit distribution is given below when r_n is (sufficiently close to) γ/ln n. We further exhibit a collection of limit distributions which can arise when r_n decays to zero in a nonsmooth manner. Continuous parameter Gaussian processes are also considered. A modified version of (1) has been given by Pickands for some continuous processes which possess sufficient asymptotic independence properties. Under a weaker form of asymptotic independence, we obtain a version of (2).     

A process of runs and its convergence to the brownian motion

Let X₁,X₂,… be i.i.d. random variables with a continuous distribution function. Let R₀=0, R_k=min{j>R_k?1, such that X_j>X_j+1}, k?1. We prove that all finite-dimensional distributions of a process $\text{W}{}^{\mathrm{(n)}}\text{(t)=}\text{(R}{}_{\mathrm{[nt]}}\text{?2[nt])}\text{2}\text{3}\text{n}\text{, t ? [0,1]}$, converge to those of the standard Brownian motion.     

A canonical decomposition of the probability measure of sets of isotropic random points in Rn

The probability measure of X = (x₀,…, x_r), where x₀,…, x_r are independent isotropic random points in $\text{R}$ⁿ (1 ≤ r ≤ n ? 1) with absolutely continuous distributions is, for a certain class of distributions of X, expressed as a product measure involving as factors the joint probability measure of (ω, ?), the probability measure of p, and the probability measure of $\text{Y}{}^1\text{= (y}{}_0{}^1\text{,…, y}{}_{\mathrm{r}}{}^1\text{)}$. Here ω is the r-subspace parallel to the r-flat η determined by X, ? is a unit vector in ω^⊥ with ‘initial’ point at the origin [ω^⊥ is the (n ? r)-subspace orthocomplementary to ω], p is the norm of the vector z from the origin to the orthogonal projection of the origin on η, and $\text{y}{}_{\mathrm{i}}{}^1\text{=}\text{(x}{}_{\mathrm{i}}\text{? z)}\text{α(p}{}^2\text{)}$, where α is a scale factor determined by p. The probability measure for ω is the unique probability measure on the Grassmann manifold of r-subspaces in $\text{R}$ⁿ invariant under the group of rotations in $\text{R}$ⁿ, while the conditional probability measure of ? given ω is uniform on the boundary of the unit (n ? r)-ball in ω^⊥ with centre at the origin. The decomposition allows the evaluation of the moments, for a suitable class of distributions of X, of the r-volume of the simplicial convex hull of {x₀,…, x_r} for 1 ≤ r ≤ n.     

Three problems on the lengths of increasing runs

Let U₁, U₂,… be a sequence of independent, uniform (0, 1) r.v.'s and let R₁, R₂,… be the lengths of increasing runs of {U_i}, i.e., X₁=R₁=inf{i:U_i+1<U_i},…, X_n=R₁+R₂+?+R_n=inf{i:i>X_n?1,U_i+1<U_i}. The first theorem states that the sequence $\text{(}\text{3}\text{2}\text{n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(X}{}_{\mathrm{n}}\text{?2n)}$ can be approximated by a Wiener process in strong sense.Let τ(n) be the largest integer for which R₁+R₂+?+R_τ(n)?n, $\text{R}{}^1{}_{\mathrm{n}}\text{=n?(R}{}_1\text{+R}{}_2\text{+?+R}{}_{\mathrm{\tau (n)}}\text{)}$ and $\text{M}{}_{\mathrm{n}}\text{=}\text{max}\text{{R}{}_1\text{,R}{}_2\text{,…,R}{}_{\mathrm{\tau (n)}}\text{,R}{}^1{}_{\mathrm{n}}\text{}}$. Here M_n is the length of the longest increasing block. A strong theorem is given to characterize the limit behaviour of M_n.The limit distribution of the lengths of increasing runs is our third problem.     

