Some inequalities on Binomial and Poisson probabilities

2020 
Let $\{X_i\}_{i\ge 1}$ be a sequence of independent random variables taking values in the set of non-negative integers and, given $k\ge 1$, set $S_k = X_1+\cdots +X_k$. Fix $k\ge 1$, and let $\alpha_{k+1}>-1$ be a real number such that $\mathbb{E}(X_{k+1})\le 1 + \alpha$. Assume further that there exists a real number $c_k>0$ such that the following two conditions hold true: \[\max_{1\le i\le k-1}\, \mathbb{P}(S_k=i) \le c_k\cdot \mathbb{P}(S_k =k) \] as well as \[ (1-c_k)\cdot \mathbb{P}(X_{k+1}=0) + c_k\cdot \sum_{j\ge k+2}\mathbb{P}(X_{k+1}\ge j) \ge c_k \cdot\alpha_{k+1} \, . \] We show that \[ \mathbb{P}(S_k\ge k)\ge\mathbb{P}(S_{k+1}\ge k+1) \, . \] Moreover, we illustrate how this result yields some known and some new inequalities for binomial as well as for Poisson random variables.
    • Correction
    • Source
    • Cite
    • Save
    • Machine Reading By IdeaReader
    10
    References
    1
    Citations
    NaN
    KQI
    []