Simple proof of a large deviation result

2001 ◽  
Vol 17 (3) ◽  
pp. 326-331
Author(s):  
Zhang Zhixiang
Keyword(s):  
Simple Proof ◽  
Large Deviation ◽  
Large Deviation Result

A Free Logarithmic Sobolev Inequality on the Circle

2006 ◽  
Vol 49 (3) ◽  
pp. 389-406 ◽  
Author(s):  
Fumio Hiai ◽  
Dénes Petz ◽  
Yoshimichi Ueda
Keyword(s):  
Fisher Information ◽  
Sobolev Inequality ◽  
Large Deviation ◽  
Logarithmic Sobolev Inequality ◽  
Approximation Procedure ◽  
Matrix Approximation ◽  
Unitary Matrices ◽  
Free Entropy ◽  
Large Deviation Result ◽  
Free Fisher Information

AbstractFree analogues of the logarithmic Sobolev inequality compare the relative free Fisher information with the relative free entropy. In the present paper such an inequality is obtained for measures on the circle. The method is based on a random matrix approximation procedure, and a large deviation result concerning the eigenvalue distribution of special unitary matrices is applied and discussed.

Stochastic scrabble: large deviations for sequences with scores

1988 ◽  
Vol 25 (1) ◽  
pp. 106-119 ◽  
Author(s):  
Richard Arratia ◽  
Pricilla Morris ◽  
Michael S. Waterman
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Law Of Large Numbers ◽  
Large Deviation ◽  
Additive Functional ◽  
Finite Alphabet ◽  
Real Numbers ◽  
Similar Treatment ◽  
Large Numbers ◽  
Large Deviation Result

A derivation of a law of large numbers for the highest-scoring matching subsequence is given. Let Xk, Yk be i.i.d. q=(q(i))i∊S letters from a finite alphabet S and v=(v(i))i∊S be a sequence of non-negative real numbers assigned to the letters of S. Using a scoring system similar to that of the game Scrabble, the score of a word w=i1 · ·· im is defined to be V(w)=v(i1) + · ·· + v(im). Let Vn denote the value of the highest-scoring matching contiguous subsequence between X1X2 · ·· Xn and Y1Y2· ·· Yn. In this paper, we show that Vn/K log(n) → 1 a.s. where K ≡ K(q,v). The method employed here involves ‘stuttering’ the letters to construct a Markov chain and applying previous results for the length of the longest matching subsequence. An explicit form for β ∊Pr(S), where β (i) denotes the proportion of letter i found in the highest-scoring word, is given. A similar treatment for Markov chains is also included.Implicit in these results is a large-deviation result for the additive functional, H ≡ Σn < τv(Xn), for a Markov chain stopped at the hitting time τ of some state. We give this large deviation result explicitly, for Markov chains in discrete time and in continuous time.

On the Survival Probability of a Branching Process in a Random Environment

1997 ◽  
Vol 29 (01) ◽  
pp. 38-55 ◽  
Author(s):  
J. C. D'souza ◽  
B. M. Hambly
Keyword(s):  
State Space ◽  
Random Environment ◽  
Survival Probability ◽  
Branching Process ◽  
Large Deviation ◽  
Convergence Results ◽  
Finite State ◽  
Large Deviation Result ◽  
Finite State Space

We derive a variety of estimates for the survival probability of a branching process in a random environment. There are three cases of interest, the critical, weakly and strongly subcritical. The large deviation result, first obtained by Dekking for the class of finite state space i.i.d. environments, is shown to hold in more general environments. We also obtain some finer convergence results.

A Large Deviation Inequality for Functions of Independent, Multi-Way Choices

1998 ◽  
Vol 7 (1) ◽  
pp. 57-63 ◽  
Author(s):  
D. A. GRABLE
Keyword(s):  
Randomized Algorithms ◽  
High Probability ◽  
Large Deviation ◽  
Individual Choice ◽  
Difference Property ◽  
Classic Result ◽  
Deviation Inequality ◽  
Large Deviation Inequality ◽  
Large Deviation Result ◽  
Independent Indicator

Often when analysing randomized algorithms, especially parallel or distributed algorithms, one is called upon to show that some function of many independent choices is tightly concentrated about its expected value. For example, the algorithm might colour the vertices of a given graph with two colours and one would wish to show that, with high probability, very nearly half of all edges are monochromatic.The classic result of Chernoff [3] gives such a large deviation result when the function is a sum of independent indicator random variables. The results of Hoeffding [5] and Azuma [2] give similar results for functions which can be expressed as martingales with a bounded difference property. Roughly speaking, this means that each individual choice has a bounded effect on the value of the function. McDiarmid [9] nicely summarized these results and gave a host of applications. Expressed a little differently, his main result is as follows.

On the Survival Probability of a Branching Process in a Random Environment

1997 ◽  
Vol 29 (1) ◽  
pp. 38-55 ◽  
Author(s):  
J. C. D'souza ◽  
B. M. Hambly
Keyword(s):  
State Space ◽  
Random Environment ◽  
Survival Probability ◽  
Branching Process ◽  
Large Deviation ◽  
Convergence Results ◽  
Finite State ◽  
Large Deviation Result ◽  
Finite State Space

We derive a variety of estimates for the survival probability of a branching process in a random environment. There are three cases of interest, the critical, weakly and strongly subcritical. The large deviation result, first obtained by Dekking for the class of finite state space i.i.d. environments, is shown to hold in more general environments. We also obtain some finer convergence results.

Sign in / Sign up

Export Citation Format

Share Document