Extremes of independent Gaussian processes

For every n ∈ ℕ, let X _1n ,..., X _nn be independent copies of a zero-mean Gaussian process X _n  = {X _n (t), t ∈ T}. We describe all processes which can be obtained as limits, as n→ ∞, of the process a _n (M _n  − b _n ), where M _n (t) =  max _{i = 1,...,n} X _in (t), and a _n , b _n are normalizing constants. We also provide an analogous characterization for the limits of the process a _n L _n , where L _n (t) =  min _{i = 1,...,n} |X _in (t)|.     

Asymptotic behavior of the ratio of tail probabilities of sum and maximum of independent random variables

For a fixed integer n ≥ 2, let X ₁ ,…, X _n be independent random variables (r.v.s) with distributions F ₁,…,F _n , respectively. Let Y be another random variable with distribution G belonging to the intersection of the longtailed distribution class and the O-subexponential distribution class. When each tail of F _i , i = 1,…,n, is asymptotically less than or equal to the tail of G, we derive asymptotic lower and upper bounds for the ratio of the tail probabilities of the sum X ₁ + ⋯ + X _n and Y. By taking different G’s, we obtain general forms of some existing results.     

The ratio of the extreme to the sum in a random sequence

For X ₁ , X ₂ , ..., X _n a sequence of non-negative independent random variables with common distribution function F(t), X _(n) denotes the maximum and S _n denotes the sum. The ratio variate R _n  = X _(n) / S _n is a quantity arising in the analysis of process speedup and the performance of scheduling. O’Brien (J. Appl. Prob. 17:539–545, 1980) showed that as n → ∞, R _n →0 almost surely iff is finite. Here we show that, provided either (1) is finite, or (2) 1 − F (t) is a regularly varying function with index ρ < − 1, then . An integral representation for the expected ratio is derived, and lower and upper asymptotic bounds are developed to obtain the result. Since is often known or estimated asymptotically, this result quantifies the rate of convergence of the ratio’s expected value. The result is applied to the performance of multiprocessor scheduling.      

Generating Random Vectors in (ℤ/<Emphasis Type="Italic">p</Emphasis>ℤ)<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript> via an Affine Random Process

This paper considers some random processes of the form X _n+1=T X _n +B _n (mod p) where B _n and X _n are random variables over (ℤ/pℤ) ^d and T is a fixed d×d integer matrix which is invertible over the complex numbers. For a particular distribution for B _n , this paper improves results of Asci to show that if T has no complex eigenvalues of length 1, then for integers p relatively prime to det (T), order (log p)² steps suffice to make X _n close to uniformly distributed where X ₀ is the zero vector. This paper also shows that if T has a complex eigenvalue which is a root of unity, then order p ^b steps are needed for X _n to get close to uniformly distributed for some positive value b≤2 which may depend on T and X ₀ is the zero vector.     

Limit theorems for a recursive maximum process with location-dependent periodic intensity-parameter

We investigate the recursive sequence Z _n : =  max {Z _n − 1,λ(Z _n − 1)X _n } where X _n is a sequence of iid random variables with exponential distributions and λ is a periodic positive bounded measurable function. We prove that the Césaro mean of the sequence λ(Z _n ) converges toward the essential minimum of λ. Subsequently we apply this result and obtain a limit theorem for the distributions of the sequence Z _n . The resulting limit is a Gumbel distribution.     

Exact Distribution of the Product of Two or More Logistic Random Variables

Without a doubt, the logistic distribution is the most popular statistical model in the social sciences and related areas. Motivated by the importance of products of random variables in these areas, we derive the exact distributions of | X ₁ X ₂ | and | X ₁ X ₂ ⋯ X _p  | when X _m are independent logistic random variables. Tabulations of the associated percentage points are provided and possible extensions discussed.     

