The Geography and Bathymetry of the Irish Sea
showing the locations of Liverpool. the Smalls Lighthouse, the port of Milford Haven, and St. David’s Head (Source: Wikimedia under CC BY-SA 3.0)
The Irish Sea fills the land basin between Ireland and Britain. It contains one of the shallowest sea waters on the planet. In some places, water depth reaches barely 40 meters even as far out as 30 miles from the coastline. Also lurking beneath the surface are vast banks of sand waiting to snare the unlucky ship, of which there have been many. Often, a floundering ship would sink vertically taking its human occupants straight down with it and get lodged in the sand, standing erect on the seabed with the tops of her masts clearly visible above the water line — a gruesome marker of the human tragedy resting just 30 meters below the surface. Such was the fate of the Pelican when she sank on March 20, 1793, right inside Liverpool Harbor, a stone’s throw from the shoreline.
The geography of the Irish sea also makes it susceptible to strong storms that come from out of nowhere and surprise you with a stunning suddenness and an insolent disregard for any nautical experience you may have had. At the lightest encouragement from the wind, the shallow waters of the sea will coil up into menacingly towering waves and produce vast clouds of blindingly opaque spray. At the slightest slip of good judgement or luck, the winds and the sea and the sands of the Irish sea will run your ship aground or bring upon a worse fate. Nimrod was, sadly, just one of the hundreds of such wrecks that litter the floor of the Irish Sea.
It stands to reason that over the years, the Irish sea has become one of the most heavily studied and minutely monitored bodies of water on the planet. From sea temperature at different depths, to surface wind speed, to carbon chemistry of the sea water, to the distribution of commercial fish, the governments of Britain and Ireland keep a close watch on hundreds of marine parameters. Dozens of sea-buoys, surveying vessels, and satellites gather data round the clock and feed them into sophisticated statistical models that run automatically and tirelessly, swallowing thousands of measurements and making forecasts of sea-conditions for several days into the future — forecasts that have made shipping on the Irish Sea a largely safe endeavor.
It’s within this copious abundance of data that we’ll study the concepts of statistical convergence of random variables. Specifically, we’ll study the following four types of convergence:
- Convergence in distribution
- Convergence in probability
- Convergence in the mean
- Almost sure convergence
There is a certain hierarchy inherent among the four types of convergences with the convergence in probability implying a convergence in distribution, and a convergence in the mean and almost sure convergence independently implying a convergence in probability.
To understand any of the four types of convergences, it’s useful to understand the concept of sequences of random variables. Which pivots us back to Nimrod’s voyage out of Liverpool.
It’s hard to imagine circumstances more conducive to a catastrophe than what Nimrod experienced. Her sinking was the inescapable consequence of a seemingly endless parade of misfortunes. If only her engines hadn’t failed, or Captain Lyall had secured a tow, or he had chosen a different port of refuge or the storm hadn’t turned into a hurricane, or the waves and rocks hadn’t broken her up, or the rescuers had managed to reach the stricken ship. The what-ifs seem to march away to a point on the distant horizon.
Nimrod’s voyage — be it a successful journey to Cork, or safely reaching one of the many possible ports of refuge, or sinking with all hands on board or any of the other possibilities limited only by how much you will allow yourself to twist your imagination — can be represented by any one of many possible sequences of events. Between the morning of February 25, 1860 and the morning of February 28, 1860, exactly one of these sequences materialized — a sequence that was to terminate in a unwholesomely bitter finality.
If you permit yourself to look at the reality of Nimrod’s fate in this way, you may find it worth your while to represent her journey as a long, theoretically infinite, sequence of random variables, with the final variable in the sequence representing the many different ways in which Nimrod’s journey could have concluded.
Let’s represent this sequence of variables as X1, X2, X3,…,Xn.
In Statistics, we regard a random variable as a function. And just like any other function, a random variable maps values from a domain to a range. The domain of a random variable is a sample space of outcomes that arise from performing a random experiment. The act of tossing a single coin is an example of a random experiment. The outcomes that arise from this random experiment are Heads and Tails. These outcomes produce the discrete sample space {Heads, Tails} which can form the domain of some random variable. A random experiment consists of one or more ‘devices’ which when when operated, together produce a random outcome. A coin is such a device. Another example of a device is a random number generator — which can be a software program — that outputs a random number from the sample space [0, 1] which, as against {Heads, Tails}, is continuous in nature and infinite in size. The range of a random variable is a set of values which are often encoded versions of things you care about in the physical world that you inhabit. Consider for example, the random variable X3 in the sequence X1, X2,X3,…,Xn. Let X3 designate the boolean event of Captain Lyall’s securing (or not securing) a tow for his ship. X3’s range could be the discrete and finite set {0, 1} where 0 could mean that Captain Lyall failed to secure a tow for his ship, while 1 could mean that he succeeded in doing so. What could be the domain of X3, or for that matter any variable in the rest of the sequence?
In the sequence X1, X2, X3,…Xk,…,Xn, we’ll let the domain of each Xk be the continuous sample space [0, 1]. We’ll also assume that the range of Xk is a set of values that encode the many different things that can theoretically happen to Nimrod during her journey from Liverpool. Thus, the variables X1, X2, X3,…,Xn are all functions of some value s ϵ [0, 1]. They can therefore be represented as X1(s), X2(s), X3(s),…,Xn(s). We’ll make the additional crucial assumption that Xn(s), which is the final (n-th) random variable in the sequence, represents the many different ways in which Nimrod’s voyage can be considered to conclude. Every time ‘s’ takes up a value in [0, 1], Xn(s) represents a specific way in which Nimrod’s voyage ended.
How might one observe a particular sequence of values? Such a sequence would be observed (a.k.a. would materialize or be realized) when you draw a value of s at random from [0, 1]. Since we don’t know anything about the how s is distributed over the interval [0, 1], we’ll take refuge in the principle of insufficient reason to assume that s is uniformly distributed over [0, 1]. Thus, each one of the infinitely uncountable numbers of real numbered values of s in the interval [0, 1] is equally probable. It’s a bit like throwing an unbiased die that has an uncountably infinite number of faces and selecting the value that it comes up as, as your chosen value of s.
Uncountable infinities and uncountably infinite-faced dice are mathematical creatures that you’ll often encounter in the weirdly wondrous world of real numbers.
So anyway, suppose you toss this fantastically chimerical die, and it comes up as some value sa ϵ [0, 1]. You will use this value to calculate the value of each Xk(s=sa) in the sequence which will yield an event that happened during Nimrod’s voyage. That would yield the following sequence of observed events:
X1(s=sa), X2(s=sa), X3(s=sa),…,Xn(s=sa).
If you toss the die again, you might get another value sb ϵ [0, 1] which will yield another possible ‘observed’ sequence:
X1(sb), X2(sb), X3(sb),…,Xn(sb).
It’s as if each time you toss your magical die, you are spawning a new universe and couched within this universe is the reality of…
