On lower bounds for tail probabilities

2007 ◽  
Vol 137 (8) ◽  
pp. 2703-2705 ◽  
Author(s):  
Valentin V. Petrov
Keyword(s):  
Lower Bounds ◽  
Tail Probabilities ◽  
Bounds For Tail Probabilities

scholarly journals Exact bounds for tail probabilities of martingales with bounded differences

2009 ◽  
Vol 50 ◽  
Author(s):  
Dainius Dzindzalieta
Keyword(s):  
Random Walks ◽  
General Principle ◽  
Isoperimetric Problem ◽  
Maximal Inequality ◽  
Tail Probabilities ◽  
Bounds For Tail Probabilities ◽  
Maximal Inequalities ◽  
Exact Bounds ◽  
Natural Classes

We consider random walks, say Wn = {0, M1, . . ., Mn} of length n starting at 0 and based on a martingale sequence Mk = X1 + ··· + Xk with differences Xm. Assuming |Xk| \leq 1 we solve the isoperimetric problem Bn(x) = supP\{Wn visits an interval [x,∞)\},  (1) where sup is taken over all possible Wn. We describe random walks which maximize the probability in (1). We also extend the results to super-martingales.For martingales our results can be interpreted as a maximalinequalitiesP\{max 1\leq k\leq n Mk   \geq x\} \leq Bn(x).The maximal inequality is optimal since the equality is achieved by martingales related to the maximizing random walks. To prove the result we introduce a general principle – maximal inequalities for (natural classes of) martingales are equivalent to (seemingly weaker) inequalities for tail probabilities, in our caseBn(x) = supP{Mn  \geq  x}.Our methods are similar in spirit to a method used in [1], where a solution of an isoperimetric problem (1), for integer x is provided and to the method used in [4], where the isoperimetric problem of type (1) for conditionally symmetric bounded martingales was solved for all x ∈ R.

Simplified bounds on the tails of compound distributions

1997 ◽  
Vol 34 (1) ◽  
pp. 127-133 ◽  
Author(s):  
G. E. Willmot ◽  
Xiaodong Lin
Keyword(s):  
Size Distribution ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Claim Size ◽  
Size Distributions ◽  
Tail Probabilities ◽  
Compound Distributions ◽  
Claim Size Distribution

Upper and lower bounds are derived for the tail probabilities of compound distributions using simple properties of the claim size distribution. General bounds are then obtained for various classes of claim size distributions. Some examples are given.

SOME NEW BOUNDS AND APPROXIMATIONS ON TAIL PROBABILITIES OF THE POISSON AND OTHER DISCRETE DISTRIBUTIONS

2018 ◽  
Vol 34 (1) ◽  
pp. 53-71
Author(s):  
Steven G. From ◽  
Andrew W. Swift
Keyword(s):  
Poisson Distribution ◽  
Lower Bounds ◽  
Discrete Distributions ◽  
Upper And Lower Bounds ◽  
Tail Probabilities ◽  
Numerical Comparisons

AbstractIn this paper, we discuss new bounds and approximations for tail probabilities of certain discrete distributions. Several different methods are used to obtain bounds and/or approximations. Excellent upper and lower bounds are obtained for the Poisson distribution. Excellent approximations (and not bounds necessarily) are also obtained for other discrete distributions. Numerical comparisons made to previously proposed methods demonstrate that the new bounds and/or approximations compare very favorably. Some conjectures are made.

scholarly journals Bounds for Tail Probabilities of the Sample Variance

2009 ◽  
Vol 2009 (1) ◽  
pp. 941936 ◽  
Author(s):  
V Bentkus ◽  
M Van Zuijlen
Keyword(s):  
Sample Variance ◽  
Tail Probabilities ◽  
Bounds For Tail Probabilities

scholarly journals Tail Bounds for the Wiener Index of Random Trees

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Tämur Ali Khan ◽  
Ralph Neininger
Keyword(s):  
Lower Bounds ◽  
Random Search ◽  
Wiener Index ◽  
Moment Generating Function ◽  
Random Trees ◽  
Upper And Lower Bounds ◽  
Search Trees ◽  
Tail Probabilities ◽  
International Audience ◽  
The Moment

International audience Upper and lower bounds for the tail probabilities of the Wiener index of random binary search trees are given. For upper bounds the moment generating function of the vector of Wiener index and internal path length is estimated. For the lower bounds a tree class with sufficiently large probability and atypically large Wiener index is constructed. The methods are also applicable to related random search trees.

Simplified bounds on the tails of compound distributions

1997 ◽  
Vol 34 (01) ◽  
pp. 127-133
Author(s):  
G. E. Willmot ◽  
Xiaodong Lin
Keyword(s):  
Size Distribution ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Claim Size ◽  
Size Distributions ◽  
Tail Probabilities ◽  
Compound Distributions ◽  
Claim Size Distribution

Upper and lower bounds are derived for the tail probabilities of compound distributions using simple properties of the claim size distribution. General bounds are then obtained for various classes of claim size distributions. Some examples are given.

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