On Suboptimal LCS-alignments for Independent Bernoulli Sequences with Asymmetric Distributions

Let X = X ₁ ... X _n and Y = Y ₁ ... Y _n be two binary sequences with length n. A common subsequence of X and Y is any subsequence of X that at the same time is a subsequence of Y; The common subsequence with maximal length is called the longest common subsequence (LCS) of X and Y. LCS is a common tool for measuring the closeness of X and Y. In this note, we consider the case when X and Y are both i.i.d. Bernoulli sequences with the parameters ϵ and 1 − ϵ, respectively. Hence, typically the sequences consist of large and short blocks of different colors. This gives an idea to the so-called block-by-block alignment, where the short blocks in one sequence are matched to the long blocks of the same color in another sequence. Such and alignment is not necessarily a LCS, but it is computationally easy to obtain and, therefore, of practical interest. We investigate the asymptotical properties of several block-by-block type of alignments. The paper ends with the simulation study, where the of block-by-block type of alignments are compared with the LCS.     

The nearness problems for symmetric centrosymmetric with a special submatrix constraint

We say that X=[x_ij]_i,j=1ⁿX=[x_{ij}]_{i,j=1}^n is symmetric centrosymmetric if x _ij  = x _ji and x _{n − j + 1,n − i + 1}, 1 ≤ i,j ≤ n. In this paper we present an efficient algorithm for minimizing ||AXA ^T  − B|| where ||·|| is the Frobenius norm, A ∈ ℝ ^m×n , B ∈ ℝ ^m×m and X ∈ ℝ ^n×n is symmetric centrosymmetric with a specified central submatrix [x _ij ] _{p ≤ i,j ≤ n − p} . Our algorithm produces a suitable X such that AXA ^T  = B in finitely many steps, if such an X exists. We show that the algorithm is stable any case, and we give results of numerical experiments that support this claim.     

Extreme values statistics for Markov chains via the (pseudo-) regenerative method

This paper is devoted to the study of specific statistical methods for extremal events in the markovian setup, based on the regenerative method and the Nummelin technique. Exploiting ideas developed in Rootzén (Adv Appl Probab 20:371–390, 1988), the principle underlying our methodology consists of first generating a random number l of approximate pseudo-renewal times τ ₁, τ ₂, ..., τ _l for a sample path X ₁, ..., X _n drawn from a Harris chain X with state space E, from the parameters of a minorization condition fulfilled by its transition kernel, and then computing submaxima over the approximate cycles thus obtained: $\max_{1+\tau_1\leq i \leq \tau_2}f(X_i),\;\ldots ,\;\max_{1+\tau_{l-1}\leq i \leq \tau_l}f(X_i)This paper is devoted to the study of specific statistical methods for extremal events in the markovian setup, based on the regenerative method and the Nummelin technique. Exploiting ideas developed in Rootzén (Adv Appl Probab 20:371–390, 1988), the principle underlying our methodology consists of first generating a random number l of approximate pseudo-renewal times τ ₁, τ ₂, ..., τ _l for a sample path X ₁, ..., X _n drawn from a Harris chain X with state space E, from the parameters of a minorization condition fulfilled by its transition kernel, and then computing submaxima over the approximate cycles thus obtained: max_{1+t1 ￡ i ￡ t2}f(X_i),  ?,  max_{1+tl-1 ￡ i ￡ tl}f(X_i)\max_{1+\tau_1\leq i \leq \tau_2}f(X_i),\;\ldots ,\;\max_{1+\tau_{l-1}\leq i \leq \tau_l}f(X_i) for any measurable function f:E→ℝ. Estimators of tail features of the sample maximum max_{1 ≤ i ≤ n} f(X _i ) are then constructed by applying standard statistical methods, tailored for the i.i.d. setting, to the submaxima as if they were independent and identically distributed. In particular, the asymptotic properties of extensions of popular inference procedures based on the conditional maximum likelihood theory, such as Hill’s method for the index of regular variation, are thoroughly investigated. Using the same approach, we also consider the problem of estimating the extremal index of the sequence {f(X _n )} _{n ∈ ℕ} under suitable assumptions. Eventually, practical issues related to the application of the methodology we propose are discussed and preliminary simulation results are displayed.     

The structure of LC-continuous functions

A function f is LC-continuous if the inverse image of any open set is a locally closed set; i.e., an intersection of an open set and a closed set. The aim of this paper is to prove the following theorem: Let f: X→Y be an LC-continuous function onto a separable metric space Y. Then X can be covered by countably many subsets T _n ⊂X such that each restriction f∣T _n is continuous at all points of T _n .