On admissible shifts of generalized white noises

Let $\text{S}$ be the Schwartz space of rapidly decreasing real functions. The dual space $\text{S}$¹ consists of the tempered distributions and the relation $\text{S}$ ? L²($\text{R}$) ? $\text{S}$¹ holds. Let γ be the Gaussian white noise on $\text{S}$¹ with the characteristic functional $\text{γ}\text{(ξ) =}\text{exp}\text{{?∥ξ∥}{}^2\text{/2}}$, ξ ∈ $\text{S}$, where ∥·∥ is the L²($\text{R}$)-norm. Let ν be the Poisson white noise on $\text{S}$¹ with the characteristic functional $\text{ν}\text{(ξ)}$ = exp?_$\text{R}$∫_$\text{R}$ {[exp(iξ(t)u)] ? 1 ? (1 + u²)^?1(iξ(t)u)} dη(u)dt), ξ ∈ $\text{S}$, where the Lévy measure is assumed to satisfy the condition ∫_$\text{R}$u²dη(u) < ∞. It is proved that γ1ν has the same dichotomy property for shifts as the Gaussian white noise, i.e., for any ω ∈ $\text{S}$¹, the shift $\text{(γ1ν)}{}_{\mathrm{\omega }}$ of γ1ν by ω and γ1ν are either equivalent or orthogonal. They are equivalent if and only if when ω ∈ L²($\text{R}$) and the Radon-Nikodym derivative is derived. It is also proved that for the Poisson white noice ν_ω is orthogonal to ν for any non-zero ω in $\text{S}$¹.     

The rate of convergence in Orey's theorem for Harris recurrent Markov chains with applications to renewal theory

We derive sufficient conditions for ∝ λ (dx)6Pⁿ(x, ·) - π6 to be of order o(ψ(n)^-1), where Pⁿ (x, A) are the transition probabilities of an aperiodic Harris recurrent Markov chain, π is the invariant probability measure, λ an initial distribution and ψ belongs to a suitable class of non-decreasing sequences. The basic condition involved is the ergodicity of order ψ, which in a countable state space is equivalent to $\text{Σ ψ(n)}\text{P}{}_{\mathrm{i}}\text{{τ}{}_{\mathrm{i}}\text{?n} <∞}$ for some i, where τ_i is the hitting time of the tate i. We also show that for a general Markov chain to be ergodic of order ψ it suffices that a corresponding condition is satisfied by a small set.We apply these results to non-singular renewal measures on $\text{R}$ providing a probabilisite method to estimate the right tail of the renewal measure when the increment distribution F satisfies ∝ tF(dt) 0; > 0 and ∝ ψ(t)(1- F(t))dt< ∞.     

On the sequence of Bayesian estimates of normal random walk process

Let (μ_t)^∞_t=0 be a k-variate (k?1) normal random walk process with successive increments being independently distributed as normal N(δ, R), and μ₀ being distributed as normal N(0, V₀). Let X_t have normal distribution N(μ_t, Σ) when μ_t is given, t = 1, 2,….Then the conditional distribution of μ_t given X₁, X₂,…, X_t is shown to be normal N(U_t, V_t) where U_t's and V_t's satisfy some recursive relations. It is found that there exists a positive definite matrix V and a constant θ, 0 < θ < 1, such that, for all t?1,

$\text{|R}\text{1}\text{2}\text{(V}{}^{\mathrm{?1}}{}_{\mathrm{t}}\text{?V}{}^{\mathrm{?1}}\text{R}\text{1}\text{2}\text{|<θ}{}^{\mathrm{t}}\text{|R}\text{1}\text{2}\text{(V}{}^{\mathrm{?1}}{}_0\text{?V}{}^{\mathrm{?1}}\text{)R}\text{1}\text{2}\text{|}$

where the norm |·| means that |A| is the largest eigenvalue of a positive definite matrix A. Thus, V_t approaches to V as t approaches to infinity. Under the quadratic loss, the Bayesian estimate of μ_t is U_t and the process {U_t}^∞_t=₀, U₀=0, is proved to have independent successive increments with normal N(θ, V_t?V_t+1+R) distribution. In particular, when V₀ =V then V_t = V for all t and {U_t}^∞_t=0 is the same as {μ_t}^∞_t=0 except that U₀ = 0 and μ₀ is random.     

Random fields associated with multiple points of the Brownian motion

Let X_t be the Brownian motion in $\text{R}$^d. The random set Γ = {(t₁,…, t_n, z): X_tl = ··· = X_tn = z} in $\text{R}$^{d + n} is empty a.s. except in the following cases: (a) n = 1, d = 1, 2,…; (b) d = 2, n = 2, 3,…; (c) d = 3, n = 2. In each of these cases, a family of random measures M_λ concentrated on Γ is constructed (λ takes values in a certain class of measures on $\text{R}$^d). Measures M_λ characterize the time-space location of self-intersections for Brownian paths. If n = d = 1, then M_λ(dt, dz) = λ(dz) N_z(dt) where N₂ is the local time at z. In the case n = 2, the set Γ can be identified with the set of Brownian loops. The measure M_λ “explodes” on the diagonal {t₁ = t₂} and, to study small loops, a random distribution which regularizes M_λ is constructed.     

