scholarly journals A Note on Rényi's ‘Record’ Problem and Engel's Series

2018 ◽  
Vol 61 (2) ◽  
pp. 363-369 ◽  
Author(s):  
Lulu Fang ◽  
Min Wu
Keyword(s):  
Number Theory ◽  
Large Deviations ◽  
Markov Processes ◽  
Classical Limit ◽  
Limit Theorems ◽  
London Math ◽  
Law Of Large Numbers ◽  
Record Times ◽  
Iterated Logarithm ◽  
Large Numbers

AbstractIn 1973, Williams [D. Williams, On Rényi's ‘record’ problem and Engel's series, Bull. London Math. Soc.5 (1973), 235–237] introduced two interesting discrete Markov processes, namely C-processes and A-processes, which are related to record times in statistics and Engel's series in number theory respectively. Moreover, he showed that these two processes share the same classical limit theorems, such as the law of large numbers, central limit theorem and law of the iterated logarithm. In this paper, we consider the large deviations for these two Markov processes, which indicate that there is a difference between C-processes and A-processes in the context of large deviations.

scholarly journals Limit Theorems for a Cox-Ingersoll-Ross Process with Hawkes Jumps

2014 ◽  
Vol 51 (03) ◽  
pp. 699-712 ◽  
Author(s):  
Lingjiong Zhu
Keyword(s):  
Stochastic Process ◽  
Large Deviations ◽  
Central Limit ◽  
Limit Theorems ◽  
Point Processes ◽  
Law Of Large Numbers ◽  
Laplace Transforms ◽  
Hawkes Process ◽  
Large Numbers ◽  
Special Case

In this paper we propose a stochastic process, which is a Cox-Ingersoll-Ross process with Hawkes jumps. It can be seen as a generalization of the classical Cox-Ingersoll-Ross process and the classical Hawkes process with exponential exciting function. Our model is a special case of the affine point processes. We obtain Laplace transforms and limit theorems, including the law of large numbers, central limit theorems, and large deviations.

Limit theorems for sums of chain-dependent processes

1974 ◽  
Vol 11 (3) ◽  
pp. 582-587 ◽  
Author(s):  
G. L. O'Brien
Keyword(s):  
Markov Chain ◽  
Limit Theorem ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Initial Distribution ◽  
Stationary Case ◽  
Strong Law ◽  
Iterated Logarithm ◽  
Large Numbers ◽  
Dependent Processes

Chain-dependent processes, also called sequences of random variables defined on a Markov chain, are shown to satisfy the strong law of large numbers. A central limit theorem and a law of the iterated logarithm are given for the case when the underlying Markov chain satisfies Doeblin's hypothesis. The proofs are obtained by showing independence of the initial distribution of the chain and by then restricting attention to the stationary case.

scholarly journals Order Book Dynamics with Liquidity Fluctuations: Limit Theorems and Large Deviations

2021 ◽  
Author(s):  
Helder Rojas ◽  
Anatoly Yambartsev ◽  
Artem Logachov
Keyword(s):  
Large Deviations ◽  
Stochastic Models ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Limit Order Book ◽  
Order Book ◽  
Limit Order ◽  
Bid Ask Spread ◽  
Large Numbers ◽  
The One

We propose a class of stochastic models for a dynamics of limit order book with different type of liquidities. Within this class of models we study the one where a spread decreases uniformly, belonging to the class of processes known as a population processes with uniform catastrophes. The law of large numbers (LLN), central limit theorem (CLT) and large deviations (LD) are proved for our model with uniform catastrophes. Our results allow us to satisfactorily explain the volatility and local trends in the prices, relevant empirical characteristics that are observed in this type of markets. Furthermore, it shows us how these local trends and volatility are determined by the typical values of the bid-ask spread. In addition, we use our model to show how large deviations occur in the spread and prices, such as those observed in flash crashes.

scholarly journals Limit Theorems for a Cox-Ingersoll-Ross Process with Hawkes Jumps

2014 ◽  
Vol 51 (3) ◽  
pp. 699-712 ◽  
Author(s):  
Lingjiong Zhu
Keyword(s):  
Stochastic Process ◽  
Large Deviations ◽  
Central Limit ◽  
Limit Theorems ◽  
Point Processes ◽  
Law Of Large Numbers ◽  
Laplace Transforms ◽  
Hawkes Process ◽  
Large Numbers ◽  
Special Case

In this paper we propose a stochastic process, which is a Cox-Ingersoll-Ross process with Hawkes jumps. It can be seen as a generalization of the classical Cox-Ingersoll-Ross process and the classical Hawkes process with exponential exciting function. Our model is a special case of the affine point processes. We obtain Laplace transforms and limit theorems, including the law of large numbers, central limit theorems, and large deviations.

