An arithmetical property of Rademacher sums

Let ε = {ε_i, i ≥ 1} be a Rademacher sequence, with partial sums S_n = ε₁ +… + ε_n, n ≥ 1. Let N_k be the k-th even integer such that N_k∥S_Nk². We prove that there exists a positive real s, of which the value is explicitly given, such that for any , almost surely.     

Some properties of nonoscillating solutions of functional-differential equations ofn-th order

A functional-differential equation ofn-th order is considered, wheren≥2,m≥1 are integers andA _t/^t: C([t₀, ∞), R)→ R, i=1,2,...,m are functionals defined for everyt∈[t ₀, ∞). Sufficient conditions have been found for which all bounded non-oscillatory solutions and all non-oscillatory solutions of the functional-differntial equation tend to zero fort→∞.     

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.     

Sojourns and extremes of Fourier sums and series with random coefficients

Let X(t) be the trigonometric polynomial Σ^k_j=0a_j(U_tcosjt+V_jsinjt), –∞< t<∞, where the coefficients U_t and V_t are random variables and a_j is real. Suppose that these random variables have a joint distribution which is invariant under all orthogonal transformations of R^2k–2. Then X(t) is stationary but not necessarily Gaussian. Put L_t(u) = Lebesgue measure {s: 0?s?t, X(s) > u}, and M(t) = max{X(s): 0?s?t}. Limit theorems for L_t(u) and $\text{P}\text{(M(t) > u)}$ for u→∞ are obtained under the hypothesis that the distribution of the random norm (Σ^k_j=0(U²_j+V²_j))^{1 2} belongs to the domain of attraction of the extreme value distribution exp{ e^–2}. The results are also extended to the random Fourier series (k=∞).     

Sur l'équivalence de certaines mesures produit

Sommaire SoitG={g _k ,kN} une suite de variables aléatoires gaussiennes centrées réduites et indépendantes; soit de plusY={y _k ,kN} une suite indépendante deG de variables aléatoires indépendantes. On étudie à quelles conditions la loi deG+Y est équivalente à celle deG. On utilise pour cela les lois zéro-un vérifiées parG en analysant leurs effets, maximaux sur la loi deY.

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Summary LetG={g _k ,kN} be a sequence of independent Gaussian centred reduced random variables; let moreoverY={y _k ,kN} be a sequence independent ofG of independent random variables: For obtaining conditions characterizing the equivalence of the distributions ofG andG+Y, we use the zero-one laws verified byG, first for the convergence of the series _k g _k or _k (g _k ² –a _k ), secundly for the asymptotic behavior of the sequence {g _k ,kN} and we analyze their maximal effects on the distribution ofY.

     

On the power of the zeros of Bessel functions

For ν≥0 let c_νk be the k-th positive zero of the cylinder functionC _v(t)=J _v(t)cosα-Y _v(t)sinα, 0≤α>π whereJ _ν(t) andY _ν(t) denote the Bessel functions of the first and the second kind, respectively. We prove thatC _v,k ^1+H(x) is convex as a function of ν, ifc _νk≥x>0 and ν≥0, whereH(x) is specified in Theorem 1.1.     

The influence of a log-type small ball factor in the study of the lower classes

Let {BH₁,H₂(t₁,t₂),t₁?0,t₂?0} be a fractional Brownian sheet with indexes 0<H₁,H₂<1. When H₁=H₂:=H, there is a logarithmic factor in the small ball function of the sup-norm statistic of BH,H. First, we state general conditions (one based on a logarithmic factor in the small ball function) on some statistics of BH,H. Then we characterize the sufficiency part of the lower classes of these statistics by an integral test. Finally, when we consider the sup-norm statistic, the influence of the log-type small ball factor in the necessity part is measured by a second integral test.     

Measure concentration for a class of random processes

Summary.   Let X={X _i } _{i =−∞} ^∞ be a stationary random process with a countable alphabet and distribution q. Let q ^∞(·|x _{− k} ⁰) denote the conditional distribution of X ^∞=(X ₁,X ₂,…,X _n ,…) given the k-length past:

On the asymptotic behaviour of the empirical random field of the brownian motion

Let ξ_t, t ? 0, be a d-dimensional Brownian motion. The asymptotic behaviour of the random field ??∫^t₀?(ξ_s) ds is investigated, where ? belongs to a Sobolev space of periodic functions. Particularly a central limit theorem and a law of iterated logarithm are proved leading to a so-called universal law of iterated logarithm.     

