Large deviations for longest runs in Markov chains

We continue our investigation on general large deviation principles (LDPs) for longest runs. Previously, a general LDP for the longest success run in a sequence of independent Bernoulli trails was derived in [Z. Liu and X. Yang, A general large deviation principle for longest runs, Statist. Probab. Lett. 110 2016, 128–132]. In the present note, we establish a general LDP for the longest success run in a two-state (success or failure) Markov chain which recovers the previous result in the aforementioned paper. The main new ingredient is to implement suitable estimates of the distribution function of the longest success run recently established in [Z. Liu and X. Yang, On the longest runs in Markov chains, Probab. Math. Statist. 38 2018, 2, 407–428].

On the longest runs in Markov chains

In the first n steps of a two-state success and failure Markov chain, the longest success run Ln has been attracting considerable attention due to its various applications. In this paper, we study Ln in terms of its two closely connected properties: moment generating function and large deviations. This study generalizes several existing results in the literature, and also finds an application in statistical inference. Our method on the moment generating function is based on a global estimate of the cumulative distribution function of Ln proposed in this paper, and the proofs of the large deviations include the Gärtner–Ellis theorem and the moment generating function.

A Study on Consumers’ Psychology on Marketing Tools

In this twenty first century, market became more competitive and existing players have to fight in the ocean and have to survive after sacrificing own blood. However, every market players wants to live in the blue ocean. Thus, they attempt to develop and market the right product for right consumers. However, this is most important to understand how consumers define their need and how they want to meet their needs. It is important to understand the difference which one is need for which customer and which one want or demand to whom. In this stage, it raises the question to have better a deep insight about consumers’ psychology. If a marketer can understand his/her consumers’ psychology, he/she might easily pick up the best weapon to hit the consumers and to get them into the spider net spread to catch them all. Today, consumers are very much informative and they know well what they want and what is going on. So, marketers need to be very careful in designing marketing strategy to offer the best brand with a sense to deliver the message or communicating the consumer about the brand without bothering or irritating the consumer. Creating the best brand depends on how well a marketer can read consumers’ mind regarding the requirement and final success depends on how well they could communicate them through different marketing techniques/tools. These all require a good sense of consumer psychology for the marketers to draw a perfect image of the consumers in their marketing strategy development. The marketing techniques used by all marketers to attract consumers, produce new product, attract through different activities like coupon, demonstration, event show, trade show, advertising, celebrity endorsement are almost same. However, success run behind the winner who can better adopt consumers’ psychology.Philosophy and Progress, Vol#55-56; No#1-2; Jan-Dec 2014

Success Run Waiting Times and Fuss-Catalan Numbers

We present power series expressions for all the roots of the auxiliary equation of the recurrence relation for the distribution of the waiting time for the first run ofkconsecutive successes in a sequence of independent Bernoulli trials, that is, the geometric distribution of orderk. We show that the series coefficients are Fuss-Catalan numbers and write the roots in terms of the generating function of the Fuss-Catalan numbers. Our main result is a new exact expression for the distribution, which is more concise than previously published formulas. Our work extends the analysis by Feller, who gave asymptotic results. We obtain quantitative improvements of the error estimates obtained by Feller.

Run Statistics Defined on the Multicolor URN Model

Recently, Makri, Philippou and Psillakis (2007b) studied the exact distribution of success run statistics defined on an urn model. They derived the exact distributions of various success run statistics for a sequence of binary trials generated by the Pólya-Eggenberger sampling scheme. In our study we derive the joint distributions of run statistics defined on the multicolor urn model using a simple unified combinatorial approach and extend some of the results of Makri, Philippou and Psillakis (2007b). As a consequence of our results, we obtain the joint distributions of success and failure runs defined on the two-color urn model. The results enable us to compute the characteristics of particular consecutive-type systems and start-up demonstration tests.

Run Statistics Defined on the Multicolor URN Model

Recently, Makri, Philippou and Psillakis (2007b) studied the exact distribution of success run statistics defined on an urn model. They derived the exact distributions of various success run statistics for a sequence of binary trials generated by the Pólya-Eggenberger sampling scheme. In our study we derive the joint distributions of run statistics defined on the multicolor urn model using a simple unified combinatorial approach and extend some of the results of Makri, Philippou and Psillakis (2007b). As a consequence of our results, we obtain the joint distributions of success and failure runs defined on the two-color urn model. The results enable us to compute the characteristics of particular consecutive-type systems and start-up demonstration tests.