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Heinrich Matzinger

Publications and source records attributed to Heinrich Matzinger.

26 records · Page 2Linked to original sources
arXiv2012

General approach to the fluctuations problem in random sequence comparison

We present a general approach to the problem of determining the asymptotic order of the variance of the optimal score between two independent random sequences defined over an arbitrary finite alphabet. Our general approach is based on identifying random variables driving the fluctuations of the optimal score and conveniently choosing functions of them which exhibit certain monotonicity properties. We show how our general approach establishes a common theoretical background for the techniques used by Matzinger et al. in a series of previous articles [6, 8, 20, 24, 26, 37] studying the same problem in especial cases. Additionally, we explicitely apply our general approach to study the fluctuations of the optimal score between two random sequences over a finite alphabet (closing the study as initiated in [26]) and of the length of the longest common subsequences between two random sequences with a certain block structure (generalizing part of [37]).

math.PR↗
arXiv2011

Information recovery from observations by a random walk having jump distribution with exponential tails

A {\it scenery} is a coloring $ξ$ of the integers. Let $\{S_t\}_{t\geq 0}$ be a recurrent random walk on the integers. Observing the scenery $ξ$ along the path of this random walk, one sees the color $χ_t:=ξ(S_t)$ at time $t$. The {\it scenery reconstruction problem} is concerned with recovering the scenery $ξ$, given only the sequence of observations $χ:=(χ_t)_{t\geq 0}$. The scenery reconstruction methods presented to date require the random walk to have bounded increments. Here, we present a new approach for random walks with unbounded increments which works when the tail of the increment distribution decays exponentially fast enough and the scenery has five colors.

math.PR↗
arXiv2010

The rate of the convergence of the mean score in random sequence comparison

We consider a general class of super-additive scores measuring the similarity of two independent sequences of $n$ i.i.d. letters from a finite alphabet. Our object of interest is the mean score by letter $l_n$. By the subadditivity $l_n$ is nondecreasing and converges to a limit $l$. We give a simple method of bounding the difference $l-l_n$ and obtaining the rate of convergence. Our result generalizes a previous result of Alexander, where only the special case of the longest common subsequence is considered.

math.PR↗
arXiv2010

Fluctuations of the Longest Common Subsequence for Sequences of Independent Blocks

The problem of the fluctuation of the Longest Common Subsequence (LCS) of two i.i.d. sequences of length $n>0$ has been open for decades. There exist contradicting conjectures on the topic. Chvatal and Sankoff conjectured in 1975 that asymptotically the order should be $n^{2/3}$, while Waterman conjectured in 1994 that asymptotically the order should be $n$. A contiguous substring consisting only of one type of symbol is called a block. In the present work, we determine the order of the fluctuation of the LCS for a special model of sequences consisting of i.i.d. blocks whose lengths are uniformly distributed on the set $\{l-1,l,l+1\}$, with $l$ a given positive integer. We showed that the fluctuation in this model is asymptotically of order $n$, which confirm Waterman's conjecture. For achieving this goal, we developed a new method which allows us to reformulate the problem of the order of the variance as a (relatively) low dimensional optimization problem.

math.PR↗
arXiv2010

Random modification effect in the size of the fluctuation of the LCS of two sequences of i.i.d. blocks

The problem of the order of the fluctuation of the Longest Common Subsequence (LCS) of two independent sequences has been open for decades. There exist contradicting conjectures on the topic, due to Chvatal - Sankoff in 1975 and Waterman in 1994. In the present article, we consider a special model of i.i.d. sequences made out of blocks. A block is a contiguous substring consisting only of one type of symbol. Our model allows only three possible block lengths, each been equiprobable picked up. In this context, we introduce a random operation (random modification) on the blocks of one of the sequences. In the present article, we develop the techniques to prove the following: if we suppose that the random modification increases the length of the LCS with high probability, then the order of the fluctuation of the LCS is as conjectured by Waterman. This result is a key technical part in the study of the size of the fluctuation of the LCS for sequences of i.i.d. blocks, developed by Matzinger and Torres.

math.PR↗
arXiv2009

Standard deviation of the longest common subsequence

Let $L_n$ be the length of the longest common subsequence of two independent i.i.d. sequences of Bernoulli variables of length $n$. We prove that the order of the standard deviation of $L_n$ is $\sqrt{n}$, provided the parameter of the Bernoulli variables is small enough. This validates Waterman's conjecture in this situation [Philos. Trans. R. Soc. Lond. Ser. B 344 (1994) 383--390]. The order conjectured by Chvatal and Sankoff [J. Appl. Probab. 12 (1975) 306--315], however, is different.

math.PR↗
arXiv2005

Reconstructing a two-color scenery by observing it along a simple random walk path

Let {ξ(n)}_{n\in Z} be a two-color random scenery, that is, a random coloring of Z in two colors, such that the ξ(i)'s are i.i.d. Bernoulli variables with parameter \tfrac12. Let {S(n)}_{n\in N} be a symmetric random walk starting at 0. Our main result shows that a.s., ξ\circ S (the composition of ξand S) determines ξup to translation and reflection. In other words, by observing the scenery ξalong the random walk path S, we can a.s. reconstruct ξup to translation and reflection. This result gives a positive answer to the question of H. Kesten of whether one can a.s. detect a single defect in almost every two-color random scenery by observing it only along a random walk path.

math.PR↗