On One Side Kolmogorov type inequalities

 

Let XOn One Side Kolmogorov type inequalities1On One Side Kolmogorov type inequalities,XOn One Side Kolmogorov type inequalities2On One Side Kolmogorov type inequalities,,XOn One Side Kolmogorov type inequalitiesnOn One Side Kolmogorov type inequalitiesOn One Side Kolmogorov type inequalities be independent random variables. Denote

SOn One Side Kolmogorov type inequalitiesnOn One Side Kolmogorov type inequalities=On One Side Kolmogorov type inequalitiesi=1On One Side Kolmogorov type inequalitiesnOn One Side Kolmogorov type inequalitiesXOn One Side Kolmogorov type inequalitiesiOn One Side Kolmogorov type inequalities.On One Side Kolmogorov type inequalities
The well known Kolmogrov inequality can stated as for all ε0On One Side Kolmogorov type inequalities
P(maxOn One Side Kolmogorov type inequalities1jnOn One Side Kolmogorov type inequalities|SOn One Side Kolmogorov type inequalitiesjOn One Side Kolmogorov type inequalities|ε)Var(SOn One Side Kolmogorov type inequalitiesnOn One Side Kolmogorov type inequalities)On One Side Kolmogorov type inequalitiesεOn One Side Kolmogorov type inequalities2On One Side Kolmogorov type inequalitiesOn One Side Kolmogorov type inequalitiesOn One Side Kolmogorov type inequalities.On One Side Kolmogorov type inequalities
  

The one side kolmogrov type ineqalites are stated as for all ε0On One Side Kolmogorov type inequalities

P(maxOn One Side Kolmogorov type inequalities1jnOn One Side Kolmogorov type inequalitiesSOn One Side Kolmogorov type inequalitiesjOn One Side Kolmogorov type inequalitiesε)Var(SOn One Side Kolmogorov type inequalitiesnOn One Side Kolmogorov type inequalities)On One Side Kolmogorov type inequalitiesεOn One Side Kolmogorov type inequalities2On One Side Kolmogorov type inequalities+Var(SOn One Side Kolmogorov type inequalitiesnOn One Side Kolmogorov type inequalities)On One Side Kolmogorov type inequalitiesOn One Side Kolmogorov type inequalities.On One Side Kolmogorov type inequalities
  We will prove this inequality in the following.

Proposition. LetXOn One Side Kolmogorov type inequalities be a random variable with Var(X)<On One Side Kolmogorov type inequalities . Then for all ε0On One Side Kolmogorov type inequalities

P(Xε)Var(X)On One Side Kolmogorov type inequalitiesεOn One Side Kolmogorov type inequalities2On One Side Kolmogorov type inequalities+Var(X)On One Side Kolmogorov type inequalitiesOn One Side Kolmogorov type inequalities.On One Side Kolmogorov type inequalities
  

Proof. Without loss of generality, we may assume that E(X)=0On One Side Kolmogorov type inequalities . Then

ε=E(ε?X)=E{(ε  ?X)IOn One Side Kolmogorov type inequalitiesX<εOn One Side Kolmogorov type inequalities}+E{(ε  ?X)IOn One Side Kolmogorov type inequalitiesXεOn One Side Kolmogorov type inequalities}E{(ε  ?X)IOn One Side Kolmogorov type inequalitiesX<εOn One Side Kolmogorov type inequalities}.On One Side Kolmogorov type inequalities
By Cauchy-Schwardz‘s inequality, We have
εOn One Side Kolmogorov type inequalities2On One Side Kolmogorov type inequalities[E{(ε  ?X)IOn One Side Kolmogorov type inequalitiesXεOn One Side Kolmogorov type inequalities}]On One Side Kolmogorov type inequalities2On One Side Kolmogorov type inequalitiesE(ε +X)On One Side Kolmogorov type inequalities2On One Side Kolmogorov type inequalitiesP(Xε)=[εOn One Side Kolmogorov type inequalities2On One Side Kolmogorov type inequalities+Var(X)][1?P(X>ε)].On One Side Kolmogorov type inequalities
Therefor,
P(Xε)Var(X)On One Side Kolmogorov type inequalitiesεOn One Side Kolmogorov type inequalities2On One Side Kolmogorov type inequalities+Var(X)On One Side Kolmogorov type inequalitiesOn One Side Kolmogorov type inequalities.On One Side Kolmogorov type inequalities

