Asymptotic normality of U-statistics based on i.i.d. or negatively associated observations by utilizing Zolotarev’s ideal metric

2021 ◽  
pp. 1-20
Author(s):  
Tasos C. Christofides ◽  
Charalambos Charalambous
Keyword(s):  
Asymptotic Normality ◽  
Negatively Associated ◽  
U Statistics

Asymptotic normality of some Graph-Related statistics

1989 ◽  
Vol 26 (1) ◽  
pp. 171-175 ◽  
Author(s):  
Pierre Baldi ◽  
Yosef Rinott
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Asymptotic Normality ◽  
Central Limit ◽  
Simple Proof ◽  
Random Function ◽  
Random Variables ◽  
Local Maxima ◽  
Colored Graphs ◽  
U Statistics

Petrovskaya and Leontovich (1982) proved a central limit theorem for sums of dependent random variables indexed by a graph. We apply this theorem to obtain asymptotic normality for the number of local maxima of a random function on certain graphs and for the number of edges having the same color at both endpoints in randomly colored graphs. We briefly motivate these problems, and conclude with a simple proof of the asymptotic normality of certain U-statistics.

scholarly journals Hidden Words Statistics for Large Patterns

10.37236/9452 ◽  
2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Svante Janson ◽  
Wojciech Szpankowski
Keyword(s):  
Intrusion Detection ◽  
Asymptotic Normality ◽  
Pattern Matching ◽  
Projection Method ◽  
Storage Systems ◽  
Asymptotically Normal ◽  
U Statistics ◽  
Special Pattern ◽  
Log Normal ◽  
Random Text

We study here the so called subsequence pattern matching also known as hidden pattern matching in which one searches for a given pattern $w$ of length $m$ as a subsequence in a random text of length $n$. The quantity of interest is the number of occurrences of $w$ as a subsequence (i.e., occurring in not necessarily consecutive text locations). This problem finds many applications from intrusion detection, to trace reconstruction, to deletion channel, and to DNA-based storage systems. In all of these applications, the pattern $w$ is of variable length. To the best of our knowledge this problem was only tackled for a fixed length $m=O(1)$. In our main result we prove that for $m=o(n^{1/3})$ the number of subsequence occurrences is normally distributed. In addition, we show that under some constraints on the structure of $w$ the asymptotic normality can be extended to $m=o(\sqrt{n})$. For a special pattern $w$ consisting of the same symbol, we indicate that for $m=o(n)$ the distribution of number of subsequences is either asymptotically normal or asymptotically log normal. After studying some special patterns (e.g., alternating) we conjecture that this dichotomy is true for all patterns. We use Hoeffding's projection method for $U$-statistics to prove our findings.

Asymptotic Normality of U-Statistics Based on Trimmed Samples.

1985 ◽  
Author(s):  
Paul Janssen ◽  
Robert Serfling ◽  
Noel Veraverbeke
Keyword(s):  
Asymptotic Normality ◽  
U Statistics

Asymptotic normality of some Graph-Related statistics

1989 ◽  
Vol 26 (01) ◽  
pp. 171-175 ◽  
Author(s):  
Pierre Baldi ◽  
Yosef Rinott
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Asymptotic Normality ◽  
Central Limit ◽  
Simple Proof ◽  
Random Function ◽  
Random Variables ◽  
Local Maxima ◽  
Colored Graphs ◽  
U Statistics

Petrovskaya and Leontovich (1982) proved a central limit theorem for sums of dependent random variables indexed by a graph. We apply this theorem to obtain asymptotic normality for the number of local maxima of a random function on certain graphs and for the number of edges having the same color at both endpoints in randomly colored graphs. We briefly motivate these problems, and conclude with a simple proof of the asymptotic normality of certain U-statistics.

scholarly journals A class of U-statistics and asymptotic normality of the number of k-clusters

1992 ◽  
Vol 43 (2) ◽  
pp. 300-330 ◽  
Author(s):  
Rabi N Bhattacharya ◽  
Jayanta K Ghosh
Keyword(s):  
Asymptotic Normality ◽  
U Statistics

Asymptotic normality of u-statistics based on trimmed samples

1987 ◽  
Vol 16 ◽  
pp. 63-74 ◽  
Author(s):  
Paul Janssen ◽  
Robert Serfling ◽  
Noël Veraverbeke
Keyword(s):  
Asymptotic Normality ◽  
U Statistics
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