Large Deviation Lower Bounds For General Sequences Of Random Variables

Exact lower bounds on the exponential moments of min(y,X) andX1{X<y} are provided given the first two moments of a random variableX. These bounds are useful in work on large deviation probabilities and nonuniform Berry-Esseen bounds, when the Cramér tilt transform may be employed. Asymptotic properties of these lower bounds are presented. Comparative advantages of the so-called Winsorization min(y,X) over the truncationX1{X<y} are demonstrated. An application to option pricing is given.

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Exact Lower Bounds on the Exponential Moments of Truncated Random Variables

Journal of Applied Probability ◽

10.1239/jap/1308662643 ◽

2011 ◽

Vol 48 (2) ◽

pp. 547-560 ◽

Cited By ~ 5

Author(s):

Iosif Pinelis

Keyword(s):

Option Pricing ◽

Lower Bounds ◽

Large Deviation ◽

Asymptotic Properties ◽

Random Variables ◽

Random Variable ◽

Exponential Moments ◽

Comparative Advantages ◽

Large Deviation Probabilities

Exact lower bounds on the exponential moments of min(y, X) and X1{X < y} are provided given the first two moments of a random variable X. These bounds are useful in work on large deviation probabilities and nonuniform Berry-Esseen bounds, when the Cramér tilt transform may be employed. Asymptotic properties of these lower bounds are presented. Comparative advantages of the so-called Winsorization min(y, X) over the truncation X1{X < y} are demonstrated. An application to option pricing is given.

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Chover-Type Laws of the Iterated Logarithm for Kesten-Spitzer Random Walks in Random Sceneries Belonging to the Domain of Stable Attraction

Discrete Dynamics in Nature and Society ◽

10.1155/2018/8968947 ◽

2018 ◽

Vol 2018 ◽

pp. 1-9

Author(s):

Wensheng Wang ◽

Anwei Zhu

Keyword(s):

Random Walk ◽

Large Deviation ◽

Sufficient Conditions ◽

Random Variables ◽

Domain Of Attraction ◽

Stable Law ◽

Iterated Logarithm ◽

Cluster Set ◽

Random Scenery ◽

Unique Integer

Let X={Xi,i≥1} be a sequence of real valued random variables, S0=0 and Sk=∑i=1kXi (k≥1). Let σ={σ(x),x∈Z} be a sequence of real valued random variables which are independent of X’s. Denote by Kn=∑k=0nσ(⌊Sk⌋) (n≥0) Kesten-Spitzer random walk in random scenery, where ⌊a⌋ means the unique integer satisfying ⌊a⌋≤a<⌊a⌋+1. It is assumed that σ’s belong to the domain of attraction of a stable law with index 0<β<2. In this paper, by employing conditional argument, we investigate large deviation inequalities, some sufficient conditions for Chover-type laws of the iterated logarithm and the cluster set for random walk in random scenery Kn. The obtained results supplement to some corresponding results in the literature.

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CLT-related large deviation bounds based on Stein's method

Advances in Applied Probability ◽

10.1239/aap/1189518636 ◽

2007 ◽

Vol 39 (3) ◽

pp. 731-752 ◽

Cited By ~ 3

Author(s):

Martin Raič

Keyword(s):

Random Graphs ◽

Finite Population ◽

Large Deviation ◽

Random Variables ◽

Stein’S Method ◽

Stein's Method ◽

Population Statistics ◽

Sums Of Random Variables ◽

Dependence Structures

Large deviation estimates are derived for sums of random variables with certain dependence structures, including finite population statistics and random graphs. The argument is based on Stein's method, but with a novel modification of Stein's equation inspired by the Cramér transform.

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