Proof of the one side Kolmogorov type inequality. Let Λ={maxOn One Side Kolmogorov type inequalities1\lejnOn One Side Kolmogorov type inequalitiesSOn One Side Kolmogorov type inequalitiesjOn One Side Kolmogorov type inequalities ε}On One Side Kolmogorov type inequalities and ΛOn One Side Kolmogorov type inequalitieskOn One Side Kolmogorov type inequalities={maxOn One Side Kolmogorov type inequalities1j<kOn One Side Kolmogorov type inequalitiesSOn One Side Kolmogorov type inequalitiesjOn One Side Kolmogorov type inequalities<ε,SOn One Side Kolmogorov type inequalitieskOn One Side Kolmogorov type inequalitiesε}On One Side Kolmogorov type inequalities , then Λ=?On One Side Kolmogorov type inequalitiesnOn One Side Kolmogorov type inequalitiesk=1On One Side Kolmogorov type inequalitiesΛOn One Side Kolmogorov type inequalitieskOn One Side Kolmogorov type inequalitiesOn One Side Kolmogorov type inequalities . Without loss of generality, we assume that E(XOn One Side Kolmogorov type inequalitiesjOn One Side Kolmogorov type inequalities)=0,j=1,,n.On One Side Kolmogorov type inequalities Then by the independence of the random variables,

εOn One Side Kolmogorov type inequalitiesOn One Side Kolmogorov type inequalities=On One Side Kolmogorov type inequalitiesOn One Side Kolmogorov type inequalitiesE[ε?SOn One Side Kolmogorov type inequalitiesnOn One Side Kolmogorov type inequalities]=E[(ε?SOn One Side Kolmogorov type inequalitiesnOn One Side Kolmogorov type inequalities)IOn One Side Kolmogorov type inequalitiesΛOn One Side Kolmogorov type inequalities]+[(ε?SOn One Side Kolmogorov type inequalitiesnOn One Side Kolmogorov type inequalities)IOn One Side Kolmogorov type inequalitiescOn One Side Kolmogorov type inequalitiesΛOn One Side Kolmogorov type inequalities]On One Side Kolmogorov type inequalitiesOn One Side Kolmogorov type inequalities=On One Side Kolmogorov type inequalitiesOn One Side Kolmogorov type inequalitiesOn One Side Kolmogorov type inequalitiesk=1On One Side Kolmogorov type inequalitiesnOn One Side Kolmogorov type inequalities[(ε?SOn One Side Kolmogorov type inequalitiesnOn One Side Kolmogorov type inequalities)IOn One Side Kolmogorov type inequalitiesΛOn One Side Kolmogorov type inequalitieskOn One Side Kolmogorov type inequalitiesOn One Side Kolmogorov type inequalities]+[(ε?SOn One Side Kolmogorov type inequalitiesnOn One Side Kolmogorov type inequalities)IOn One Side Kolmogorov type inequalitiesΛOn One Side Kolmogorov type inequalitiescOn One Side Kolmogorov type inequalitiesOn One Side Kolmogorov type inequalities]=On One Side Kolmogorov type inequalitiesk=1On One Side Kolmogorov type inequalitiesnOn One Side Kolmogorov type inequalitiesE[{(ε?SOn One Side Kolmogorov type inequalitieskOn One Side Kolmogorov type inequalities)?(SOn One Side Kolmogorov type inequalitiesnOn One Side Kolmogorov type inequalities?SOn One Side Kolmogorov type inequalitieskOn One Side Kolmogorov type inequalities)}IOn One Side Kolmogorov type inequalitiesΛOn One Side Kolmogorov type inequalitieskOn One Side Kolmogorov type inequalitiesOn One Side Kolmogorov type inequalities]+[(ε?SOn One Side Kolmogorov type inequalitiesnOn One Side Kolmogorov type inequalities)IOn One Side Kolmogorov type inequalitiesΛOn One Side Kolmogorov type inequalitiescOn One Side Kolmogorov type inequalitiesOn One Side Kolmogorov type inequalities]On One Side Kolmogorov type inequalitiesOn One Side Kolmogorov type inequalities=On One Side Kolmogorov type inequalitiesOn One Side Kolmogorov type inequalitiesOn One Side Kolmogorov type inequalitiesk=1On One Side Kolmogorov type inequalitiesnOn One Side Kolmogorov type inequalitiesE[(ε?SOn One Side Kolmogorov type inequalitieskOn One Side Kolmogorov type inequalities)IOn One Side Kolmogorov type inequalitiesΛOn One Side Kolmogorov type inequalitieskOn One Side Kolmogorov type inequalitiesOn One Side Kolmogorov type inequalities]+On One Side Kolmogorov type inequalitiesk=1On One Side Kolmogorov type inequalitiesnOn One Side Kolmogorov type inequalities[E(SOn One Side Kolmogorov type inequalitiesnOn One Side Kolmogorov type inequalities?SOn One Side Kolmogorov type inequalitieskOn One Side Kolmogorov type inequalities)IOn One Side Kolmogorov type inequalitiesΛOn One Side Kolmogorov type inequalitieskOn One Side Kolmogorov type inequalitiesOn One Side Kolmogorov type inequalities]+E[(ε?SOn One Side Kolmogorov type inequalitiesnOn One Side Kolmogorov type inequalities)IOn One Side Kolmogorov type inequalitiesΛOn One Side Kolmogorov type inequalitiescOn One Side Kolmogorov type inequalitiesOn One Side Kolmogorov type inequalities]On One Side Kolmogorov type inequalitiesOn One Side Kolmogorov type inequalities=On One Side Kolmogorov type inequalitiesOn One Side Kolmogorov type inequalitiesOn One Side Kolmogorov type inequalitiesk=1On One Side Kolmogorov type inequalitiesnOn One Side Kolmogorov type inequalitiesE[(ε?SOn One Side Kolmogorov type inequalitieskOn One Side Kolmogorov type inequalities)IOn One Side Kolmogorov type inequalitiesΛOn One Side Kolmogorov type inequalitieskOn One Side Kolmogorov type inequalitiesOn One Side Kolmogorov type inequalities]+E[(ε?SOn One Side Kolmogorov type inequalitiesnOn One Side Kolmogorov type inequalities)IOn One Side Kolmogorov type inequalitiesΛOn One Side Kolmogorov type inequalitiescOn One Side Kolmogorov type inequalitiesOn One Side Kolmogorov type inequalities]On One Side Kolmogorov type inequalitiesOn One Side Kolmogorov type inequalitiesOn One Side Kolmogorov type inequalitiesOn One Side Kolmogorov type inequalitiesE[(ε?SOn One Side Kolmogorov type inequalitiesnOn One Side Kolmogorov type inequalities)IOn One Side Kolmogorov type inequalitiesΛOn One Side Kolmogorov type inequalitiescOn One Side Kolmogorov type inequalitiesOn One Side Kolmogorov type inequalities.On One Side Kolmogorov type inequalitiesOn One Side Kolmogorov type inequalitiesOn One Side Kolmogorov type inequalities

