Small Deviations of Probabilities for Weighted Sum of Independent Positive Random Variables with a Common Distribution That Decreases at Zero Not Faster than a Power

2016 ◽  
Vol 60 (1) ◽  
pp. 142-150 ◽  
Author(s):  
L. V. Rozovsky
Keyword(s):  
Random Variables ◽  
Weighted Sum ◽  
Small Deviations ◽  
Common Distribution

The central limit problem for trimmed sums

1987 ◽  
Vol 102 (2) ◽  
pp. 329-349 ◽  
Author(s):  
Philip S. Griffin ◽  
William E. Pruitt
Keyword(s):  
Distribution Function ◽  
Random Variables ◽  
Random Variable ◽  
Limit Problem ◽  
Absolute Value ◽  
Trimmed Sums ◽  
Common Distribution ◽  
Common Distribution Function ◽  
Central Limit Problem ◽  
Trimmed Sum

Let X, X1, X2,… be a sequence of non-degenerate i.i.d. random variables with common distribution function F. For 1 ≤ j ≤ n, let mn(j) be the number of Xi satisfying either |Xi| > |Xj|, 1 ≤ i ≤ n, or |Xi| = |Xj|, 1 ≤ i ≤ j, and let (r)Xn = Xj if mn(j) = r. Thus (r)Xn is the rth largest random variable in absolute value from amongst X1, …, Xn with ties being broken according to the order in which the random variables occur. Set (r)Sn = (r+1)Xn + … + (n)Xn and write Sn for (0)Sn. We will refer to (r)Sn as a trimmed sum.

scholarly journals Distribution of Geometrically Weighted Sum of Bernoulli Random Variables

2011 ◽  
Vol 02 (11) ◽  
pp. 1382-1386 ◽  
Author(s):  
Deepesh Bhati ◽  
Phazamile Kgosi ◽  
Ranganath Narayanacharya Rattihalli
Keyword(s):  
Random Variables ◽  
Weighted Sum ◽  
Bernoulli Random Variables

Optimal selection of stochastic intervals under a sum constraint

1987 ◽  
Vol 19 (2) ◽  
pp. 454-473 ◽  
Author(s):  
E. G. Coffman ◽  
L. Flatto ◽  
R. R. Weber
Keyword(s):  
Decision Rule ◽  
Selection Process ◽  
Random Variables ◽  
Optimal Threshold ◽  
Optimal Selection ◽  
Expected Number ◽  
Common Distribution ◽  
The Common ◽  
Image Position ◽  
Selection Of

We model a selection process arising in certain storage problems. A sequence (X1, · ··, Xn) of non-negative, independent and identically distributed random variables is given. F(x) denotes the common distribution of the Xi′s. With F(x) given we seek a decision rule for selecting a maximum number of the Xi′s subject to the following constraints: (1) the sum of the elements selected must not exceed a given constant c > 0, and (2) the Xi′s must be inspected in strict sequence with the decision to accept or reject an element being final at the time it is inspected.We prove first that there exists such a rule of threshold type, i.e. the ith element inspected is accepted if and only if it is no larger than a threshold which depends only on i and the sum of the elements already accepted. Next, we prove that if F(x) ~ Axα as x → 0 for some A, α> 0, then for fixed c the expected number, En(c), selected by an optimal threshold is characterized by Asymptotics as c → ∞and n → ∞with c/n held fixed are derived, and connections with several closely related, well-known problems are brought out and discussed.

Probability maximizing approach to a secretary problem by random change-point of the distribution law of the observed process

1984 ◽  
Vol 21 (1) ◽  
pp. 98-107 ◽  
Author(s):  
Minoru Yoshida
Keyword(s):  
Distribution Function ◽  
Stopping Time ◽  
Random Variables ◽  
Secretary Problem ◽  
Independent Random Variables ◽  
Distribution Law ◽  
Common Distribution ◽  
Common Distribution Function ◽  
Random Moment ◽  
The Moment

Before some random moment θ, independent identically distributed random variables x1, · ··, xθ–1 with common distribution function μ (dx) appear consecutively. After the moment θ, independent random variables xθ, xθ+1, · ·· have another common distribution function f (x)μ (dx). Our information about θ can be constructed only by successively observed values of the x's.In this paper we find an optimal stopping policy by which we can maximize the probability that the quantity associated with the stopping time is the largest of all θ + m – 1 quantities for a given integer m.

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