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Exponential-type upper bounds are formulated for the probability that the maximum of the partial sample sums of discrete random variables having ﬁnite equispaced support exceeds or di ﬀ ers from the population mean by a speciﬁed positive constant. The new inequalities extend the work of Serﬂing (1974). An example of the results are given to demonstrate their e ﬃ cacy.


Introduction
Serfling (1974) has obtained upper bounds for the probability that the sum of observations sampled without replacement from a finite population exceeds its expected value by a specified quantity.Serfling (1974) also noted that his bound is crude due to the incorporation of the coarse variance upper bound σ 2 < (b − a) 2 /4 and has suggested that "it would be desirable to obtain a sharpening of this result involving the quantity σ 2 in place of the quantity (b − a) 2 /4."While this problem remains unsolved, we attempt to at least partially fulfill Serfling's suggestion.In order to do this we tighten his inequality bound by restricting ourselves to a particular class of discrete distributions.The general problem which we address may be stated as follows.
Consider a finite population of size N whose members are not necessarily distinct.Let the set Ω N = {x 1 , x 2 , . . ., x N } be the set representation of this population.Denote by X 1 , X 2 , . . ., X n the values of a sample of size n drawn without replacement from Ω N .Define the statistics and let the sampling fractions be f n = (n − 1)/(N − 1) and g n = (n − 1)/N.We are concerned with the behavior of the sum In particular, we derive a new parameter-free upper bounds on the probabilities where ε > 0.
The most familiar upper bound for (1.3) is the Bienayme-Chebyshev inequality, which is of the form (1.5) Serfling (1974) has derived an alternative upper bound for (1.3), which may be expressed as and an alternative bound for a two-sided version of (1.4), which is given as where r is a positive integer.
We then compare our new under bound with these under the following scenario, which is similar to an example presented by Savage (1961).Suppose one wishes to study the average height of a finite population of people.
Assume that all individuals in the population are between 60 and 78 inches and their heights are measured to the nearest inch.The question we wish to answer is, "What is the probability that the average height of a sample of 100 from a population of 4,000 individuals is within two inches of the population mean height?" The main vehicle we utilize for the sharpening of inequalities (1.6) and (1.7) is the additional assumption that the random variables given by (X i |X 1 , . . ., X i−1 ) for i = 1, . . ., n are discrete random variables with probability functions having finite, equispaced support, and whose variance is bounded above by the discrete uniform variance.The remainder of the paper is as follows.In Section 2 we give some mathematical preliminaries while in Section 2.1 we derive the main inequality results.Finally, in Section 3 we present the before mentioned application of the newly-derived probability bounds.

The Set V a,b,J
From this point forward we work almost exclusively with an equispaced set of J points in the interval [a, b] beginning at a and ending at b.We denote the set as where b = a + (J − 1)c and c = b−a J−1 .We refer to J as the support size.As before X 1 , X 2 , . . ., X n denote the values of a sample of size n drawn without replacement from Ω a,b,J and S k = k i=1 X i .To sharpen the inequalities (1.6) and (1.7) we work with probability distributions that have a variance bounded by the variance of a discrete uniform probability distribution.This leads us to the following definition.
Definition 2.1 Let V a,b,J be the set of probability functions f with support on Ω a,b,J such that for a random variable X having probability function f the variance is bounded as per Remark 2.2 For f ∈ V a,b,J , the above bound on Var f (X) is simply the variance of a discrete uniform probability function on Ω a,b,J .
Remark 2.3 The definition for V a,b,J , although appearing somewhat restrictive, still allows for a broad and rich range of distributions.In particular, it applies to a broad range of discrete unimodal distributions.

Probability Inequalities for V a,b,J
In this section we derive a new maximal probability inequality for sums of discrete unimodal random variables sampled without replacement from a set of probability functions belonging to V a,b,J .We shall need the following lemmas, theorems, and corollaries to develop the new maximal probability inequality.We now develop two lemmas which are used in the proof of the main theorem.
Lemma 2.4 Let X be a random variable with probability function in V a,b,J and let E(X) = μ.Then for any λ ≥ 0 where d = b − a and α = J+1 Proof.Let Z = X − μ and notice that (2.1) since E[Z] = 0. Now let f be the probability function for X.Then for any j ≥ 2 we have 12 into (2.1),we get Corollary 2.5 Let X be a random variable with probability function in V a,b,J and let E(X) = μ.Then for any λ ≥ 0 Proof.By Lemma 2.4, we have Thus, exp 1 12 The following lemma uses an argument similar to Theorem 2.2 in Sefling's paper (1974).
If the probability function of X 1 and the conditional probability functions of (X k |X 1 , X 2 , . . ., X k−1 ), k = 2, . . ., n are in V a,b,J , then, for any λ ≥ 0 and any n ∈ {1, 2, . . ., N} we have where S n = n k=1 X k .Proof.For λ = 0, the result is obvious.Given λ > 0 let Notice λ k is increasing in k up to λ as k goes from 1 to n.

Main Result
We now give the main result of the paper in the following theorem.
Noting that P n (ε) ≤ R n (ε) and applying Theorem 2.8 we get the following corollary.
Corollary 2.9 For any ε > 0 we have From (1.6) we have Observe that Applying Theorem 2.8 to the above we get the following corollary.
Corollary 2.10 For any ε > 0, For any ε > 0 and λ > 0 we have Proof we get the desired result.

Finite Population Drawn Without Replacement
Here, we present some results which will aid us in our quest for sharpening of inequalities (1.6) and (1.7).These results were first used without proof in Serfling's paper (1974).We give proofs here for the sake of completeness.
We work under the same set-up as presented in Section 1.That is Ω N = {x 1 , x 2 , . . ., x N } is a finite population of size N, the members of which are not necessarily distinct.Also X 1 , X 2 , . . ., X n denote the values of a sample of size n drawn without replacement from Ω N and S k is the sum of the first k samples, as in (1.2).We also take μ as in (1.1).
Proposition 5 Let μ k ≡ (X k − μ|X 1 , X 2 , . . ., X k−1 ).Then, One can easily check that T k is a reverse martingale and T * k are martingales Serfling (1974).That is, To prove (A.2) note that . By Remark 3, {e λZ k } is a reverse submartingale.Now using c = e λε in Proposition 2 we have n≤k≤N e λZ k ≥ e λε = P max n≤k≤N