Modes of convergence of Markov chain transition probabilities

Suppose {Pⁿ(x, A)} denotes the transition law of a general state space Markov chain {X_n}. We find conditions under which weak convergence of {X_n} to a random variable X with law L (essentially defined by ∝ Pⁿ(x, dy) g(y) → ∝ L(dy) g(y) for bounded continuous g) implies that {X_n} tends to X in total variation (in the sense that $\text{∥ P}{}^{\mathrm{n}}\text{(x,}{}^{\mathrm{.}}\text{) ? L ∥ → 0)}$, which then shows that L is an invariant measure for {X_n}. The conditions we find involve some irreducibility assumptions on {X_n} and some continuity conditions on the one-step transition law {P(x, A)}.     

On the square root of a positive B(X,X1)-valued function

Let (Ω, $\text{B}$, μ) be a measure space, $\text{X}$ a separable Banach space, and $\text{X}$¹ the space of all bounded conjugate linear functionals on $\text{X}$. Let f be a weak¹ summable positive B($\text{X, X}$¹)-valued function defined on Ω. The existence of a separable Hilbert space $\text{K}$, a weakly measurable B($\text{X, K}$)-valued function Q satisfying the relation $\text{Q}{}^1\text{(ω)Q(ω) = f(ω)}$ is proved. This result is used to define the Hilbert space L_2,f of square integrable operator-valued functions with respect to f. It is shown that for B⁺($\text{X, X}$¹)-valued measures, the concepts of weak¹, weak, and strong countable additivity are all the same. Connections with stochastic processes are explained.     

A note on the conditional moments of a multivariate normal distribution confined to a convex set

Let Y be an N(μ, Σ) random variable on R^m, 1 ≤ m ≤ ∞, where Σ is positive definite. Let C be a nonempty convex set in R^m with closure $\text{C}$. Let (·,-·) be the Eculidean inner product on R^m, and let μ_c be the conditional expected value of Y given Y ∈ C. For v ∈ R^m and s ≥ 0, let β_s(v) be the expected value of |(v, Y) ? (v, μ)|^s and let γ_s(v) be the conditional expected value of |(v, Y) ? (v, μ_c)|^s given Y ∈ C. For s ≥ 1, γ_s(v) < β_s(v) if and only if $\text{C}\text{+ Σ v ≠}\text{C}$, and γ_s(v) < β_s(v) for all v ≠ 0 if and only if $\text{C}\text{+ v ≠}\text{C}$ for any v ∈ R^m such that v ≠ 0.     

Duality between some linear preserver problems. II. Isometries with respect to c-special norms and matrices with fixed singular values

Let $\text{F}{}_{\mathrm{m\times n}}$ (m?n) denote the linear space of all m × n complex or real matrices according as $\text{F}$=$\text{C}$ or $\text{R}$. Let c=(c₁,…,c_m)≠0 be such that c₁???c_m?0. The c-spectral norm of a matrix A?$\text{F}$_m×n is the quantity

$\text{6A6}{}_{\mathrm{c}}\text{∑}\text{i=I}\text{m}\text{c}{}_{\mathrm{i}}\text{σ}{}_{\mathrm{i}}\text{(A)}$

. where σ₁(A)???σ_m(A) are the singular values of A. Let d=(d₁,…,d_m)≠0, where d₁???d_m?0. We consider the linear isometries between the normed spaces $\text{(}\text{F}{}_{\mathrm{m\times }}{}_{\mathrm{n}}\text{,∥·∥}{}_{\mathrm{<mtext>c</mtext>}}\text{)}$ and $\text{(}\text{F}{}_{\mathrm{m\times }}{}_{\mathrm{n}}\text{,∥·∥}{}_{\mathrm{d}}\text{)}$, and prove that they are dual transformations of the linear operators which map $\text{L}$(d) onto $\text{L}$(c), where

$\text{L}\text{(c)= {X?}\text{F}{}_{\mathrm{m\times n}}\text{:X}\text{has singular values}\text{c}{}_1\text{,…,c}{}_{\mathrm{m}}\text{}}$

.     

A functional central limit theorem for ϱ-mixing sequences

In this note a functional central limit theorem for ?-mixing sequences of I. A. Ibragimov (Theory Probab. Appl.20 (1975), 135–141) is generalized to nonstationary sequences (X_n)_n ∈ $\text{N}$, satisfying some assumptions on the variances and the moment condition $\text{E |X}{}_{\mathrm{n}}\text{|}{}^{\mathrm{2\; +\; b}}\text{= O(n}{}^{\mathrm{<mtext>b</mtext><mtext>2</mtext><mtext>??</mtext>}}\text{)}$ for some b > 0, ? > 0.