scholarly journals Generalized Galois Numbers, Inversions, Lattice Paths, Ferrers Diagrams and Limit Theorems

10.37236/2188 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Svante Janson
Keyword(s):  
Limit Theorems ◽  
Random Variables ◽  
Lattice Paths ◽  
Random Lattice ◽  
Ferrers Diagrams

Bliem and Kousidis recently considered a family of random variables whose distributions are given by the generalized Galois numbers (after normalization). We give probabilistic interpretations of these random variables, using inversions in random words, random lattice paths and random Ferrers diagrams, and use these to give new proofs of limit theorems as well as some further limit results.

scholarly journals On the Number of Turns in Reduced Random Lattice Paths

2013 ◽  
Vol 50 (2) ◽  
pp. 499-515
Author(s):  
Yunjiang Jiang ◽  
Weijun Xu
Keyword(s):  
Asymptotic Expansion ◽  
Limit Theorems ◽  
Lattice Paths ◽  
Lattice Path ◽  
Large Deviations Principle ◽  
Random Lattice ◽  
Symmetric Random Walk ◽  
Mean And Variance ◽  
The Mean ◽  
Lower Order Terms

We consider the tree-reduced path of a symmetric random walk on ℤd. It is interesting to ask about the number of turns Tn in the reduced path after n steps. This question arises from inverting the signatures of lattice paths: Tn gives an upper bound of the number of terms in the signature needed to reconstruct a ‘random’ lattice path with n steps. We show that, when n is large, the mean and variance of Tn in the asymptotic expansion have the same order as n, while the lower-order terms are O(1). We also obtain limit theorems for Tn, including the large deviations principle, central limit theorem, and invariance principle. Similar techniques apply to other finite patterns in a lattice path.

scholarly journals On the Number of Turns in Reduced Random Lattice Paths

2013 ◽  
Vol 50 (02) ◽  
pp. 499-515
Author(s):  
Yunjiang Jiang ◽  
Weijun Xu
Keyword(s):  
Asymptotic Expansion ◽  
Limit Theorems ◽  
Lattice Paths ◽  
Lattice Path ◽  
Large Deviations Principle ◽  
Random Lattice ◽  
Symmetric Random Walk ◽  
Mean And Variance ◽  
The Mean ◽  
Lower Order Terms

We consider the tree-reduced path of a symmetric random walk on ℤ d . It is interesting to ask about the number of turns T n in the reduced path after n steps. This question arises from inverting the signatures of lattice paths: T n gives an upper bound of the number of terms in the signature needed to reconstruct a ‘random’ lattice path with n steps. We show that, when n is large, the mean and variance of T n in the asymptotic expansion have the same order as n, while the lower-order terms are O(1). We also obtain limit theorems for T n, including the large deviations principle, central limit theorem, and invariance principle. Similar techniques apply to other finite patterns in a lattice path.

A central limit theorem by remainder term for renewal processes

1992 ◽  
Vol 24 (2) ◽  
pp. 267-287 ◽  
Author(s):  
Allen L. Roginsky
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Remainder Term ◽  
Limit Theorems ◽  
Random Variables ◽  
Central Limit Theorems ◽  
Renewal Processes

Three different definitions of the renewal processes are considered. For each of them, a central limit theorem with a remainder term is proved. The random variables that form the renewal processes are independent but not necessarily identically distributed and do not have to be positive. The results obtained in this paper improve and extend the central limit theorems obtained by Ahmad (1981) and Niculescu and Omey (1985).

Central Limit Theorems for Interchangeable Processes

1958 ◽  
Vol 10 ◽  
pp. 222-229 ◽  
Author(s):  
J. R. Blum ◽  
H. Chernoff ◽  
M. Rosenblatt ◽  
H. Teicher
Keyword(s):  
Stochastic Process ◽  
Probability Measure ◽  
Central Limit ◽  
Joint Distribution ◽  
Limit Theorems ◽  
Random Variables ◽  
Central Limit Theorems ◽  
Probability Measures ◽  
Image Position ◽  
Positive Integers

Let {Xn} (n = 1, 2 , …) be a stochastic process. The random variables comprising it or the process itself will be said to be interchangeable if, for any choice of distinct positive integers i 1, i 2, H 3 … , ik, the joint distribution of depends merely on k and is independent of the integers i 1, i 2, … , i k. It was shown by De Finetti (3) that the probability measure for any interchangeable process is a mixture of probability measures of processes each consisting of independent and identically distributed random variables.

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