An initial value problem for a singular parabolic equation of even order

At first Cauchy-problem for the equation: \(L[u(X,t)] \equiv \sum\limits_{i = 1}^n {\frac{{\partial ^2 u}}{{\partial x_1^2 }} + \frac{{2v}}{{\left| X \right|^2 }}} \sum\limits_{i = 1}^n {x_i \frac{{\partial u}}{{\partial x_i }} - \frac{{\partial u}}{{\partial t}} = 0} \) wheren≥1,v—an arbitrary constant,t>0,X=(x ₁, …, x_n)∈E ⁿ/{0}, |X|= =(x ₁ ² +…+x _n ² )^1/2, with 0 being a centre of coordinate system, is studied. Basing on the above, the solution of Cauchy-Nicolescu problem is given which consist in finding a solution of the equationL ^p [u (X, t)]=0, withp∈N subject the initial conditions \(\mathop {\lim }\limits_{t \to \infty } L^k [u(X,t)] = \varphi _k (X)\) ,k=0, 1,…,p?1 and ?_k(X) are given functions.     

Parity results for 9-regular partitions

Let t≥2 be an integer. We say that a partition is t-regular if none of its parts is divisible by t, and denote the number of t-regular partitions of n by b _t (n). In this paper, we establish several infinite families of congruences modulo 2 for b ₉(n). For example, we find that for all integers n≥0 and k≥0, $$b_9 \biggl(2^{6k+7}n+ \frac{2^{6k+6}-1}{3} \biggr)\equiv 0 \quad (\mathrm{mod}\ 2 ). $$      

An Erdős-Révész type law of the iterated logarithm for stationary Gaussian processes

Summary Let {X(t),t 0} be a stationary Gaussian process withEX(t)=0,EX ²(t)=1 and covariance function satisfying (i)r(t) = 1 2212;C |t | + o (|t|)ast0 for someC>0, 0<2; (ii)r(t)=0(t ^–2) as t for some >0 and (iii) sup_ts|r(t)|<1 for eachs>0. Put (t)= sup {s:0 s t,X(s) (2logs)^1/2}. The law of the iterated logarithm implies a.s. This paper gives the lower bound of (t) and obtains an Erds-Rèvèsz type LIL, i.e., a.s. if 0<<2 and . Applications to infinite series of independent Ornstein-Uhlenbeck processes and to fractional Wiener processes are also given.Research supported by the Fok Yingtung Education Foundation of China and by Charles Phelps Taft Postdoctoral Fellowship of the University of Cincinnati     

Asymptotic behavior of solutions of linear delay equations

The delay differential equations of the formx′(t)=?a(t)x(t?1),t≥0 are considered, wherea(t)≥0 is locally integrable on [0,∞). The main result: Let 0<c(t)≤a(t)≤k(t) for large ∫^∞, andc(t)≤Mc(t′) fort, t′≥T,|t?t′|≤l with some constantsl>0,M>1,T≥0. Then the condition \(k(t) \leqslant \frac{3}{2} + \alpha c(t), t \geqslant T\) with some constant α>0 dependent onl, M, ensures that all solutions of (*) tend to zero ast→∞.     

On the interrelation of almost sure invariance principles for certain stochastic adaptive algorithms and for partial sums of random variables

Since the novel work of Berkes and Philipp⁽³⁾ much effort has been focused on establishing almost sure invariance principles of the form (1) $$\left| {\sum\limits_{i = 1}^{|\_t\_|} {x_1 - X_t } } \right| \ll t^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} - \gamma } $$ where {x _i ,i=1,2,3,...} is a sequence of random vectors and {X _t ,t>-0} is a Brownian motion. In this note, we show that if {A _k ,k=1,2,3,...} and {b _k ,k=1,2,3,...} are processes satisfying almost-sure bounds analogous to Eq. (1), (where {X _t ,t≥0} could be a more general Gauss-Markov process) then {h _k ,k=1,2,3...}, the solution of the stochastic approximation or adaptive filtering algorithm (2) $$h_{k + 1} = h_k + \frac{1}{k}(b_k - A_k h_k )for{\text{ }}k{\text{ = 1,2,3}}...$$ also satisfies and almost sure invariance principle of the same type.     

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.     

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.     

Circulation of lifted processes on trivial principal bundles

A Brownian motion {x _t } _t?0 on a compact Riemannian manifold M with a drift vector field X can be lifted to a diffusion process $\left\{ {\tilde x_t } \right\}_{t \ge 0} $ on M × T^k corresponding to an ?^k valued smooth differential one-form A on M. The circulations (rotation numbers) of the lifted process $\left\{ {\tilde x_t } \right\}_{t \ge 0} $ around the k circles of T^k are studied. By choosing a certain ?^k -valued differential one-form A, these circulations give the hidden circulation of {x _t } _t?0 in M and the rotation numbers of {x _t } _t?0 around some closed curves in M which generalize the first homology group H₁(M,?) of M.     

Investigation of solvability of the multidimensional two-phase Stefan and the nonstationary filtration Florin problems for second order parabolic equations in weighted Hölder spaces of functions

Multidimensional two-phase Stefan (k=1) and nonstationary filtration Florin (k=0) problems for second order parabolic equations in the case when the free boundary is a graph of a functionx _n =Ψ _k (x′t),x′∈ ^n?1 ,n≥2,t∈(0,T) are studied. A unique solvability theorem in weighted Hölder spaces of functions with time power weight is proved, coercive estimates for solutions are obtained. Bibliography: 30 titles.