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The payoff in the Chow–Robbins coin-tossing game is the proportion of heads when you stop. Stopping to maximize expectation was addressed by Chow and Robbins (1965), who proved there exist integers 
${k_n}$ such that it is optimal to stop at n tosses when heads minus tails is 
${k_n}$. Finding 
${k_n}$ was unsolved except for finitely many cases by computer. We prove an 
$o(n^{-1/4})$ estimate of the stopping boundary of Dvoretsky (1967), which then proves 
${k_n} = \left\lceil {\alpha \sqrt n \,\, - 1/2\,\, + \,\,\frac{{\left( { - 2\zeta (\! -1/2)} \right)\sqrt \alpha }}{{\sqrt \pi }}{n^{ - 1/4}}} \right\rceil $ except for n in a set of density asymptotic to 0, at a power law rate. Here, 
$\alpha$ is the Shepp–Walker constant from the Brownian motion analog, and 
$\zeta$ is Riemann’s zeta function. An 
$n^{ - 1/4}$ dependence was conjectured by Christensen and Fischer (2022). Our proof uses moments involving Catalan and Shapiro Catalan triangle numbers which appear in a tree resulting from backward induction, and a generalized backward induction principle. It was motivated by an idea of Häggström and Wästlund (2013) to use backward induction of upper and lower Value bounds from a horizon, which they used numerically to settle a few cases. Christensen and Fischer, with much better bounds, settled many more cases. We use Skorohod’s embedding to get simple upper and lower bounds from the Brownian analog; our upper bound is the one found by Christensen and Fischer in another way. We use them first for yet many more examples and a conjecture, then algebraically in the tree, with feedback to get much sharper Value bounds near the border, and analytic results. Also, we give a formula that gives the exact optimal stop rule for all n up to about a third of a billion; it uses the analytic result plus terms arrived at empirically.
In this paper, we consider a joint drift rate control and two-sided impulse control problem in which the system manager adjusts the drift rate as well as the instantaneous relocation for a Brownian motion, with the objective of minimizing the total average state-related cost and control cost. The system state can be negative. Assuming that instantaneous upward and downward relocations take a different cost structure, which consists of both a setup cost and a variable cost, we prove that the optimal control policy takes an 
$\left\{ {\!\left( {{s^{\ast}},{q^{\ast}},{Q^{\ast}},{S^{\ast}}} \right),\!\left\{ {{\mu ^{\ast}}(x)\,:\,x \in [ {{s^{\ast}},{S^{\ast}}}]} \right\}} \right\}$ form. Specifically, the optimal impulse control policy is characterized by a quadruple 
$\left( {{s^{\ast}},{q^{\ast}},{Q^{\ast}},{S^{\ast}}} \right)$, under which the system state will be immediately relocated upwardly to 
${q^{\ast}}$ once it drops to 
${s^{\ast}}$ and be immediately relocated downwardly to 
${Q^{\ast}}$ once it rises to 
${S^{\ast}}$; the optimal drift rate control policy will depend solely on the current system state, which is characterized by a function 
${\mu ^{\ast}}\!\left( \cdot \right)$ for the system state staying in 
$[ {{s^{\ast}},{S^{\ast}}}]$. By analyzing an associated free boundary problem consisting of an ordinary differential equation and several free boundary conditions, we obtain these optimal policy parameters and show the optimality of the proposed policy using a lower-bound approach. Finally, we investigate the effect of the system parameters on the optimal policy parameters as well as the optimal system’s long-run average cost numerically.
We establish higher moment formulae for Siegel transforms on the space of affine unimodular lattices as well as on certain congruence quotients of 
$\mathrm {SL}_d({\mathbb {R}})$. As applications, we prove functional central limit theorems for lattice point counting for affine and congruence lattices using the method of moments.
In the classical gambler’s ruin problem, the gambler plays an adversary with initial capitals z and 
$a-z$, respectively, where 
$a>0$ and 
$0< z < a$ are integers. At each round, the gambler wins or loses a dollar with probabilities p and 
$1-p$. The game continues until one of the two players is ruined. For even a and 
$0<z\leq {a}/{2}$, the family of distributions of the duration (total number of rounds) of the game indexed by 
$p \in [0,{\frac{1}{2}}]$ is shown to have monotone (increasing) likelihood ratio, while for 
${a}/{2} \leq z<a$, the family of distributions of the duration indexed by 
$p \in [{\frac{1}{2}}, 1]$ has monotone (decreasing) likelihood ratio. In particular, for 
$z={a}/{2}$, in terms of the likelihood ratio order, the distribution of the duration is maximized over 
$p \in [0,1]$ by 
$p={\frac{1}{2}}$. The case of odd a is also considered in terms of the usual stochastic order. Furthermore, as a limit, the first exit time of Brownian motion is briefly discussed.
