To let $ G = (V, E) $ be a graph with $ n $ Corners. Consider a few simple random walks $ (X, Y) $ in the graph the length $ L $ starting from a node $ v in V $, We refer to a length$ L $ aimless walk $ X $ as a tuple in $ V ^ L $, as $ (X_1, ldots, X_L) $, Now consider an estimate of the number of crossings in such a pair of random walks given by

begin {align}

T (X, Y) = sum_ {j = 1} ^ L sum_ {k = 1} ^ L mathbb {I} _ { {X_j = Y_k }}

end

from where $ mathbb {I} _ { { cdot }} $ is the display function of the event $ { cdot } $, My question is, can the random variable be $ T (X, Y) $ shown as martingale (plus some memory terms)?

# Tag: Walks

## Reference Request – Common drunkard walks

The drunkard walk is a game in which two players play $ a $ and $ b $ Dollars, and they play a series of fair games (both risking a dollar in each game) until one of them goes bankrupt.

My question is: was this taken into account in a graph? More specifically, we get a (finite, connected) simple graph, and each vertex is assigned a positive integer. In each round, we randomly choose an edge randomly and throw a fair coin to decide which vertex gives the other a dollar. The game is played until someone goes broke.

(I have reasonably good estimates for the expected runtime if the graph is a cycle.)

## App Development Cost Calculator

Mobile App Development Company is extremely competitive with several million mobile apps on the market. The question of how to reach your target users and how to achieve your ROI goals is really difficult. Businesses are looking to find a way to calculate the average cost of app development in advance. To meet this demand, several online calculators have been developed, of which we will highlight only the most popular ones.

**Otreva machines **

This calculator was developed by Scranton, an Otreva mobile apps development company in Pennsylvania. The calc provides statistics that cover multiple platforms, app and admin functions, average cost of building these platforms, and average costs. It also allows estimating the cost of developing iOS, iOS, Windows Phone or even web app. It takes users through two steps, when users need to choose a platform and then create certain features that they want in mobile apps, and then create a quote in USD.

**Imason machines**

, Just like Otreva calc, it is built on a three-level structure. As a first step, users can choose between app type for iPhone, iPad, phone, tablet, Windows Phone or Windows tablet and, in a second step, select certain features they need for a development offer such as login, social login, second language support and more ,

**Kinvey machines**

Unlike two previous calculators, Kinvey has gone through more steps for the user and allows for a more detailed app development cost estimate. The steps are – choosing platforms, choosing internal / external team, using or not using any cloud tech, size, user management, type of data to store, selection of data sources, using / not using location data, third-party cloud APIs, Selection of user contact channels and more. After all this information has been provided, users can provide their e-mail address for a quote as well.

**Enterprise Mobile App Calculator**

It takes company users through many steps to provide an estimate. The steps are – Platform, B2C / B2B / B2E, requires authentic or non-type of app (access to corporate data / re-launch of a business process / mobilization of a new business process), UA complexity, features using push notifications or not, number of corporate networks to which a connection can be made and more.

,

## Survival probability for casual walks

The probability of survival for a hiker starting at the origin is defined as the probability that the hiker remains positive by n steps. Thanks to the Sparre-Andersen theorem, I know that this PDF is from

```
plot[Binomial[2 n, n]* 2 ^ {- 2 n}, {n, 0, 100}]
```

However, I would like to confirm this empirically.

My attempt to confirm this for n = 100:

```
FoldList[
If[#2 < 0, 0, #1 + #2] &,
prefixing[Accumulate[RandomVariate[NormalDistribution[0, 1], 100]]0]]
```

I want`FoldList`

to stop if `# 2 <0`

not just replace in 0.

## Probability or Statistics – Modeling Snowfall with Random Walks with Drift

I try to simulate a (very) simple model of snowfall / accumulation with random walks in the following way:

```
sf = accumulate[RandomVariate[BernoulliDistribution[0.2], 100]*
RandomVariate[GammaDistribution[1, 2], 100]/. {0th -> -0,4}]ListLinePlot[sf]
```

I generate Bernoulli trials with a probability of 0.2 to simulate days when it snows. On a day when it does not snow, instead of a simple 0 entry, I insert a negative drift duration of -0.4 to simulate the melting of the snow.

Where I have problems is that you can never have negative snowfall. I want the Walker to always be greater than or equal to 0. However, I can not send all negative entries to 0 as this would eliminate the data of the days it is snowing, but the drift term is greater than the snowfall.

Many Thanks.