By Cauchy-Schwardz‘s inequality, we have

εOn One Side Kolmogorov type inequalities2On One Side Kolmogorov type inequalities{E[(ε?SOn One Side Kolmogorov type inequalitiesnOn One Side Kolmogorov type inequalities)IOn One Side Kolmogorov type inequalitiesΛOn One Side Kolmogorov type inequalitiescOn One Side Kolmogorov type inequalitiesOn One Side Kolmogorov type inequalities]}On One Side Kolmogorov type inequalities2On One Side Kolmogorov type inequalitiesE[(ε?SOn One Side Kolmogorov type inequalitiesnOn One Side Kolmogorov type inequalities)]On One Side Kolmogorov type inequalities2On One Side Kolmogorov type inequalitiesP(IOn One Side Kolmogorov type inequalitiesΛOn One Side Kolmogorov type inequalitiescOn One Side Kolmogorov type inequalitiesOn One Side Kolmogorov type inequalities)=[εOn One Side Kolmogorov type inequalities2On One Side Kolmogorov type inequalities+Var(SOn One Side Kolmogorov type inequalities2On One Side Kolmogorov type inequalitiesnOn One Side Kolmogorov type inequalities)][1?P(Λ)].On One Side Kolmogorov type inequalities
Therefore,

P(Xε)Var(X)On One Side Kolmogorov type inequalitiesεOn One Side Kolmogorov type inequalities2On One Side Kolmogorov type inequalities+Var(X)On One Side Kolmogorov type inequalitiesOn One Side Kolmogorov type inequalitiesOn One Side Kolmogorov type inequalities
as the inequality claimed.

The inequality is also true for martingale difference sequence, the proof is samilar. 

On One Side Kolmogorov type inequalities

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