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 23h comment How to obtain accumulated counts of past events by time $t$? can you represent $\tau_i$ in terms of my $f$? How about $g(t) = |\{s \in [0,t], f(s) = 1\}|$? Is this or yours a more elegant way than some integral or sum if exists? (I guess no) 23h comment How to obtain accumulated counts of past events by time $t$? Thanks. Can you formulate what you mean by "introduce random times for the events and add up only terms corresponding to those times"? Jul 28 comment Is this a game theory problem or optimization problem? @shane: (1) What are some ways to create such a utility function out of $f$ and $g$? (2) The utility function is not supposed to let the seller or manufacture know, but only to the customer, me, for buying a product of a given type. Is my problem an optimization problem rather than a game theory one? Jul 28 comment Is this a game theory problem or optimization problem? @Shane: As a customer, I am trying to choose a product of a given type to buy, but have to balance between product quality and price. I will make the choice by choosing the product model (thus manufacturer), seller, payment method (cashback credit cards, gift cards, and which card if i have multiple), timing, and any other factors that can affect the product price and quality. Jul 28 comment Is this a game theory problem or optimization problem? @joriki: I am still in the process of understanding and formulating the problem, so I am not sure if my problem is well defined. My post is asking whether there have been well-defined/formulated problems similar to mine. In my question, a solution is one $x$ that makes the $f(x)$ and $g(x)$ fall in equilibirum in some sense similar to the equilibrium solution to a minmax problem in game theory. I don't know if there have been different possibilities to characterize "sense" already. I also like to know if there have been problems in game theory which are close to my problem. Jul 28 comment Is this a game theory problem or optimization problem? @Marc: (1) about using $\max$ and $\min$ to formulate an optimization problem, what you said makes sense to me. Do you have some references that mention that? (2) I am not just maximizing $f(x)$, but try to find some $x$ so that $f(x)$ and $g(x)$ are in equilibrium similar to the solution to a minmax problem in game theory, yet subject to the two constraints. So the two constraints are not superficial. If the constraints make the problem too complicated, we can start with simplifying the problem by ignoring the constraints. (see my update) Jul 9 comment Does a counting process allow a state change by more than 1 in a single transition? thanks. do you know the names for those special counting processes that can change by 1 in a single transition? Jul 8 comment Is there a concept called the cross derivative between two functions? thanks. yes.......... Feb 8 comment Definition of $\rho$-mixing and its relation to strong mixing? Thanks! Very surprised and pleased to receive a reply from you. My post was a while ago, and hope someday I could have a chance to pick up the topic again. Dec 20 comment How can we write this in math? @mvw: From the statement, I guess $n_\epsilon$ may depend on $y$, and not on $z$. Am I right? (I don't write the statement myself, all I am doing is trying to understand it) Dec 20 comment How can we write this in math? Thanks. I should add that $f$ is nonnegative. Are the two in my post the same/equivalent? Dec 20 comment How can we write this in math? @ClementC: what you wrote is in math. I also wonder if the single equation I gave is equivalent to the original statement? Or something similar to the equation? Dec 20 comment How can we write this in math? What do you mean by "same"? Dec 19 comment Approximate the unit ball in an infinite-dimensional Hilbert space, by compact sets? @GEdgar: I updated my post with my specific question. Dec 19 comment Approximate the unit ball in an infinite-dimensional Hilbert space, by compact sets? @AlexanderShamov: I updated my post with my specific question. Dec 18 comment Approximate the unit ball in an infinite-dimensional Hilbert space, by compact sets? @GEdgar: Thanks. Do you have some link/reference for me to read in detail? Dec 18 comment Approximate the unit ball in an infinite-dimensional Hilbert space, by compact sets? @user8268:Are there some examples for proving such generalization in your way? Dec 18 comment Approximate the unit ball in an infinite-dimensional Hilbert space, by compact sets? @GEdgar: Are there some examples for proving such generalization in your way? Dec 18 comment Approximate the unit ball in an infinite-dimensional Hilbert space, by compact sets? @AlexanderShamov: Are there some examples for proving such generalization in your way? Dec 5 comment Application of Jensen's inequality Thanks. How do you prove my second inequality? @Timbuc?