Rough large-deviation probability asymptotics of some functionals of the norm kind

2000 ◽  
Vol 99 (2) ◽  
pp. 1161-1172 ◽  
Author(s):  
O. A. Podkoryatova
Keyword(s):  
Large Deviation ◽  
Large Deviation Probability ◽  
Deviation Probability

scholarly journals The Lower Tail of the Random Minimum Spanning Tree

10.37236/1004 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Abraham D. Flaxman
Keyword(s):  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Large Deviation ◽  
Random Variable ◽  
Independent Random Variables ◽  
Large Deviation Probability ◽  
Deviation Probability ◽  
Lower Tail ◽  
The Subject ◽  
The Cost

Consider a complete graph $K_n$ where the edges have costs given by independent random variables, each distributed uniformly between 0 and 1. The cost of the minimum spanning tree in this graph is a random variable which has been the subject of much study. This note considers the large deviation probability of this random variable. Previous work has shown that the log-probability of deviation by $\varepsilon$ is $-\Omega(n)$, and that for the log-probability of $Z$ exceeding $\zeta(3)$ this bound is correct; $\log {\rm Pr}[Z \geq \zeta(3) + \varepsilon] = -\Theta(n)$. The purpose of this note is to provide a simple proof that the scaling of the lower tail is also linear, $\log {\rm Pr}[Z \leq \zeta(3) - \varepsilon] = -\Theta(n)$.

scholarly journals Secretary problem with vanishing objects

2020 ◽  
Vol 12 (2) ◽  
pp. 63-81
Author(s):  
Сергей Иванович Доценко ◽  
Sergey Dotsenko ◽  
Георгий Шевченко ◽  
Georgiy Shevchenko
Keyword(s):  
Large Deviation ◽  
Geometric Distribution ◽  
Secretary Problem ◽  
Independent Interest ◽  
Selection Strategy ◽  
Large Deviation Probability ◽  
Auxiliary Result ◽  
Deviation Probability ◽  
Independent Variables ◽  
Probability Estimates

We consider a version of the secretary problem where elements may vanish during the selection and become unchoosable. We construct a selection strategy and identify the probability to select the best element, which turns out to be asymptotically maximal as number of elements increases indefinitely. As an auxiliary result of independent interest we establish large deviation probability estimates for sums of independent variables with distinct geometric distribution.

scholarly journals Probabilities of very large deviations

1978 ◽  
Vol 25 (3) ◽  
pp. 332-347 ◽  
Author(s):  
Stephen A. Book
Keyword(s):  
Generating Functions ◽  
Large Deviation ◽  
Random Variables ◽  
Large Deviation Probability ◽  
Deviation Probability ◽  
Absolutely Continuous ◽  
The Central Limit Theorem ◽  
The Subject ◽  
Elementary Form ◽  
Independent Identically Distributed

If {Xn: 1 ≦ n < ∞} are independent, identically distributed random variables having E(X1) = 0 and Var(X1) = 1, the most elementary form of the central limit theorem implies that P(n-½Sn≧ zn) → 0 as n → ∞, where Sn = Σnk=1 X,k, for all sequences {zn:1 ≧ n gt; ∞} for which zn → ∞. The probability P(n-½ Sn ≧ zn) is called a “large deviation probability”, and the rate at which it converges to 0 has been the subject of much study. The objective of the present article is to complement earlier results by describing its asymptotic behavior when n-½zn → ∞ as n → ∞, in the case of absolutely continuous random variables having moment-generating functions.

Optimal leverage ratio estimate of various models for leveraged ETFs to exceed a target: Probability estimates of large deviations

2018 ◽  
Vol 05 (02) ◽  
pp. 1850016
Author(s):  
Nian Yao
Keyword(s):  
Large Deviation ◽  
Garch Model ◽  
Tail Probability ◽  
Deviation Probability ◽  
Leverage Ratio ◽  
Probability Estimates ◽  
Volatility Models ◽  
Optimal Leverage ◽  
Model Inverse ◽  
Leveraged Etfs

In this paper, we study the deviation probability estimate for a leveraged exchanged-traded fund (LETF). By large deviation principle, we derive explicitly the logarithmic limit of the tail probability when the price of a LETF exceeds a given reference asset, which allows us to compute the underlying leverage ratio. Then we apply our results to various existing models, including the geometric Brownian motion (GBM) model, generalized autoregressive conditional heteroskedasticity (GARCH) model, inverse GARCH model, extended Cox–Ingersoll–Ross (CIR) model, 3/2 model, as well as the Heston and 3/2 stochastic volatility models, and to present their corresponding optimal leverage ratios, respectively.

scholarly journals A note on long-term optimal portfolios under drawdown constraints

2006 ◽  
Vol 38 (3) ◽  
pp. 673-692 ◽  
Author(s):  
Jun Sekine
Keyword(s):  
Optimization Problem ◽  
Large Deviation ◽  
Quadratic Model ◽  
General Situation ◽  
Maximization Problem ◽  
Large Deviation Probability ◽  
Linear Quadratic ◽  
Linear Quadratic Model ◽  
Portfolio Optimization Problem

The maximization of the long-term growth rate of expected utility is considered under drawdown constraints. In a general situation, the value and the optimal strategy of the problem are related to those of another ‘standard’ risk-sensitive-type portfolio optimization problem. Furthermore, an upside-chance maximization problem of a large deviation probability is stated as a ‘dual’ optimization problem. As an example, a ‘linear-quadratic’ model is studied in detail: the conditions to ensure the solvabilities of the problems are discussed and explicit expressions for the solutions are presented.

scholarly journals A note on long-term optimal portfolios under drawdown constraints

2006 ◽  
Vol 38 (03) ◽  
pp. 673-692 ◽  
Author(s):  
Jun Sekine
Keyword(s):  
Optimization Problem ◽  
Large Deviation ◽  
Quadratic Model ◽  
General Situation ◽  
Maximization Problem ◽  
Large Deviation Probability ◽  
Linear Quadratic ◽  
Linear Quadratic Model ◽  
Portfolio Optimization Problem

The maximization of the long-term growth rate of expected utility is considered under drawdown constraints. In a general situation, the value and the optimal strategy of the problem are related to those of another ‘standard’ risk-sensitive-type portfolio optimization problem. Furthermore, an upside-chance maximization problem of a large deviation probability is stated as a ‘dual’ optimization problem. As an example, a ‘linear-quadratic’ model is studied in detail: the conditions to ensure the solvabilities of the problems are discussed and explicit expressions for the solutions are presented.

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