scholarly journals Normalizing constants of log-concave densities

2018 ◽  
Vol 12 (1) ◽  
pp. 851-889 ◽  
Author(s):  
Nicolas Brosse ◽  
Alain Durmus ◽  
Éric Moulines
Keyword(s):  
Normalizing Constants

scholarly journals Normalization of the Kolmogorov–Smirnov and Shapiro–Wilk tests of normality

2015 ◽  
Vol 52 (2) ◽  
pp. 85-93 ◽  
Author(s):  
Zofia Hanusz ◽  
Joanna Tarasińska
Keyword(s):  
Sample Size ◽  
P Values ◽  
Normalizing Constants ◽  
Kolmogorov Smirnov ◽  
Tests For Normality

Abstract Two very well-known tests for normality, the Kolmogorov-Smirnov and the Shapiro- Wilk tests, are considered. Both of them may be normalized using Johnson’s (1949) SB distribution. In this paper, functions for normalizing constants, dependent on the sample size, are given. These functions eliminate the need to use non-standard statistical tables with normalizing constants, and make it easy to obtain p-values for testing normality.

Limit theorems for threshold-stopped random variables with applications to optimal stopping

1990 ◽  
Vol 22 (2) ◽  
pp. 396-411 ◽  
Author(s):  
Douglas P. Kennedy ◽  
Robert P. Kertz
Keyword(s):  
Asymptotic Behaviour ◽  
Optimal Stopping ◽  
Limit Theorems ◽  
Random Variables ◽  
Limit Distributions ◽  
Joint Distributions ◽  
Normalizing Constants

The extremal types theorem identifies asymptotic behaviour for the maxima of sequences of i.i.d. random variables. A parallel theorem is given which identifies the asymptotic behaviour of sequences of threshold-stopped random variables. Three new types of limit distributions arise, but normalizing constants remain the same as in the maxima case. Limiting joint distributions are also given for maxima and threshold-stopped random variables. Applications to the optimal stopping of i.i.d. random variables are given.

Stability of maxima of random variables defined on a Markov chain

1972 ◽  
Vol 4 (2) ◽  
pp. 285-295 ◽  
Author(s):  
Sidney I. Resnick
Keyword(s):  
Markov Chain ◽  
Regularity Condition ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Finite Markov Chain ◽  
Finite Product ◽  
Normalizing Constants ◽  
Necessary And Sufficient

Consider maxima Mn of a sequence of random variables defined on a finite Markov chain. Necessary and sufficient conditions for the existence of normalizing constants Bn such that are given. The problem can be reduced to studying maxima of i.i.d. random variables drawn from a finite product of distributions πi=1mHi(x). The effect of each factor Hi(x) on the behavior of maxima from πi=1mHi is analyzed. Under a mild regularity condition, Bn can be chosen to be the maximum of the m quantiles of order (1 - n-1) of the H's.

Limit laws for maxima of a sequence of random variables defined on a Markov chain

1970 ◽  
Vol 2 (2) ◽  
pp. 323-343 ◽  
Author(s):  
Sidney I. Resnick ◽  
Marcel F. Neuts
Keyword(s):  
Markov Chain ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Value Distribution ◽  
Extreme Value Distributions ◽  
Limit Laws ◽  
Limiting Behavior ◽  
Normalizing Constants

Consider the bivariate sequence of r.v.'s {(Jn, Xn), n ≧ 0} with X0 = - ∞ a.s. The marginal sequence {Jn} is an irreducible, aperiodic, m-state M.C., m < ∞, and the r.v.'s Xn are conditionally independent given {Jn}. Furthermore P{Jn = j, Xn ≦ x | Jn − 1 = i} = pijHi(x) = Qij(x), where H1(·), · · ·, Hm(·) are c.d.f.'s. Setting Mn = max {X1, · · ·, Xn}, we obtain P{Jn = j, Mn ≦ x | J0 = i} = [Qn(x)]i, j, where Q(x) = {Qij(x)}. The limiting behavior of this probability and the possible limit laws for Mn are characterized.Theorem. Let ρ(x) be the Perron-Frobenius eigenvalue of Q(x) for real x; then:(a)ρ(x) is a c.d.f.;(b) if for a suitable normalization {Qijn(aijnx + bijn)} converges completely to a matrix {Uij(x)} whose entries are non-degenerate distributions then Uij(x) = πjρU(x), where πj = limn → ∞pijn and ρU(x) is an extreme value distribution;(c) the normalizing constants need not depend on i, j;(d) ρn(anx + bn) converges completely to ρU(x);(e) the maximum Mn has a non-trivial limit law ρU(x) iff Qn(x) has a non-trivial limit matrix U(x) = {Uij(x)} = {πjρU(x)} or equivalently iff ρ(x) or the c.d.f. πi = 1mHiπi(x) is in the domain of attraction of one of the extreme value distributions. Hence the only possible limit laws for {Mn} are the extreme value distributions which generalize the results of Gnedenko for the i.i.d. case.

LIMIT LAWS FOR ERGODIC PROCESSES

2012 ◽  
Vol 12 (01) ◽  
pp. 1150012 ◽  
Author(s):  
JEAN-PAUL THOUVENOT ◽  
BENJAMIN WEISS
Keyword(s):  
Probability Theory ◽  
Random Variables ◽  
Arbitrary Distribution ◽  
Stationary Processes ◽  
Limit Laws ◽  
Ergodic Processes ◽  
Normalizing Constants

The study of limit laws for normalized sums of a sequence of random variables is one of the classical topics in probability theory. We give here two results showing that ergodic stationary processes can admit an arbitrary distribution as the limit of normalized sums. In the first, we take the usual average of the first n variables, but the process is not integrable. In the second, the variables take on only two values and the sequence of normalizing constants is constructed inductively. In both cases the processes can be defined as factors of an arbitrary aperiodic measure preserving system.

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