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probability inequalities

Source: miklos schweitzer 2005 q12

August 28, 2021
inequalitiesprobability and stats

Problem Statement

Let x1,x2,⋯ ,xnx_1,x_2,\cdots,x_n be iid rv. Sn=∑xkS_n=\sum x_k (a) let P(∣x1∣≤1)=1P(|x_1|\leq 1)=1 , E[x1]=0E[x_1]=0 , E[x12]=σ2>0E[x_1^2]=\sigma^2>0 Prove that ∃C>0\exists C>0 , ∀u≥2nσ2\forall u\geq 2n\sigma^2 P(Sn≥u)≤e−Culog⁡(u/nσ2)P(S_n\geq u)\leq e^{-C u \log(u/n\sigma^2)}
(b) let P(x1=1)=P(x1=−1)=σ2/2P(x_1=1)=P(x_1=-1)=\sigma^2/2 , P(x1=0)=1−σ2P(x_1=0)=1-\sigma^2 Prove that ∃B1<1,B2>1,B3>0\exists B_1<1,B_2>1,B_3>0 , ∀u≥1,B1n≥u≥B2nσ2\forall u\geq1, B_1 n\geq u\geq B_2 n\sigma^2 P(Sn≥u)>e−B3ulog⁡(u/nσ2)P(S_n\geq u)>e^{-B_3 u \log(u/n\sigma^2)}
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