Robert W. Chen

A remark on the tail probability of a distribution

Let {Xn}n>=1 be a sequence of independent and identically distributed random variables. For each integer n >= 1 and positive constants r, t, and [epsilon], let Sn = [Sigma]j=1n Xj and E{N[infinity](r, t, [epsilon])} = [Sigma]n=1[infinity] nr-2P{Sn > [epsilon]nr/t}. In this paper, we prove that (1) lim[epsilon]-->0+ [epsilon][alpha](r-1)E{N[infinity](r, t, [epsilon])} =K(r, t) if E(X1) = 0, Var(X1) = 1, and E( X1 t) 0+ G(t, [epsilon])/H(t, [epsilon]) = 0 if 2 0, and E(X1t) [epsilon]n} --> [infinity] as [epsilon] --> 0+ and H(t, [epsilon]) = E{N[infinity](t, t, [epsilon])} = [Sigma]n=1[infinity] nt-2P{ Sn > [epsilon]n2/t} --> [infinity] as [epsilon] --> 0+, i.e., H(t, [epsilon]) goes to infinity much faster than G(t, [epsilon]) as [epsilon] --> 0+ if 2 0, and E( X1 t)

Failure Distributions of Consecutive-k-out-of-n:F Systems

Most literature on consecutive-k-out-of-n:F systems gives recursive equations for computing the system reliability. This paper gives a formula for computing the system reliability directly. Using this formula, the system failure distribution is derived. Upper and lower bounds for the system reliability or the system failure distribution are given. Some numerical examples are included.

On fair coin-tossing games

Let [Omega] be a finite set with k elements and for each integer n [greater, double equals] 1 let [Omega]n = [Omega] - [Omega] - ... - [Omega] (n-tuple) and 11 and aj [not equal to] aj+1 for some 1 [less, double equals] j [less, double equals] n - 1}. Let {Ym} be a sequence of independent and identically distributed random variables such that P(Y1 = a) = k-1 for all a in [Omega]. In this paper, we obtain some very surprising and interesting results about the first occurrence of elements in [Omega]n and in [Omega]n with respect to the stochastic process {Ym}. The results here provide us with a better and deeper understanding of the fair coin-tossing (k-sided) process.

On almost sure convergence in a finitely additive setting

In their book “How To Gable If You Must”, Dubins and Savage introduced finitely additive stochastic processes in discrete time and they obtained some results about finitely additive probability measures on infinite product spaces. In the paper, “Some Finitely Additive Probability”, Purves and Sudderth showed how to extend these finitely additive probability measures and it thus became possible to consider many of the classical strong convergence theorems. In this paper, we extend many of the classical strong convergence theorems to a finitely additive setting. Since all proofs in this paper are valid for a countably additive setting if we consider the problem on a coordinate representation process, the results in this paper are generalizations of the classical results on such a process. Some examples are also provided for contrasting a finitely additive setting with a countably additive setting.

Subfair “Red-and-Black” in the presence of inflation

SummaryThe gambling problem of “Red-and-Black” casinos in the presence of inflation is introduced. The optimality of the bold strategy is shown when the lottery is subfair or fair. The non-optimality of the bold strategy is also shown when the lottery is superfair.

Subfair primitive casino with a discount factor

It is known that, for a subfair primitive casino, if a gambler has an initial fortune in (0, 1) and wishes to reach 1, then he should play boldly. Now if the game is modified by adding a discount factor which is used to motivate the gambler recognizing the time value of fortune and completing the game as quickly as is reasonably consistent with reaching the goal, then one would intuitively suspect that the gambler should still play boldly. However, this intuitive conjecture is false and the result of this paper goes in the opposite direction and states that the bold strategy need not be optimal for a subfair primitive casino with a discount factor.

Some inequalities for randomly stopped variables with applications to pointwise convergence

In this paper, first, we prove some inequalities for randomly stopped variables, which arise naturally in the gambling theory, then we show that a theorem of Chacon and some pointwise convergence theorems, which imply the submartingale convergence theorem, are immediate consequences of these inequalities.

Stronger players win more balanced knockout tournaments

A player is said to be stronger than another player if he has a better chance of beating the other player than vice versa and his chance of beating any third player is at least as good as that of the other player. Recently, Israel gave an example which shows that a stronger player can have a smaller probability of winning a knockout tournament than a weaker one when players are randomly assigned to starting positions. In this paper we prove that this anomaly cannot happen if the tournament plan is a balanced one.

Limiting distributions of two random sequences

For fixed p (0 = 2, with probability p let {Ln, Rn} = {Ln - 1, Xn} or = {Xn, Rn - 1} according as , with probability 1 - p let {Ln, Rn} = {Xn, Rn - 1} or = {Ln - 1, Xn} according as , and let Xn + 1 be a uniform random variable over {Ln, Rn}. By this iterated procedure, a random sequence {Xn}n >= 1 is constructed, and it is easy to see that Xn converges to a random variable Yp (say) almost surely as n --> [infinity]. Then what is the distribution of Yp? It is shown that the Beta, (2, 2) distribution is the distribution of Y1; that is, the probability density function of Y1 is g(y) = 6y(1 - y) I0,1(y). It is also shown that the distribution of Y0 is not a known distribution but has some interesting properties (convexity and differentiability).

Some finitely additive versions of the strong law of large numbers

LetX be a non-empty set,H= X{su\t8, \gs = \lj{in1}x\lj{in2}x,σ=γ1×γ2×… be an independent strategy onH, and {Yn} be a sequence of coordinate mappings onH. The following strong law in a finitely additive setting is proved: For some constantr≧1, if \GSn=1\t8{\GS(\vbYn\vb2r)n1+n < \t8 andσ(Yn)=0 for alln=1, 2, …, then \1n\gS{inj-1}/{sun} Y{inj}Yjconverges to 0 withσ-measure 1 asn → ∞. The theorem is a generalization of Chung’s strong law in a coordinate representation process. Finally, Kolmogorov’s strong law in a finitely additive setting is proved by an application of the theorem.