Limit theorems for sums of chain-dependent processes

1974 ◽  
Vol 11 (03) ◽  
pp. 582-587 ◽  
Author(s):  
G. L. O'Brien
Keyword(s):  
Markov Chain ◽  
Limit Theorem ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Initial Distribution ◽  
Stationary Case ◽  
Strong Law ◽  
Iterated Logarithm ◽  
Large Numbers ◽  
Dependent Processes

Chain-dependent processes, also called sequences of random variables defined on a Markov chain, are shown to satisfy the strong law of large numbers. A central limit theorem and a law of the iterated logarithm are given for the case when the underlying Markov chain satisfies Doeblin's hypothesis. The proofs are obtained by showing independence of the initial distribution of the chain and by then restricting attention to the stationary case.

On record and inter-record times for a sequence of random variables defined on a Markov chain

1975 ◽  
Vol 7 (01) ◽  
pp. 195-214 ◽  
Author(s):  
Gary Lee Guthrie ◽  
Paul T. Holmes
Keyword(s):  
Markov Chain ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Finite Markov Chain ◽  
Record Times ◽  
Strong Law ◽  
The Central Limit Theorem ◽  
Iterated Logarithm ◽  
Continuous Parameter ◽  
Large Numbers

The familiar three theorems of Rényi concerning the record times in an i.i.d. sequence of random variables are extended to the record times and inter-record times of a sequence of dependent, non-identically distributed random variables defined on a finite Markov chain. These theorems are the Central Limit Theorem (C.L.T.), the Strong Law of Large Numbers (S.L.L.N.) and the Law of the Iterated Logarithm (L.I.L.). Similar results are also obtained for m-record times, inter-m-record times, and for the continuous parameter situation when observations are taken at the epochs of a Poisson process.

scholarly journals Limit Theorems for Empirical Density of Greatest Common Divisors

2016 ◽  
Vol 161 (3) ◽  
pp. 517-533 ◽  
Author(s):  
BEHZAD MEHRDAD ◽  
LINGJIONG ZHU
Keyword(s):  
Number Theory ◽  
Limit Theorem ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Greatest Common Divisor ◽  
Classic Result ◽  
The Central Limit Theorem ◽  
Large Numbers ◽  
Empirical Density ◽  
Greatest Common Divisors

AbstractThe law of large numbers for the empirical density for the pairs of uniformly distributed integers with a given greatest common divisor is a classic result in number theory. We study the large deviations of the empirical density. We will also obtain a rate of convergence to the normal distribution for the central limit theorem. Some generalisations are provided.

On record and inter-record times for a sequence of random variables defined on a Markov chain

1975 ◽  
Vol 7 (1) ◽  
pp. 195-214 ◽  
Author(s):  
Gary Lee Guthrie ◽  
Paul T. Holmes
Keyword(s):  
Markov Chain ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Finite Markov Chain ◽  
Record Times ◽  
Strong Law ◽  
The Central Limit Theorem ◽  
Iterated Logarithm ◽  
Continuous Parameter ◽  
Large Numbers

The familiar three theorems of Rényi concerning the record times in an i.i.d. sequence of random variables are extended to the record times and inter-record times of a sequence of dependent, non-identically distributed random variables defined on a finite Markov chain. These theorems are the Central Limit Theorem (C.L.T.), the Strong Law of Large Numbers (S.L.L.N.) and the Law of the Iterated Logarithm (L.I.L.). Similar results are also obtained for m-record times, inter-m-record times, and for the continuous parameter situation when observations are taken at the epochs of a Poisson process.

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